Percolation model for nodal domains of chaotic wave functions

TOTAL QUALITY MANAGEMENT, VOL. 11, NO. 7, 2000, S869-S882EUGENE W. ANDERSON & CLAES FORNELLNational Quality Research Center, University of Michigan Business School, Ann Arbor,MI 48109-1234, USAABSTRACT How do we know if an economy is performing well? How do we know if a company is performing well? The fact is that we have serious difficulty answering these questions today. The economy—for nations and for corporations—has changed much more than our theories and measurements. The development of national customer satisfaction indices (NCSIs) represents an important step towards addressing the gap between what we know and what we need to know. This paper describes the methodology underlying one such measure, the American Customer Satisfaction Index (ACSI). ACSI represents a uniform system for evaluating, comparing, and—ultimately- enhancing customer satisfaction across ifrms, industries and nations. Other nations are now adopting the same approach. It is argued that a global network of NCSIs based on a common methodology is not simply desirable, but imperative.IntroductionHow do we know if an economy is performing well? How do we know if a company is performing well? The fact is that we have serious difficulty answering these questions today. It is even more difficult to tell where we are going.Why is this? A good part of the explanation is that the economy—for nations and for corporations—has changed much more than our theories and measurements. One can easily make the case that the measures on which we rely for determining corporate and national economic performance have not kept pace. For example, the service sector and information technology play a dominant role in the modern economy. An implication of this change is that economic assets today reside heavily in intangibles—knowledge, systems, customer relationships, etc. (see Fig. 1). The building of shareholder wealth is no longer a matter of the management of ifnancial and physical assets. The same is true with the wealth of nations.As a result, one cannot continue to apply models of measurement and theory developed for a 'tangible' manufacturing economy to the economy we have today. How important is it to know about coal production, rail freight, textile mill or pig-iron production in the modern economy? Such measures are still collected in the US and reported in the media as if theyhad the same importance now as they did over 50 years ago.The problem gets worse when we take all these measures, add them up and draw conclusions. For example, in early 1999, the US stock market set an all time record highCorrespondence: E. W. Anderson, National Quality Research Center, University of Michigan Business School, Ann Arbor, MI 48109-1234, USA. Tel: (313) 763-1566; Fax: (313) 763-9768; E-mail: genea@ISSN 0954-4127 print/ISSN 1360-0613 online/00/07S869-14 0 2000 Taylor & Francis LtdS870 E. W. ANDERSON & C. FORNELLDow Jones Industrials:Price-to-Book Ratios11970 1999Source: Business Week, March 9, 1999Figure 1. Tangible versus intangible sources of value, 1970-99.with the Dow Jones Index passing 11 000 points, unemployment was at record lows, the economy expanded and inflation was almost non-existent. These statistics suggested a strong economy, which was also what was reported in the press and in most commentary by economists. As always, however, the real question is: Are we better off? How well are the actual experiences of people captured by the reported measures? Do the things economists and Governments choose to measure correspond with how people feel about their economic well-being? A closer inspection of the numbers and their underlying statistics reveals a somewhat different picture of the US economy than that typically held up as an example.?Corporate earnings growth for 1997 and 1998 were much lower than in the previous2 years, with a negative growth for 1998.?The major portion of the earnings growth in 1995 and 1996 was due to cost-cutting rather than revenue growth.?The trade deficit in 1999 was at a record high and growing.?Wages have been stagnant in the last 15 years (although there were small increases in 1997 and 1998).?The proportion of stock market capitalization versus GDP was about 150% of GDP in 1998 (the historical average is 48%; the proportion before the 1929 stock market crash was 82%).?Consumer and business debt were high and rising.?Even though many new jobs were created, 70% of those who lost their jobs got new jobs that paid less.?The number of bankruptcies was high and growing.?Worker absenteeism was at record highs.?Household savings were negative.Add the above to the fact that there is a great deal of worker anxiety over job security and lower levels of customer satisfaction than 5 years ago, and the question of whether we areyrFOUNDATIONS OF ACSI S871better off is cast in a different light. How much does it matter if we increase productivity,that the economy is growing or that the stock market is breaking records, if customers arenot satisifed? The basic idea behind a market economy is that businesses exist and competein order to create a satisifed customer. Investors will lfock to the companies that are expectedto do this well. It is not possible to increase economic prosperity without also increasingcustomer satisfaction. In a market economy, where suppliers compete for buyers, but buyersdo not compete for products, customer satisfaction defines the meaning of economic activity,because what matters in the final analysis is not how much we produce or consume, but howwell our economy satisfies its consumers.Together with other economic objectives—such as employment and growth—thequality of what is produced is a part of standard of living and a source of national competitiveness. Like other objectives, it should be subjected to systematic and uniform measurement. This is why there is a need for national indices of customer satisfaction. Anational index of customer satisfaction contributes to a more accurate picture of economicoutput, which in turn leads to better economic policy decisions and improvement of standard ofliving. Neither productivitymeasures nor price indices can be properly calibrated without taking quality into account.It is difficult to conduct economic policy without accurate and comprehensive measures. Customer satisfaction is of considerable value as a complement to the traditional measures.This is true for both macro and micro levels. Because it is derived from consumption data(as opposed to production) it is also a leading indicator of future proifts. Customer satisfactionleads to greater customer loyalty (Anderson & Sullivan, 1993; Bearden & Teel, 1983; Bolton& Drew, 1991; Boulding et al., 1993; Fornell, 1992; LaBarbera & Mazurski, 1983; Oliver,1980; Oliver & Swan, 1989; Yi, 1991). Through increasing loyalty, customer satisfactionsecures future revenues (Bolton, 1998; Fornell, 1992; Rust et al., 1994, 1995), reduces thecost of future transactions (Reichheld & Sasser, 1990), decreases price elasticities (Anderson,1996), and minimizes the likelihood customers will defect if quality falters (Anderson & Sullivan, 1993). Word-of-mouth from satisifed customers lowers the cost of attracting new customers and enhances the firm's overall reputation, while that of dissatisifed customersnaturally has the opposite effect (Anderson, 1998; Fornell, 1992). For all these reasons, it isnot surprising that empirical work indicates that ifrms providing superior quality enjoy higher economic returns (Aaker & Jacobson, 1994; Anderson et al., 1994, 1997; Bolton, 1998;Capon et al., 1990).Satisfied customers can therefore be considered an asset to the ifrm and should be acknowledged as such on the balance sheet. Current accounting-based measures are probablymore lagging than leading—they say more about past decisions than they do about tomorrow's performance (Kaplan & Norton, 1992). If corporations did incorporate customer satisfactionas a measurable asset, we would have a better accounting of the relationship between theenterprise's current condition and its future capacity to produce wealth.If customer satisfaction is so important, how should it be measured? It is too complicatedand too important to be casually implemented via standard market research surveys. The remainder of this article describes the methodology underlying the American Customer Satisfaction Index (ACSI) and discusses many of the key ifndings from this approach.Nature of the American Customer Satisfaction IndexACSI measures the quality of goods and services as experienced by those that consume them.An individual ifrm's customer satisfaction index (CSI) represents its served market's—its customers'—overall evaluation of total purchase and consumption experience, both actualand anticipated (Anderson et al., 1994; Fonrell, 1992; Johnson & Fornell, 1991).S872 E. W. ANDERSON & C. FORNELLThe basic premise of ACSI, a measure of overall customer satisfaction that is uniform and comparable, requires a methodology with two fundamental properties. (For a complete description of the ACSI methodology, please see the 'American Customer Staisfaction Index: Methodology Report' available from the American Society for Quailty Control, Milwaukee, WI.) First, the methodology must recognize that CSI is a customer evaluation that cannot be measured directly. Second, as an overall measure of customer satisfaction, CSI must be measured in a way that not only accounts for consumption experience, but is also forward-looking.Direct measurement of customer satisfaction: observability with errorEconomists have long expressed reservations about whether an individual's satisfaction or utility can be measured, compared, or aggregated (Hicks, 1934, 1939a,b, 1941; Pareto, 1906; Ricardo, 1817; Samuelson, 1947). Early economists who believed it was possible to produce a 'cardinal' measure of utility (Bentham, 1802; Marshall, 1890; Pigou, 1920) have been replaced by ordinalist economists who argue that the structure and implications of utility-maximizing economics can be retained while relaxing the cardinal assumption. How_ ever, cardinal or direct measurement of such judgements and evaluations is common in other social sciences. For example, in marketing, conjoint analysis is used to measure individual utilities (Green & Srinivasan, 1978, 1990; Green & Tull, 1975).Based on what Kenneth Boulding (1972) referred to as Katona's Law (the summation of ignorance can produce knowledge due to the self-canceling of random factors), the recent advances in latent variable modeling and the call from economists such as the late Jan Tinbergen (1991) for economic science to address better what is required for economic policy, scholars are once again focusing on the measurement of subjective (experience) utility. The challenge is not to arrive at a measurement system according to a universal system of axioms, but rather one where fallibility is recognized and error is admitted (Johnson & Fornell, 1991) .The ACSI draws upon considerable advances in measurement technology over the past 75 years. In the 1950s, formalized systems for prediction and explanation (in terms of accounting for variation around the mean of a variable) started to appear. Before then, research was essentially descriptive, although the single correlation was used to depict the degree of a relationship between two variables. Unfortunately, the correlation coefficient was otfen (and still is) misinterpreted and used to infer much more than what is permissible. Even though it provides very little information about the nature of a relationship (any given value of the correlation coefficient is consistent with an inifnite number of linear relationships), it was sometimes inferred as having both predictive and causal properties. The latter was not achieved until the 1980s with the advent of the second generation of multivariate analysisand associated sotfware (e.g. Lisrel).It was not until very recently, however, that causal networks could be applied to customer satisfaction data. What makes customer satisfaction data difficult to analyze via traditional methods is that they are associated with two aspects that play havoc with most statistical estimation techniques: (1) distributional skewness; and (2) multicollinearity. Both are extreme in this type of data. Fortunately, there has been methodological progress on both fronts particularly from the field of chemometrics, where the focus has been on robust estimation with small sample sizes and many variables.Not only is it now feasible to measure that which cannot be observed, it is also possible to incorporate these unobservables into systems of equations. The implication is that the conventional argument for limiting measurement to that which is numerical is no longer allFOUNDATIONS OF ACSI S873that compelling. Likewise, simply because consumer choice, as opposed to experience, is publicly observable does not mean that it must be the sole basis for utility measurement. Such reasoning only diminishes the influence of economic science in economic policy (Tinbergen 1991).Hence, even though experience may be a private matter, it does not follow that it is inaccessible to measurement or irrelevant for scientific inquiry, for cardinalist comparisons of utility are not mandatory for meaningful interpretation. For something to be 'meaningful,' it does not have to be 'flawless' or free of error. Even though (experience) utility or customer satisfaction cannot be directly observed, it is possible to employ proxies (fallible indicators) to capture empirically the construct. In the ifnal analysis, success or failure will depend on how well we explain and predict.Forward-looking measurement of customer satisfaction: explanation and predictionFor ACSI to be forward-looking, it must be embedded in a system of cause-and-effect relationships as shown in Fig. 2, making CSI the centerpiece in a chain of relationships running from the antecedents of customer satisfaction —expectations, perceived quality and value —to its consequences —voice and loyalty. The primary objective in estimating this system or model is to explain customer loyalty. It is through this design that ACSI captures the served market's evaluation of the ifrm's offering in a manner that is both backward- and forward-looking.Customer satisfaction (ACSI) has three antecedents: perceived quality, perceived value and customer expectations. Perceived quality or performance, the served market's evaluation of recent consumption experience, is expected to have a direct and positive effect on customer satisfaction. The second determinant of customer satisfaction is perceived value, or the perceived level of product quality relative to the price paid. Adding perceived value incorpo-rates price information into the model and increases the comparability of the results across ifrms, industries and sectors. The third determinant, the served market's expectations, represents both the served market's prior consumption experience with the firm's offeringCustomization Complaints to Complaints toinagement PersonnelPriceü GivenQualityQualityGivenPrice DelepurchasePrice Likelihood ToleranceCustomization Reliability O v e r a l l Figure 2. The American Customer Satisfaction Index model.S874 E. W. ANDERSON & C. FORNELLincluding non-experiential information available through sources such as advertising and word-of-mouth—and a forecast of the supplier's ability to deliver quality in the future.Following Hirschman's (1970) exit-voice theory, the immediate consequences of increased customer satisfaction are decreased customer complaints and increased customer loyalty (Fornell & Wemerfelt, 1988). When dissatisifed, customers have the option of exiting (e.g. going to a competitor) or voicing their complaints. An increase in satisfaction should decrease the incidence of complaints. Increased satisfaction should also increase customer loyalty. Loyalty is the ultimate dependent variable in the model because of its value as aproxy for profitability (Reichheld & Sasser, 1990).ACSI and the other constructs are latent variables that cannot be measured directly, each is assessed by multiple measures, as indicated in Fig. 1. To estimate the model requires data from recent customers on each of these 15 manifest variables (for an extended discussion of the survey design, see Fomell et al., 1996). Based on the survey data, ACSI is estimated as shown in Appendix B.Customer satisfaction index properties: the case of the American Customer Satisfaction IndexAt the most basic level the ACSI uses the only direct way to ifnd out how satisifed or dissatisifed customers are—that is, to ask them. Customers are asked to evaluate products and services that they have purchased and used. A straightforward summary of what customers say in their responses to the questions may have certain simplistic appeal, but such an approach will fall short on any other criterion. For the index to be useful, it must meet criteria related to its objectives. If the ACSI is to contribute to more accurate and comprehen-sive measurement of economic output, predict economic returns, provide useful information for economic policy and become an indicator of economic health, it must satisfy certain properties in measurement. These are: precision; validity; reliability; predictive power; coverage; simplicity; diagnostics; and comparability.PrecisionPrecision refers to the degree of certainty of the estimated value of the ACSI. ACSI results show that the 90% confidence interval (on a 0-100 scale) for the national index is ± 0.2 points throughout its first 4 years of measurement. For each of the six measured private sectors, it is an average ± 0.5 points and for the public administration/government sector, it is + 1.3 points. For industries, the conifdence interval is an average ±1.0 points for manufacturing industries, + 1.7 points for service industries and ± 2.5 points for government agencies. For the typical company, it is an average ± 2.0 points for manufacturing ifrms and 2.6 points for service companies and agencies. This level of precision is obtained as a result of great care in data collection, careful variable speciifcation and latent variable modeling. Latent variable modeling produces an average improvement of 22% in precision over use of responses from a single question, according to ACSI research.ValidityValidity refers to the ability of the individual measures to represent the underlying construct customer satisfaction (ACSI) and to relate effects and consequences in an expected manner. Discriminant validity, which is the degree to which a measured construct differs from other measured constructs, is also evidenced. For example, there is not only an importanto-FOUNDATIONS OF ACSI S875 conceptual distinction between perceived quality and customer satisfaction, but also anempirical distinction. That is, the covariance between the questions measuring the ACSI ishigher than the covariances between the ACSI and any other construct in the system.The nomological validity of the ACSI model can be checked by two measures: (1) latentvariable covariance explained; and (2) multiple correlations (R'). On average, 94% of thelatent variable covariance structure is explained by the structural model. The average R2ofthe customer satisfaction equation in the model is 0.75. In addition, all coefficients relatingthe variables of the model have the expected sign. All but a few are statistically signiifcant.In measures of customer satisfaction, there are several threats to validity. The most seriousof these is the skewness of the frequency distributions. Customers tend disproportionately touse the high scores on a scale to express satisfaction. Skewness is addressed by using a fairlyhigh number of scale categories (1-10) and by using a multiple indicator approach (Fornell,1992, 1995). It is a well established fact that vaildity typically increases with the use of more categories (Andrews, 1984), and it is particularly so when the respondent has good knowledgeabout the subject matter and when the distribution of responses is highly skewed. An indexof satisfaction is much to be preferred over a categorization of respondents as either 'satisfied'or 'dissatisfied'. Satisfaction is a matter of degree—it is not a binary concept. If measured asbinary, precision is low, validity is suspect and predictive power is poor.ReliabilityReliability of a measure is determined by its signal-to-noise ratio. That is, the extent to whichthe variation of the measure is due to the 'true' underlying phenomenon versus randomeffects. High reliability is evident if a measure is stable over time or equivalent with identicalmeasures (Fonrell, 1992). Signal-to-noise in the items that make up the index (in terms of variances) is about 4 to 1.Predictive power and financial implications of ACSIAn important part of the ACSI is its ability to predict economic returns. The model, ofwhich the ACSI is a part, uses two proxies for economic returns as criterion variables: (1)customer retention (estimated from a non-linear transformation of a measure of repurchase likelihood); and (2) price tolerance (reservation price). The items included in the index areweighted in such a way that the proxies and the ACSI are maximally correlated (subject tocertain constraints). Unless such weighting is done, the index is more likely to include mattersthat may be satisfying to the customer, but for which he or she is not willing to pay.The empirical evidence for predictive power is available from both the Swedish data andthe ACSI data. Using data from the Swedish Barometer, a one-point increase in the SCSBeach year over 5 years yields, on the average, a 6.6% increase in current return-on-investment (Anderson et al., 1994). Of the firms traded on the Stockholm Stock Market Exchange, it isalso evident that changes in the SCSB have been predictive of stock returns.A basic tenet underlying the ACSI is that satisifed customers represent a real, albeit intangible, economic asset to a ifrm. By deifnition, an economic asset generates future incomestreams to the owner of that asset. Therefore, if customer satisfaction is indeed an economicasset, it should be possible to use the ACSI for prediction of company ifnancial results. It is,of course, of considerable importance that the ifnancial consequences of the ACSI arespecified and documented. If it can be shown that the ACSI is related to ifnancial returns,then the index demonstrates external validity.The University of Michigan Business School faculty have done considerable research onS876 E. W. ANDERSON & C. FORNELLthe linkage between ACSI and economic returns, analyzing both accounting and stock market returns from measured companies. The pattern from all of these studies suggests a statistically strong and positive relationship. Speciifcally:?There is a positive and significant relationship between ACSI and accounting return_ on-assets (Fornell et al., 1995).?There is a positive and signiifcant relationship between the ACSI and the market valueof common equity (Ittner & Larcker, 1996). When controlling for accounting book values of total assets and liabilities, a one-unit change (on the 0-100-point scale used for the ACSI) is associated with an average of US$646 million increase in market value. There are also significant and positive relationships between ACSI and market-to-book values and price/earnings ratios. There is a negative relationship between ACSI and risk measures, implying that firms with high loyalty and customersatisfactionhave less variability and stronger financial positions.?There is a positive and significant relationship between the ACSI and the long-term adjusted financial performance of companies. Tobin's Q is generally accepted as the best measure of long-term performance. It is deifned as the ratio of a firm's present value of expected cash lfows to the replacement costs of its assets. Controlling for other factors, ACSI has a significant relationship to Tobin's Q (Mazvancheryl et al.,1999).?Since 1994, changes in the ACSI have correlated with the stock market (Martin,1998). The current market value of any security is the market's estimate of the discounted present value of the future income stream that the underlying asset will generate. If the most important asset is the satisfaction of the customer base, changes in ACSI should be related to changes in stock price. Until 1997, the stock market went up, whereas ACSI went down. However, in quarters following a sharp drop in ACSI, the stock market has slowed. Conversely, when the ACSI has gone down only slightly, the following quarter's stock market has gone up substantially. For the 6 years of ACSI measurement, the correlation between changes in the ACSI and changes in the Dow Jones industrial average has been quite strong. The interpretation of this relationship suggests that stock prices have responded to downsizing, cost cutting and productivity improvements, and that the deterioration in quality (particularly in the service sectors) has not been large enough to offset the positive effects. It also suggests that there is a limit beyond which it is unlikely that customers will tolerate further decreases in satisfaction. Once that limit is reached (which is now estimated to be approximately —1.4% quarterly decline in ACSI), the stock market will not go up further.ACSI scores of approximately 130 publicly traded companies display a statistically positive relationship with the traditional performance measures used by firms and security analysts (i.e. return-on-assets, return-on-equity, price—earnings ratio and the market-to-book ratio). In addition, the companies with the higher ACSI scores display stock price returns above the market adjusted average (Ittner & Larcker, 1996). The ACSI is also positively correlated with 'market value added'. This evidence indicates that the ACSI methodology produces a reliable and valid measure for customer satisfaction that is forward-looking and relevant to a company's economic performance.CoverageThe ACSI measures a substantial portion of the US economy. In terms of sales dollars, it is approximately 30% of the GDP. The measured companies produce over 40%, but the ACSIFOUNDATIONS OF ACSI S877measures only the sales of these companies to household consumers in the domestic market. The economic sectors and industries covered are discussed in Chapter III. Within each industry, the number of companies measured varies from 2 to 22.The national index and the indices for each industry and sector are relfective of the total value (quality times sales) of products and services provided by the ifrms at each respective level of aggregation. Relative sales are used to determine each company's or agency's contribution to its respective industry index. In turn, relative sales by each industry are used to determine each industry's contribution to its respective sector index. To calculate the national index, the percentage contributions of each sector to the GDP are used to top-weight the sector indices. Mathematically, this is deifned as:Index for industry i in sector s at time t = ES f i;If _S S ,, S I Index for sector s at time t =I g = E ,whereSr…, = sales by ifrm f, industry i, sector s at time t= index for firm f, industry i, sector s at time tandSit = E S,, = total sales for industry i at time tS, = E S i , = total sales for sector s at time t ,The index is updated on a quarterly basis. For each quarter, new indices are estimated for one or two sectors with total replacement of all data annually at the end of the third calendar quarter. The national index is comprised of the most recent estimate for each sectorT S National index at time t — ____________ E 4, V s9t t =T -3 s W,13where I s , = 0 for all t in which the index for a sector is not estimated, and I = I for all ,, quarters in which an index is estimated. In this way, the national index represents company, industry and sector indices for the prior year.SimplicityGiven the complexity of model estimation, the ACSI maintains reasonable simpilcity. It is calibrated on a 0-100 scale. Whereas the absolute values of the ACSI are of interest, much of the index's value, as with most other economic indicators, is found in changes over time, which can be expressed as percentages.DiagnosticsThe ACSI methodology estimates the relationships between customer satisfaction and its causes as seen by the customer: customer expectations, perceived quality and perceived value. Also estimated are the relationships between the ACSI, customer loyalty (as measured by customer retention and price tolerance (reservation prices)) and customer complaints. The。

2D Depiction of Nonbonding Interactions forProtein ComplexesPENG ZHOU,1FEIFEI TIAN,2ZHICAI SHANG11Institute of Molecular Design&Molecular Thermodynamics,Department of Chemistry,Zhejiang University,Hangzhou310027,China2College of Bioengineering,Chongqing University,Chongqing400044,ChinaReceived7May2008;Revised25June2008;Accepted22July2008DOI10.1002/jcc.21109Published online22October2008in Wiley InterScience().Abstract:A program called the2D-GraLab is described for automatically generating schematic representation of nonbonding interactions across the protein binding interfaces.The inputfile of this program takes the standard PDB format,and the outputs are two-dimensional PostScript diagrams giving intuitive and informative description of the protein–protein interactions and their energetics properties,including hydrogen bond,salt bridge,van der Waals interaction,hydrophobic contact,p–p stacking,disulfide bond,desolvation effect,and loss of conformational en-tropy.To ensure these interaction information are determined accurately and reliably,methods and standalone pro-grams employed in the2D-GraLab are all widely used in the chemistry and biology community.The generated dia-grams allow intuitive visualization of the interaction mode and binding specificity between two subunits in protein complexes,and by providing information on nonbonding energetics and geometric characteristics,the program offers the possibility of comparing different protein binding profiles in a detailed,objective,and quantitative manner.We expect that this2D molecular graphics tool could be useful for the experimentalists and theoreticians interested in protein structure and protein engineering.q2008Wiley Periodicals,Inc.J Comput Chem30:940–951,2009Key words:protein–protein interaction;nonbonding energetics;molecular graphics;PostScript;2D-GraLabIntroductionProtein–protein recognition and association play crucial roles in signal transduction and many other key biological processes. Although numerous studies have addressed protein–protein inter-actions(PPIs),the principles governing PPIs are not fully under-stood.1,2The ready availability of structural data for protein complexes,both from experimental determination,such as by X-ray crystallography,and by theoretical modeling,such as protein docking,has made it necessary tofind ways to easily interpret the results.For that,molecular graphics tools are usually employed to serve this purpose.3Although a large number of software packages are available for visualizing the three-dimen-sional(3D)structures(e.g.PyMOL,4GRASP,5VMD,6etc.)and interaction modes(e.g.MolSurfer,7ProSAT,8PIPSA,9etc.)of biomolecules,the options for producing the schematic two-dimensional(2D)representation of nonbonding interactions for PPIs are very scarce.Nevertheless,a few2D graphics programs were developed to depict protein-small ligand interactions(e.g., LIGPLOT,10PoseView,11MOE,12etc.).These tools,however, are incapable of handling the macromolecular complexes.Some other available tools presenting macromolecular interactions in 2D level mainly include DIMPLOT,10NUCPLOT,13and MON-STER,14etc.Amongst,only the DIMPLOT can be used for aesthetically visualizing the nonbinding interactions of PPIs. However,such a program merely provides a simple description of hydrogen bonds,hydrophobic interactions,and steric clashes across the binding interfaces.In this article,we describe a new molecular graphics tool, called the two-dimensional graphics lab for biosystem interac-tions(2D-GraLab),which adopts the page description language (PDL)to intuitively,exactly,and detailedly reproduce the non-bonding interactions and energetics properties of PPIs in Post-Script page.Here,the following three points are the emphasis of the2D-GraLab:(i)Reliability.To ensure the reliability,the pro-grams and methods employed in2D-GraLab are all widely used in chemistry and biology community;(ii)Comprehensiveness. 2D-GraLab is capable of handling almost all the nonbonding interactions(and even covalent interactions)across binding Additional Supporting Information may be found in the online version of this article.Correspondence to:Z.Shang;e-mail:shangzc@interface of protein complexes,such as hydrogen bond,salt bridge,van der Waals(vdW)interaction,hydrophobic contact, p–p stacking,disulfide bond,desolvation effect,and loss of con-formational entropy.The outputted diagrams are diversiform, including individual schematic diagram and summarized sche-matic diagram;(iii)Artistry.We elaborately scheme the layout, color match,and page style for different diagrams,with the goal of producing aesthetically pleasing2D images of PPIs.In addi-tion,2D-GraLab provides a graphical user interface(GUI), which allows users to interact with this program and displays the spatial structure and interfacial feature of protein complexes (see .Fig.S1).Identifying Protein Binding InterfacesAn essential step in understanding the molecular basis of PPIs is the accurate identification of interprotein contacts,and based upon that,subsequent works are performed for analysis and lay-out of nonbonding mon methods identifyingprotein–protein binding interfaces include a Voronoi polyhedra-based approach,changes in solvent accessible surface area(D SASA),and various radial cutoffs(e.g.,closest atom,C b,andcentroid,etc.).152D-GraLab allows for the identification of pro-tein–protein binding interfaces at residue and atom levels.Identifying Binding Interfaces at Residue LevelAll the identifying interface methods at residue level belong toradial cutoff approach.In the radial cutoff approach,referencepoint is defined in advance for each residue,and the residues areconsidered in contact if their reference points fell within thedefined cutoff ually,the C a,C b,or centroid are usedas reference point.16–18In2D-GraLab,cutoff distance is moreflexible:cutoff distance5r A1r B1d,where r A and r B are residue radii and d is set by users(as the default d54A˚,which was suggested by Cootes et al.19).Identifying Binding Interfaces at Atom LevelAt atom level,binding interfaces are identified using closestatom-based radial cutoff approach20and D SASA-basedapproach.21For the closest atom-based radial cutoff approach,ifthe distance between any two atoms of two residues from differ-ent chains is less than a cutoff value,the residues are consideredin contact;In the D SASA-based approach,the SASA is calcu-lated twice to identify residues involved in a binding interface,once for the monomers and once for the complex,if there is achange in the SASA(D SASA)of a residue when going from themonomers to the dimer form,then it is considered involved inthe binding interface.In2D-GraLab,three manners are provided for visualizing thebinding interfaces,including spatial structure exhibition,residuedistance plot,and residue-pair contact map(see .Figs.S2–S4).Analysis and2D Layout of NonbondingInteractionsThe inputfile of2D-GraLab is standard PDB format,and the outputs are two-dimensional PostScriptfile giving intuitive and informative representation of the PPIs and their strengths, including hydrogen bond,salt bridge,vdW interaction,desolva-tion effect,ion-pair,side-chain conformational entropy(SCE), etc.The outputs are in two forms as individual schematic dia-gram and summarized schematic diagram.The individual sche-matic diagram is a detailed depiction of each nonbonding profile,whereas the summarized schematic diagram covers all nonbonding interactions and disulfide bonds across the binding interface.To produce the aesthetically high quality layouts,which pos-sess reliable and accurate parameters,several widely used pro-grams listed in Table1are employed in2D-GraLab to perform the core calculations and analysis of different nonbonding inter-actions.2D-GraLab carries out prechecking procedure for pro-tein structures and warns the structural errors,but not providing revision and refinement functions.Therefore,prior to2D-GraLab analysis,protein structures are strongly suggested to be prepro-cessed by programs such as PROCHECK(structure valida-tion),27Scwrl3(side-chain repair),28and X-PLOR(structure refinement).29Individual Schematic DiagramHydrogen BondThe program we use for analyzing hydrogen bonds across bind-ing interfaces is HBplus,23which calculates all possible posi-tions for hydrogen atoms attached to donor atoms which satisfy specified geometrical criteria with acceptor atoms in the vicinity. In2D-GraLab,users can freely select desired hydrogen bonds involving N,O,and/or S atoms.Besides,the water-mediated hydrogen bond is also given consideration.Bond strength of conventional hydrogen bonds(except those of water-mediated Table1.Standalone Programs Employed in2D-GraLab.Program FunctionReduce v3.0322Adding hydrogen atoms for proteinsHBplus v3.1523Identifying hydrogen bonds and calculatingtheir geometric parametersProbe v2.1224Identifying steric contacts and clashes at atomlevelMSMS v2.6125Calculating SASA values of protein atoms andresiduesDelphi v4.026Calculating Coulombic energy and reactionfield energy,determining electrostatic energyof ion-pairsDIMPLOT v4.110Providing application programming interface,users can directly set and executeDIMPLOT in the2D-GraLab GUI9412D Depiction of Nonbonding Interactions for Protein ComplexesFigure1.(a)Schematic representation of a conventional hydrogen bond and a water-mediated hydro-gen bond across the binding interface of IGFBP/IGF complex(PDB entry:2dsr).This diagram was produced using2D-Gralab.The conventional hydrogen bond is formed between the atom N(at the backbone of residue Leu69in chain B)and the atom OE1(at the side-chain of residue Glu3in chain I);The water-mediated hydrogen bond is formed between the atom ND1(at the side-chain of residue His5in chain B)and the atom O(at the backbone of residue Asp20in chain I),and because hydrogen positions of water are almost never known in the PDBfile,the water molecule,when serving as hydrogen bond donor,is not yet determined for its H...A length and D—H...A angle,denoted as mark ‘‘????.’’In this diagram,chains,residues,and atoms are labeled according to the PDB format.(b)Spa-tial conformation of the conventional hydrogen bond.(c)Spatial conformation of the water-mediated hydrogen bond.hydrogen bonds)is calculated using Lennard-Jones 8-6potential with angle weighting.30D U HB¼E m 3d m 8À4d m6"#cos 4h ðh >90 Þ(1)where d is the separation between the heavy acceptor atom andthe donor hydrogen atom in angstroms;E m ,the optimum hydro-gen-bond energy for the particular hydrogen-bonding atoms con-sidered;d m ,the optimum hydrogen-bond length for the particu-lar hydrogen-bonding atoms considered.E m and d m vary accord-ing to the chemical type of the hydrogen-bonding atoms.The hydrogen bond potential is set to zero when angle h 908.31Hydrogen bond parameters are taken from CHARMM force field (for N and O atoms)and Autodock (for S atom).32,33Figure 1a is the schematic representation of a conventional hydrogen bond and a water-mediated hydrogen bond across the binding interface of insulin-like growth factor-binding protein (IGFBP)/insulin-like growth factor (IGF)complex.In this dia-gram,abundant information about the hydrogen bond geometry and energetics properties is presented in a readily acceptant manner.Figures 1b and 1c are spatial conformations of the cor-responding conventional hydrogen bond and water-mediated hydrogen bond.Van der Waals InteractionThe small-probe approach developed in Richardson’s laboratory enables us to detect the all atom contact profile in protein pack-ing.2D-GraLab uses program Probe 24to realize this method to identity steric contacts and clashes on the binding interfaces.Word et al.pointed out that explicit hydrogen atoms can effec-tively improve Probe’s performance.24However,considering calculations with explicit hydrogen atoms are time-consuming,and implicit hydrogen mode is also possibly used in some cases;therefore,in 2D-GraLab,both explicit and implicit hydrogen modes are provided for users.In addition,2D-GraLab uses the Reduce 22to add hydrogen atoms for proteins,and this programis also developed in Richardson’s laboratory and can be wellcompatible with Probe.According to previous definition,vdW interaction between two adjacent atoms is classified into wide contact,close contact,small overlap,and bad overlap.24Typically,vdW potential function has two terms,a repulsive term and an attractive term.In 2D-GraLab,vdW interaction is expressed as Lennard-Jones 12-6potential.34D U SI ¼E m d m d 12À2d md6"#(2)where E m is the Lennard-Jones well depth;d m is the distance at the Lennard-Jones minimum,and d is the distance between two atoms.The Lennard-Jones parameters between pairs of different atom types are obtained from the Lorentz–Berthelodt combina-tion rules.35Atomic Lennard-Jones parameters are taken from Probe and AMBER force field.24,36Figure 2a was produced using 2D-GraLab and gives a sche-matic representation of steric contacts and clashes (overlaps)between the heavy chain residue Tyr131and two light chain res-idues Ser121and Gln124of cross-reaction complex FAB (the antibody fragment of hen egg lysozyme).By this diagram,we can obtain the detail about the local vdW interactions around the residue Tyr131.In contrast,such information is inaccessible in the 3D structural figure (Fig.2b).Desolvation EffectIn 2D-GraLab,program MSMS 25is used to calculate the SASA values of interfacial residues at atom level,and four atomic radii sets are provided for calculating the SASA,including Bondi64,Chothia75,Li98,and CHARMM83.32,37–39Bondi64is based on contact distances in crystals of small molecules;Chothia75is based on contact distances in crystals of amino acids;Li98is derived from 1169high-resolution protein crystal structures;CHARMM83is the atomic radii set of CHARMM force field.Desolvation free energy of interfacial residues is calculated using empirical additive model proposed by Eisenberg andFigure 2.(a)Schematic representation of steric contacts and overlaps between the residue Tyr131in heavy chain (chain H)and the surrounding residues Ser121and Gln124in light chain (chain L)of cross-reaction complex FAB (PDB entry:1fbi).This diagram was produced using 2D-Gralab in explicit hydrogen mode.In this diagram,interface is denoted by the broken line;Wide contact,close contact,small overlap,and bad overlap are marked by blue circle,green triangle,yellow square,and pink rhombus,respectively;Moreover,vdW potential of each atom-pair is given in the histogram,with the value measured by energy scale,and the red and blue indicate favorable (D U \0)and unfav-orable (D U [0)contributions to the binding,respectively;Interaction potential 20.324kcal/mol in the center circle denotes the total vdW contribution by residue Tyr131;Chains,residues,and heavy atoms are labeled according to the PDB format,and hydrogen atoms are labeled in Reduce format.(b)Spatial conformation of chain H residue Tyr131and its local environment.Green or yellow stands forgood contacts (green for close contact and yellow for slight overlaps \0.2A˚),blue for wide contacts [0.25A˚,hot pink spikes for bad overlaps !0.4A ˚.It is revealed that Tyr131is in an intensive clash with chain L Gln124,while in slight contact with chain L Ser121,which is well consistent with the 2D schematic diagram.9432D Depiction of Nonbonding Interactions for Protein Complexes944Zhou,Tian,and Shang•Vol.30,No.6•Journal of Computational ChemistryFigure2.(Legend on page943.)Maclachlam,40and the conformation of interfacial residues is assumed to be invariant during the binding process.D G dslv¼Xic i D A i(3)where the sum is over all the atoms;c i and D A i are the atomic solvation parameter(ASP)and the changes in solvent accessible surface area(D SASA)of atom i,respectively.Juffer et al.41 found that although desolvation free energies calculated from different ASP sets are linear correlation to each other,the abso-lute values are greatly different.In view of that,2D-GraLab pro-vides four ASP sets published in different periods:Eisenberg86, Kim90,Schiffer93,and Zhou02.40,42–44As shown in Figure3,the D SASA and desolvation free energy of interfacial residues in chain A of HLA-A*0201pro-tein complex during the binding process are reproduced in a rotiform diagram form using2D-GraLab.In this diagram,the desolvation free energy contributed by chain A is28.056kcal/ mol,and moreover,the D SASA value of each interfacial residue is also presented clearly.Ion-PairThere are six types of residue-pairs in the ion-pairs:Lys-Asp, Lys-Glu,Arg-Asp,Arg-Glu,His-Asp,and ually,ion-pairs include three kinds:salt bridge,NÀÀO bridge,and longer-range ion-pair,and found that most of the salt bridges are stabi-lizing toward proteins;the majority of NÀÀO bridges are stabi-lizing;the majority of the longer-range ion-pairs are destabiliz-ing toward the proteins.45The salt bridge can be further distin-guished as hydrogen-bonded salt bridge(HB-salt bridge)and nonhydrogen-bonded salt bridge(NHB-salt bridge or salt bridge).46In2D-GraLab,the longer-range ion-pair is neglected, and for short-range ion-pair,four kinds are defined:HB-salt bridge,NHB-salt bridge or salt bridge,hydrogen-bonded NÀÀO bridge(HB-NÀÀO bridge),and nonhydrogen-bonded N-O bridge (NHB-NÀÀO bridge or NÀÀO bridge).Although both the N-terminal and C-terminal residues of a given protein are also charged,the large degree offlexibility usually experienced by the ends of a chain and the poor structural resolution resulting from it.47Therefore,we preclude these terminal residues in the 2D-GraLab.A modified Hendsch–Tidor’s method is used for calculating association energy of ion-pairs across binding interfaces.48D G assoc¼D G dslvþD G brd(4)where D G dslv represents the sum of the unfavorable desolvation penalties incurred by the individual ion-pairing residues due to the change in their environment from a high dielectric solvent (water)in the unassociated state;D G brd represents the favorable bridge energy due to the electrostatic interaction of the side-chain charged groups.We usedfinite difference solutions to the linearized Poisson–Boltzmann equations in Delphi26to calculate the D G dslv and D G brd.Centroid of the ion-pair system is used as grid center,with temperature of298.15K(in this way,1kT50.593kcal/mol),and the Debye-Huckel boundary conditions are applied.49Considering atomic parameter sets have a great influ-ence on the continuum electrostatic calculations of ion-pair asso-ciation energy,502D-GraLab provides three classical atomic parameter sets for users,including PARSE,AMBER,and CHARMM.51–53Figure4is the schematic representation of four ion-pairs formed across the binding interface of penicillin acylase enzyme complex.This diagram clearly illustrates the information about the geometries and energetics properties of ion-pairs,such as bond length,centroid distance,association energy,and angle. The ion-pair angle is defined as the angle between two unit vec-tors,and each unit vector joins a C a atom and a side-chain charged group centroid in an ion-pairing residue.54In this dia-gram,the four ion-pairs,two HB-salt bridges,and two HB-NÀÀO bridges formed across the binding interface are given out. Association energies of the HB-salt bridges are both\21.5 kcal/mol,whereas that of the HB-NÀÀO bridges are all[20.5 kcal/mol.Therefore,it is believed that HB-salt bridge is more stable than HB-NÀÀO bridge,which is well consistent with the conclusion of Kumar and Nussinov.45,46Side-Chain Conformational EntropyIn general,SCE can be divided into the vibrational and the con-formational.55Comparison of several sets of results using differ-ent techniques shows that during protein folding process,the mean conformational free energy change(T D S)is1kcal/mol per side-chain or0.5kcal/mol per bond.Changes in vibrational entropy appear to be negligible compared with the entropy change resulted from the loss of accessible rotamers.56SCE(S) can be calculated quite simply using Boltzmann’s formulation.57S¼ÀRXip i ln p i(5)where R is the universal gas constant;The sum is taken over all conformational states of the system and p i is the probability of being in state i.Typical methods used for SCE calculations, include self-consistent meanfield theory,58molecular dynam-ics,59Monte Carlo simulation,60etc.,that are all time-consum-ing,thus not suitable for2D-GraLab.For that,the case is sim-plified,when we calculate the SCE of an interfacial residue,its local surrounding isfixed(adopting crystal conformation).In this way,SCE of each interfacial residue is calculated in turn.For the20coded amino acids,Gly,Ala,Pro,and Cys in disulfide bonds are excluded.57For other cases,each residue’s side-chain conformation is modeled as a rotamer withfinite number of discrete states.61The penultimate rotamer library used was developed by Lovell et al.,62as recommended by Dun-brack for the study of SCE.63For an interfacial residue,the potential E i of each rotamer i is calculated in both binding state and unbinding state,and subsequently,rotamer’s probability dis-tribution(p)of this residue is resulted by Boltzmann’s distribu-tion law,then the SCE in different states are solved out using eq.(5).The situation of rotamer i is defined as serious clash or nonclash:serious clash is the clash score of rotamer i more than a given threshold value,and then E i511;whereas for the9452D Depiction of Nonbonding Interactions for Protein Complexes946Zhou,Tian,and Shang•Vol.30,No.6•Journal of Computational ChemistryFigure3.Schematic representation of desolvation effect for interfacial residues in chain A of HLA-A*0201complex(PDB entry:1duz).This diagram was produced using2D-GraLab.In this diagram,the pie chart is equally divided,with each section indicates an interfacial residue in chain A;In a sec-tor,red1blue is the SASA of corresponding residue in unbinding state,the blue is in binding state,and the red is thus of D SASA;The green polygonal line is made by linking desolvation free energy ofeach interfacial residue,and at the purple circle,desolvation free energy is0(D U50),beyond thiscircle indicates unfavorable contributions to binding(D U[0),otherwise is favorable(D U\0);Inthe periphery,residue symbols are colored in red,blue,and black in terms of favorable,unfavorable,and neutral contributions to the binding,respectively;The SASA and desolvation free energy for eachinterfacial residue can be measured qualitatively by the horizontally black and green scales.[Colorfigure can be viewed in the online issue,which is available at .]Figure4.Four ion-pairs formed across the binding interface of penicillin acylase enzyme complex (PDB entry:1gkf).In thisfigure,left is2D schematic diagram produced using2D-GraLab,and posi-tively and negatively charged residues are colored in blue and red,respectively;Bridge-bonds formed between the charged atoms of ion-pairs are colored in green,blue,and yellow dashed lines for the hydrogen-bonded bridge,nonhydrogen-bonded bridge,and long-range interactions,respectively;The three parameters in bracket are ion-pair type,angle,and association energy.The right in thisfigure is the spatial conformations of corresponding ion-pairs.[Colorfigure can be viewed in the online issue, which is available at .]Figure5.(a)Loss of side-chain conformational entropy of chain B interfacial residues in HIV-1 reverse transcriptase complex(PDB entry:1rt1).This diagram was produced using2D-GraLab.In this diagram,the pie chart is equally divided,with each section indicates an interfacial residue in chain B; In a sector,side-chain conformational entropies in unbinding and binding state are colored in yellow and blue,respectively;The green polygonal line is made by linking conformational free energy of each interfacial residue;The conformational entropy and conformational free energy for each interfa-cial residue can be measured qualitatively by the horizontally black and green scales,respectively;In the periphery,residue symbols are colored in yellow,blue,and black in terms of favorable,unfavora-ble,and neutral contributions to binding,respectively.(b)The rotamers of chain B interfacial residues Lys20,Lys22,Tyr56,Asn136,Ile393,and Trp401in HIV-1reverse transcriptase complex.These rotamers were generated using2D-GraLab.[Colorfigure can be viewed in the online issue,which is available at .]9472D Depiction of Nonbonding Interactions for Protein Complexes948Zhou,Tian,and Shang•Vol.30,No.6•Journal of Computational ChemistryFigure5.(Legend on page947.)Figure6.The summarized schematic diagram of nonbonding interactions and disulfide bond across the interface of AIV hemagglutinin H5complex(PDB entry:1jsm).Length of chain A and chain B are321and160,represented as two bold horizontal lines.Interface parts in the bold lines are colored in orange,and residue-pairs in interactions are linearly linked;Conventional hydrogen bond,water-mediated hydrogen bond,ionpair,hydrophobic force,steric clash,p–p stacking,and disulfide bond are colored in aqua,bottle green,red,blue,purple,yellow,and brown,respectively;In the‘‘dumbbell shape’’symbols,residue-pair types and distances are also presented.[Colorfigure can be viewed in the online issue,which is available at .]9492D Depiction of Nonbonding Interactions for Protein Complexescase of nonclash,four potential functions are used in2D-Gra-Lab:(i)E i5E0,a constant61;(ii)statistical potential,the poten-tial energy E i of rotamer i is calculated from database-derived probability61;(iii)coarse-grained model,E i of rotamer i is esti-mated by atomic contact energies(ACE)64;and(iv)Lennard-Jones potential.58Loss of binding entropy of chain B interfacial residues in HIV-1reverse transcriptase complex is schematically repre-sented in Figure5a.Similar to desolvation effect diagram,loss of binding entropy is also presented in a rotiform diagram form. This diagram reveals that during the process of forming HIV-1 reverse transcriptase complex,the total loss of conformational free energy of chain B is9.14kcal/mol,indicating a strongly unfavorable contribution to binding(D G[0),and the average loss of conformational free energy for each residue is about0.3 kcal/mol,much less than those in protein folding(about1kcal/ mol56).Figure5b shows the rotamers of six interfacial residues in chain B.Summarized Schematic DiagramFigure6illustrates nonbonding interactions and disulfide bond formed across the binding interface of avian influenza virus (AIV)hemagglutinin H5.This protein is a dimer linked by a disulfide bond.In this diagram,conventional hydrogen bond, water-mediated hydrogen bond,ion-pair,hydrophobic force, steric clash,p–p stacking,and disulfide bond are represented in different colors.Hydrogen bonds,colored in aqua,are calculated by program HBplus.23Data in this diagram are the separation between the acceptor atom and the heavy donor atom.Water-mediated hydrogen bonds are colored in bottle green, also calculated by HBplus.23Ion-pairs,colored in red,include salt bridge and NÀÀO bridge,determined by the Kumar’s rule.45,46Data in this dia-gram are centroid distance of ion-pair.Hydrophobic forces are colored in blue.According to the D SASA rule,if the two apolar and/or aromatic interfacial resi-dues(Leu,Ala,Val,Ile,Met,Cys,Pro,Tyr,Phe,and Trp)are within the distance d\r A1r B12.8(r A and r B are side-chain radii,2.8is the diameter of water molecule),they are considered in hydrophobic contact.Data in this diagram are centroid–cent-roid separation between the two residues.Steric clashes are colored in purple.Here,only bad overlaps calculated by Probe24are presented.In2D-GraLab,explicit and implicit hydrogen modes are provided,hydrogen atoms in explicit hydrogern mode are added using Reduce.22Data in this diagram are the centroid–centroid separation when the two atoms are badly overlapped.p–p stacking are colored in yellow.Presently,studies on pro-tein stacking interactions are in lack.In2D-GraLab,p–p stack-ing is identified using the McGaughey’s rule,65i.e.,if the cent-roid–centroid separation between two aromatic rings is within 7.5A˚,they are regarded as p–p stacking(aromatic residues are Phe,Tyr,Trp,and His).This rule has been successfully adopted to study the p–p stacking across protein interfaces by Cho et al.66Besides,2D-GraLab also sets the constraints of stacking angle(dihedral angel between the planes of two aromatic rings).Data in this diagram are centroid–centroid separations between two aromatic rings in stacking state.Disulfide bonds are colored in brown,taken from the PDB records.Data in this diagram are the separations of two sulfide atoms.ConclusionsMost,if not all,biological processes are regulated through asso-ciation and dissociation of protein molecules and essentially controlled by nonbonding energetics.67Graphically-intuitive vis-ualization of these nonbonding interactions is an important approach for understanding the mechanism of a complex formed between two proteins.Although a large number of software packages are available for visualizing the3D structures,the options for producing schematic2D summaries of nonbonding interactions for a protein complex are comparatively few.In practice,the2D and3D visualization methods are complemen-tary.In this article,we have described a new2D molecular graphics tool for analyzing and visualizing PPIs from spatial structures,and the intended goal is to schematically present the nonbonding interactions stabilizing the macromolecular complex in a graphically-intuitive manner.We anticipate that renewed in-terest in automated generation of2D diagrams will significantly reduce the burden of protein structure analysis and make insights into the mechanism of PPIs.2D-GraLab is written in C11and OpenGL,and the output-ted2D schematic diagrams of nonbinding interactions are described in PostScript.Presently,2D-GraLab v1.0is available to academic users free of charge by contacting us. References1.Chothia,C.;Janin,J.Nature1974,256,705.2.Jones,S.;Thornton,J.M.Proc Natl Acad Sci USA1996,93,13.3.Luscombe,N.M.;Laskowski,R.A.;Westhead,D.R.;Milburn,D.;Jones,S.;Karmirantzoua,M.;Thornton,J.M.Acta Crystallogr D 1998,54,1132.4.DeLano,W.L.The PyMOL Molecular Graphics System;DeLanoScientific:San Carlos,CA,2002.5.Petrey,D.;Honig,B.Methods Enzymol2003,374,492.6.Humphrey,W.;Dalke,A.;Schulten,K.J Mol Graphics1996,14,33.7.Gabdoulline,R.R.;Wade,R.C.;Walther,D.Nucleic Acids Res2003,31,3349.8.Gabdoulline,R.R.;Hoffmann,R.;Leitner,F.;Wade,R.C.Bioin-formatics2003,19,1723.9.Wade,R. C.;Gabdoulline,R.R.;De Rienzo, F.Int J QuantumChem2001,83,122.10.Wallace, A. C.;Laskowski,R. A.;Thornton,J.M.Protein Eng1995,8,127.11.Stierand,K.;Maaß,P.C.;Rarey,M.Bioinformatics2006,22,1710.12.Clark,A.M.;Labute,P.J Chem Inf Model2007,47,1933.13.Luscombe,N.M.;Laskowski,R. A.;Thorntonm J.M.NucleicAcids Res1997,25,4940.14.Salerno,W.J.;Seaver,S.M.;Armstrong,B.R.;Radhakrishnan,I.Nucleic Acids Res2004,32,W566.15.Fischer,T.B.;Holmes,J.B.;Miller,I.R.;Parsons,J.R.;Tung,L.;Hu,J.C.;Tsai,J.J Struct Biol2006,153,103.950Zhou,Tian,and Shang•Vol.30,No.6•Journal of Computational Chemistry。

percolation methodThe percolation method is a mathematical model used to study the behavior of interconnected systems. It is primarily used in the field of physics and applied to various phenomena, such as the flow of fluids through porous materials, the spread of information in social networks, and the behavior of networks in computer science.In the percolation method, a system is represented by a lattice or network of interconnected nodes. Each node can be in one of two states: occupied or unoccupied. The nodes are randomly assigned these states, and the percolation process involves studying how a property or behavior spreads through the system.One common application of the percolation method is in studying the flow of fluids through porous materials, such as the movement of water through soil or the flow of gas through a network of interconnected pores. By assigning occupied states to the nodes representing open pores and unoccupied states to closed pores, researchers can simulate the movement of fluids and analyze properties like permeability and conductivity.In social networks, the percolation method can be used to study the spread of information or influence. By assigning occupied states to nodes representing individuals who have adopted a new behavior oridea, researchers can analyze how this behavior spreads through the network and identify key influencers or critical thresholds.In computer science, the percolation method is used to study the behavior of networks, such as the robustness of communication networks or the spread of computer viruses. By assigning occupied states to nodes representing active or infected computers, researchers can simulate the spread of viruses or analyze the resilience of the network to node failures.Overall, the percolation method provides a framework for studying the behavior and properties of interconnected systems, allowing researchers to analyze various phenomena and make predictions about their behavior.。

Package‘nmw’May10,2023Version0.1.5Title Understanding Nonlinear Mixed Effects Modeling for PopulationPharmacokineticsDescription This shows how NONMEM(R)software works.NONMEM's classical estimation meth-ods like'First Order(FO)approximation','First Order Conditional Estima-tion(FOCE)',and'Laplacian approximation'are explained.Depends R(>=3.5.0),numDerivByteCompile yesLicense GPL-3Copyright2017-,Kyun-Seop BaeAuthor Kyun-Seop BaeMaintainer Kyun-Seop Bae<********>URL https:///package=nmwNeedsCompilation noRepository CRANDate/Publication2023-05-1003:40:02UTCR topics documented:nmw-package (2)AddCox (3)CombDmExPc (4)CovStep (5)EstStep (6)InitStep (7)TabStep (9)TrimOut (10)Index1112nmw-package nmw-package Understanding Nonlinear Mixed Effects Modeling for PopulationPharmacokineticsDescriptionThis shows how NONMEM(R)</innovation/nonmem/>software works. DetailsThis package explains’First Order(FO)approximation’method,’First Order Conditional Estima-tion(FOCE)’method,and’Laplacian(LAPL)’method of NONMEM software.Author(s)Kyun-Seop Bae<********>References1.NONMEM Users guide2.Wang Y.Derivation of various NONMEM estimation methods.J Pharmacokinet Pharmaco-dyn.2007.3.Kang D,Bae K,Houk BE,Savic RM,Karlsson MO.Standard Error of Empirical BayesEstimate in NONMEM(R)VI.K J Physiol Pharmacol.2012.4.Kim M,Yim D,Bae K.R-based reproduction of the estimation process hidden behind NON-MEM Part1:First order approximation method.2015.5.Bae K,Yim D.R-based reproduction of the estimation process hidden behind NONMEM Part2:First order conditional estimation.2016.ExamplesDataAll=Theophcolnames(DataAll)=c("ID","BWT","DOSE","TIME","DV")DataAll[,"ID"]=as.numeric(as.character(DataAll[,"ID"]))nTheta=3nEta=3nEps=2THETAinit=c(2,50,0.1)OMinit=matrix(c(0.2,0.1,0.1,0.1,0.2,0.1,0.1,0.1,0.2),nrow=nEta,ncol=nEta) SGinit=diag(c(0.1,0.1))LB=rep(0,nTheta)#Lower boundUB=rep(1000000,nTheta)#Upper boundFGD=deriv(~DOSE/(TH2*exp(ETA2))*TH1*exp(ETA1)/(TH1*exp(ETA1)-TH3*exp(ETA3))*(exp(-TH3*exp(ETA3)*TIME)-exp(-TH1*exp(ETA1)*TIME)),AddCox3 c("ETA1","ETA2","ETA3"),function.arg=c("TH1","TH2","TH3","ETA1","ETA2","ETA3","DOSE","TIME"),func=TRUE,hessian=TRUE)H=deriv(~F+F*EPS1+EPS2,c("EPS1","EPS2"),function.arg=c("F","EPS1","EPS2"),func=TRUE) PRED=function(THETA,ETA,DATAi){FGDres=FGD(THETA[1],THETA[2],THETA[3],ETA[1],ETA[2],ETA[3],DOSE=320,DATAi[,"TIME"]) Gres=attr(FGDres,"gradient")Hres=attr(H(FGDres,0,0),"gradient")if(e$METHOD=="LAPL"){Dres=attr(FGDres,"hessian")Res=cbind(FGDres,Gres,Hres,Dres[,1,1],Dres[,2,1],Dres[,2,2],Dres[,3,])colnames(Res)=c("F","G1","G2","G3","H1","H2","D11","D21","D22","D31","D32","D33") }else{Res=cbind(FGDres,Gres,Hres)colnames(Res)=c("F","G1","G2","G3","H1","H2")}return(Res)}#######First Order Approximation Method#Commented out for the CRAN CPU time#InitStep(DataAll,THETAinit=THETAinit,OMinit=OMinit,SGinit=SGinit,LB=LB,UB=UB,#Pred=PRED,METHOD="ZERO")#(EstRes=EstStep())#4sec#(CovRes=CovStep())#2sec#PostHocEta()#Using e$FinalPara from EstStep()#TabStep()########First Order Conditional Estimation with Interaction Method#InitStep(DataAll,THETAinit=THETAinit,OMinit=OMinit,SGinit=SGinit,LB=LB,UB=UB,#Pred=PRED,METHOD="COND")#(EstRes=EstStep())#2min#(CovRes=CovStep())#1min#get("EBE",envir=e)#TabStep()########Laplacian Approximation with Interacton Method#InitStep(DataAll,THETAinit=THETAinit,OMinit=OMinit,SGinit=SGinit,LB=LB,UB=UB,#Pred=PRED,METHOD="LAPL")#(EstRes=EstStep())#4min#(CovRes=CovStep())#1min#get("EBE",envir=e)#TabStep()AddCox Add a Covariate Column to an Existing NONMEM datasetDescriptionA new covariate column can be added to an existing NONMEM dataset.4CombDmExPcUsageAddCox(nmData,coxData,coxCol,dateCol="DATE",idCol="ID")ArgumentsnmData an existing NONMEM datasetcoxData a data table containing a covariate columncoxCol the covariate column name in the coxData tabledateCol date column name in the NONMEM dataset and the covariate data tableidCol ID column name in the NONMEM dataset and the covariate data tableDetailsItfirst carry forward for the missing data.If NA is remained,it carry backward.ValueA new NONMEM dataset containing the covariate columnAuthor(s)Kyun-Seop Bae<********>CombDmExPc Combine the demographics(DM),dosing(EX),and DV(PC)tables intoa new NONMEM datasetDescriptionA new NONMEM dataset can be created from the demographics,dosing,and DV tables.UsageCombDmExPc(dm,ex,pc)Argumentsdm A demographics table.It should contain a row per subject.ex An exposure table.Drug administration(dosing)history table.pc A DV(dependent variable)or PC(drug concentration)tableDetailsCombining a demographics,a dosing,and a concentration table can produce a new NONMEM dataset.CovStep5ValueA new NONMEM datasetAuthor(s)Kyun-Seop Bae<********>CovStep Covariance StepDescriptionIt calculates standard errors and various variance matrices with the e$FinalPara after estimation step.UsageCovStep()DetailsBecause EstStep uses nonlinear optimization,covariance step is separated from estimation step.It calculates variance-covariance matrix of estimates in the original scale.ValueTime consumed timeStandard Error standard error of the estimates in the order of theta,omega,and sigmaCovariance Matrix of Estimatescovariance matrix of estimates in the order of theta,omega,and sigma.This isinverse(R)x S x inverse(R)by default.Correlation Matrix of Estimatescorrelation matrix of estimates in the order of theta,omega,and sigma Inverse Covariance Matrix of Estimatesinverse covariance matrix of estimates in the order of theta,omega,and sigma Eigen Values eigen values of covariance matrixR Matrix R matrix of NONMEM,the second derivative of log likelihood function with respect to estimation parametersS Matrix S matrix of NONMEM,sum of individual cross-product of thefirst derivative of log likelihood function with respect to estimation parametersAuthor(s)Kyun-Seop Bae<********>6EstStepReferencesNONMEM Users GuideSee AlsoEstStep,InitStepExamples#Only after InitStep and EstStep#CovStep()EstStep Estimation StepDescriptionThis estimates upon the conditions with InitStep.UsageEstStep()DetailsIt does not have arguments.All necessary arguments are stored in the e environment.It assumes "INTERACTION"between eta and epsilon for"COND"and"LAPL"options.The output is basically same to NONMEM output.ValueInitial OFV initial value of the objective functionTime time consumed for this stepOptim the raw output from optim functionFinal Estimatesfinal estimates in the original scaleAuthor(s)Kyun-Seop Bae<********>ReferencesNONMEM Users GuideSee AlsoInitStepExamples#Only After InitStep#EstStep()InitStep Initialization StepDescriptionIt receives parameters for the estimation and stores them into e environment.UsageInitStep(DataAll,THETAinit,OMinit,SGinit,LB,UB,Pred,METHOD)ArgumentsDataAll Data for all subjects.It should contain columns which Pred function uses.THETAinit Theta initial valuesOMinit Omega matrix initial valuesSGinit Sigma matrix initial valuesLB Lower bounds for theta vectorUB Upper bounds for theta vectorPred Prediction function nameMETHOD one of the estimation methods"ZERO","COND",or"LAPL"DetailsPrediction function should return not only prediction values(F or IPRED)but also G(first derivative with respect to etas)and H(first derivative of Y with respect to epsilon).For the"LAPL",prediction function should return second derivative with respect to eta also."INTERACTION"is TRUE for "COND"and"LAPL"option,and FALSE for"ZERO".Omega matrix should be full block one.Sigma matrix should be diagonal one.ValueThis does not return values,but stores necessary values into the environment e.Author(s)Kyun-Seop Bae<********>ReferencesNONMEM Users GuideExamplesDataAll=Theophcolnames(DataAll)=c("ID","BWT","DOSE","TIME","DV")DataAll[,"ID"]=as.numeric(as.character(DataAll[,"ID"]))nTheta=3nEta=3nEps=2THETAinit=c(2,50,0.1)#Initial estimateOMinit=matrix(c(0.2,0.1,0.1,0.1,0.2,0.1,0.1,0.1,0.2),nrow=nEta,ncol=nEta)OMinitSGinit=diag(c(0.1,0.1))SGinitLB=rep(0,nTheta)#Lower boundUB=rep(1000000,nTheta)#Upper boundFGD=deriv(~DOSE/(TH2*exp(ETA2))*TH1*exp(ETA1)/(TH1*exp(ETA1)-TH3*exp(ETA3))*(exp(-TH3*exp(ETA3)*TIME)-exp(-TH1*exp(ETA1)*TIME)),c("ETA1","ETA2","ETA3"),function.arg=c("TH1","TH2","TH3","ETA1","ETA2","ETA3","DOSE","TIME"),func=TRUE,hessian=TRUE)H=deriv(~F+F*EPS1+EPS2,c("EPS1","EPS2"),function.arg=c("F","EPS1","EPS2"),func=TRUE) PRED=function(THETA,ETA,DATAi){FGDres=FGD(THETA[1],THETA[2],THETA[3],ETA[1],ETA[2],ETA[3],DOSE=320,DATAi[,"TIME"]) Gres=attr(FGDres,"gradient")Hres=attr(H(FGDres,0,0),"gradient")if(e$METHOD=="LAPL"){Dres=attr(FGDres,"hessian")Res=cbind(FGDres,Gres,Hres,Dres[,1,1],Dres[,2,1],Dres[,2,2],Dres[,3,])colnames(Res)=c("F","G1","G2","G3","H1","H2","D11","D21","D22","D31","D32","D33") }else{Res=cbind(FGDres,Gres,Hres)colnames(Res)=c("F","G1","G2","G3","H1","H2")}return(Res)}#########First Order Approximation MethodInitStep(DataAll,THETAinit=THETAinit,OMinit=OMinit,SGinit=SGinit,LB=LB,UB=UB, Pred=PRED,METHOD="ZERO")#########First Order Conditional Estimation with Interaction MethodInitStep(DataAll,THETAinit=THETAinit,OMinit=OMinit,SGinit=SGinit,LB=LB,UB=UB, Pred=PRED,METHOD="COND")#########Laplacian Approximation with Interacton MethodInitStep(DataAll,THETAinit=THETAinit,OMinit=OMinit,SGinit=SGinit,LB=LB,UB=UB,TabStep9 Pred=PRED,METHOD="LAPL")TabStep Table StepDescriptionThis produces standard table.UsageTabStep()DetailsIt does not have arguments.All necessary arguments are stored in the e environment.This is similar to other standard results table.ValueA table with ID,TIME,DV,PRED,RES,WRES,derivatives of G and H.If the estimation methodis other than’ZERO’(First-order approximation),it includes CWRES,CIPREDI(formerly IPRED), CIRESI(formerly IRES).Author(s)Kyun-Seop Bae<********>ReferencesNONMEM Users GuideSee AlsoEstStepExamples#Only After EstStep#TabStep()10TrimOut TrimOut Trimming and beutifying NONMEM original OUTPUTfileDescriptionTrimOut removes unnecessary parts from NONMEM original OUTPUTfile.UsageTrimOut(inFile,outFile="PRINT.OUT")ArgumentsinFile NONMEM original untidy OUTPUTfile nameoutFile Outputfile name to be writtenDetailsNONMEM original OUTPUTfile contains unnecessary parts such as CONTROLfile content, Start/End Time,License Info,Print control characters such as"+","0","1".This function trims those.ValueoutFile will be written in the current working folder or designated folder.Ths returns TRUE if the process was smooth.Author(s)Kyun-Seop Bae<********>Index∗Covariance StepCovStep,5∗Data PreparationAddCox,3CombDmExPc,4∗Estimation StepEstStep,6∗Initialization StepInitStep,7∗NONMEM OUTPUTTrimOut,10∗Nonlinear Mixed Effects Modelingnmw-package,2∗Population Pharmacokineticsnmw-package,2∗Tabulation StepTabStep,9AddCox,3CombDmExPc,4CovStep,5EstStep,6,6,9InitStep,6,7nmw(nmw-package),2nmw-package,2TabStep,9TrimOut,1011。

临界问题物理经典模型必修一Physics is a fundamental science that seeks to understand the natural world through observations, experimentation, and mathematical models. One classic model in physics is the critical phenomena, which deal with the behavior of physical systems at critical points. These critical points are characterized by certain properties such as diverging correlation lengths and scaling symmetries, which are essential for studying phase transitions.物理学是一门基础科学，通过观察、实验和数学模型来理解自然界。

在物理学中有一个经典模型，就是临界现象，这个模型处理物理系统在临界点的行为。

这些临界点的特征包括发散的相关长度和尺度对称性，这些特征对于研究相变非常重要。

The study of critical phenomena plays a critical role in various fields of physics, such as condensed matter physics and statistical mechanics. Understanding how systems behave at critical points can provide valuable insights into the nature of phase transitions, as well as the universal behavior of physical systems near criticality. The critical phenomena model has been successfully applied to explain awide range of physical phenomena, from the behavior of magnets to the properties of liquid-gas transitions.对临界现象的研究在物理学的多个领域中都起着至关重要的作用，比如凝聚态物理学和统计力学。

LPS 与ATP 共同诱导小鼠原代腹腔巨噬细胞焦亡模型的建立①刘慧玲 吴传新② 龙贤梨 李丽 李飞 郭晖 孙航（重庆医科大学附属第二医院病毒性肝炎研究所，重庆 400010）中图分类号 R392.1 文献标志码 A 文章编号 1000-484X （2023）10-2028-06［摘要］ 目的：探索脂多糖（LPS ）和三磷酸腺苷（ATP ）共同诱导小鼠原代腹腔巨噬细胞焦亡模型的最佳条件。

方法：采用流式细胞仪F4/80和CD -11b 染色检测巨噬细胞纯度，Annexin V -PE/7-AAD 双染色法筛选出LPS 和ATP 共同诱导细胞焦亡的最适浓度及时间。

巨噬细胞随机分为control 组、LPS 组、ATP 组和LPS+ATP 组；Western blot 检测GSDMD 、caspase -1、caspase -11、NLRP3、ASC 、pro -IL -1β、pro -IL -18和HMGB1蛋白表达水平；ELISA 检测培养上清中IL -1β和TNF -α表达水平；透射电镜（TEM ）和扫描电镜（SEM ）观察巨噬细胞焦亡形态。

结果：巨噬细胞的纯度达到90%；500 ng/ml LPS 24 h+5 mmol/L ATP 4 h 为诱导巨噬细胞焦亡的最佳组合方式；LPS+ATP 组的GSDMD 、caspase -1、caspase -11、NLRP3、ASC 、pro -IL -1β、pro -IL -18和HMGB1的蛋白表达量明显高于对照组（P <0.05）；培养上清中IL -1β和TNF -α表达量显著高于对照组（P <0.05）；电镜下可观察到明显的焦亡特征。

结论：成功建立了LPS 和ATP 共同诱导小鼠原代腹腔巨噬细胞的焦亡模型，为深入探讨免疫细胞焦亡的分子机制提供了稳定的细胞模型。

［关键词］ LPS ；ATP ；细胞焦亡；原代腹腔巨噬细胞；脓毒症Establishment of pyroptosis model on primary peritoneal macrophages induced by LPS and ATPLIU Huiling ， WU Chuanxin ， LONG Xianli ， LI Li ， LI Fei ， GUO Hui ， SUN Hang. Institute for Viral Hepatitis , the Second Affiliated Hospital ， Chongqing Medical University ， Chongqing 400010， China［Abstract ］ Objective ：To explore optimal condition of a model of pyroptosis on primary peritoneal macrophages induced by thelipopolysaccharide （LPS ） and adenosine triphosphate （ATP ）. Methods ：Purity of macrophages was detected by flow cytometric with F4/80 and CD11-b ， and Annexin V -PE/7-AAD double staining was used to detect pyroptosis cell for screening the optimum concentra‑tion and time of pyroptotic cells induced by LPS and ATP. Macrophages were randomly divided into control group ， LPS group ， ATP group and LPS+ATP group. Expressions of GSDMD ， caspase -1， caspase -11， NLRP3， ASC ， pro -IL -1β， pro -IL -18 and HMGB1 proteins were detected by Western blot. Levels of IL -1β and TNF -α in culture supernatant were measured by ELISA. Structure of pyroptosis macrophages was observed by transmission electron microscope （TEM ） and scan electron microscope （SEM ）. Results ：Purity of primary peritoneal macrophages could be 90%； 500 ng/ml LPS 24 h and 5 mmol/L ATP 4 h was the optimal combination of inducing macrophages pyroptosis. Compared with control group ， LPS and ATP group had significantly increased protein expressions of GSDMD ， caspase -1， caspase -11， NLRP3， ASC ， pro -IL -1β， pro -IL -18 and HMGB1 （P <0.05）， and levels of IL -1β and TNF -α in culture supernatant were significantly higher than that in control group （P <0.05）； structure of pyroptosis macrophages could be obviously observed by TEM and SEM. Conclusion ：Pyroptosis model of primary peritoneal macrophages induced by LPS and ATP is successfully established ， whichprovides a cell model for exploring the molecular mechanism of pyroptosis on immune cells in the future.［Key words ］ LPS ；ATP ；Pyroptosis ；Primary peritoneal macrophages ；Sepsis细胞焦亡是一种依赖半胱天冬蛋白酶（caspase -1/-4/-5/-11）活化的炎症细胞死亡方式，其形态介于细胞凋亡和细胞坏死之间，且细胞焦亡的发生机制和调控机制与凋亡和坏死大不相同［1］。

The Copula-GARCH model of conditionaldependencies:An international stockmarket applicationEric Jondeau,Michael Rockinger *Swiss Finance Institute and University of Lausanne,Lausanne,SwitzerlandAbstractModeling the dependency between stock market returns is a difficult task when returns follow a com-plicated dynamics.When returns are non-normal,it is often simply impossible to specify the multivariate distribution relating two or more return series.In this context,we propose a new methodology based on copula functions,which consists in estimating first the univariate distributions and then the joining distri-bution.In such a context,the dependency parameter can easily be rendered conditional and time varying.We apply this methodology to the daily returns of four major stock markets.Our results suggest that con-ditional dependency depends on past realizations for European market pairs only.For these markets,de-pendency is found to be more widely affected when returns move in the same direction than when they move in opposite directions.Modeling the dynamics of the dependency parameter also suggests that de-pendency is higher and more persistent between European stock markets.Ó2006Published by Elsevier Ltd.JEL classification:C51;F37;G11Keywords:Stock indices;International correlation;Dependency;GARCH model;Skewed Student-t distribution;Copula function*Corresponding author.University of Lausanne,Ecole des HEC,Department of Finance and Insurance,1015Lausanne,Switzerland.Tel.:þ41216923348;fax:þ41216923435.E-mail addresses:eric.jondeau@unil.ch (E.Jondeau),michael.rockinger@unil.ch (M.Rockinger).0261-5606/$-see front matter Ó2006Published by Elsevier Ltd.doi:10.1016/j.jimonfin.2006.04.007Journal of International Money and Finance 25(2006)827e853828 E.Jondeau,M.Rockinger/Journal of International Money and Finance25(2006)827e8531.IntroductionAn abundant literature has investigated how the correlation between stock market returns varies when markets become agitated.In a multivariate GARCH framework,for instance, Hamao et al.(1990),Susmel and Engle(1994),and Bekaert and Harvey(1995)have measured the interdependence of returns and volatilities across stock markets.More specifically,Longin and Solnik(1995)have tested the hypothesis of a constant conditional correlation between a large number of stock markets.They found that correlation generally increases in periods of high-volatility of the U.S.market.In addition,in a similar context,tests of a constant cor-relation have been proposed by Bera and Kim(2002)and Tse(2000).Recent contributions by Kroner and Ng(1998),Engle and Sheppard(2001),Engle(2002),and Tse and Tsui(2002) have developed GARCH models with time-varying covariances or correlations.As an alterna-tive approach,Ramchand and Susmel(1998)and Ang and Bekaert(2002)have estimated a mul-tivariate Markov-switching model and tested the hypothesis of a constant international conditional correlation between stock markets.They obtained that correlation is generally higher in the high-volatility regime than in the low-volatility regime.In this context,an important issue is how dependency between stock markets can be mea-sured when returns are non-normal.In the GARCH framework,some recent papers have focused on multivariate distributions which allow for asymmetry as well as fat tails.For instance,multivariate skewed distributions,and in particular the skewed Student-t distribution, have been studied by Sahu et al.(2001)and Bauwens and Laurent(2002).In addition,in the Markov-switching context,Chesnay and Jondeau(2001)have tested for a constant correlation between stock returns,while allowing for Student-t innovations.1For most types of univariate distributions,however,it is simply impossible to specify a multivariate extension that would allow the dependency structure to be captured.In this paper,we present a new methodology to measure conditional dependency in a GARCH context.Our methodology builds on so-called ‘‘copula’’functions.These functions provide an interesting tool to model a multivariate distri-bution when only marginal distributions are known.Such an approach is,thus,particularly use-ful in situations where multivariate normality does not hold.An additional interesting feature of copulas is the ease with which the associated dependency parameter can be conditioned and rendered time varying,even when complicated marginal dynamics are estimated.We use this methodology to investigate the impact of certain joint stock return realizations on the subsequent dependency of international markets.Many univariate models have been pro-posed to specify the dynamics of returns.However,given the focus of this work,we draw on recent advances in the modeling of conditional returns that allow second,third,and fourth moments to vary over time.Our univariate model builds on Hansen’s(1994)seminal paper. In that paper,a so-called skewed Student-t distribution is derived.This distribution allows for a control of asymmetry and fat-tailedness.By rendering these characteristics conditional, it is possible to obtain time-varying higher moments.2This model,therefore,extends Engle’s (1982)ARCH and Bollerslev’s(1986)GARCH models.In an extension to Hansen(1994), 1Some papers also considered how correlation varies when stock market indices are simultaneously affected by very large(positive or negative)fluctuations.Longin and Solnik(2001),using extreme value theory,found that dependency increases more during downside movements than during upside movements.Poon et al.(2004)adopted an alternative statistical framework to test conditional dependency between extreme returns and showed that such a tail dependency may have been overstated once the time-variability of volatility is accounted for.2Higher moments refer to the standardized third and fourth central moments.Jondeau and Rockinger (2003a,b)determine the expression of skewness and kurtosis of the skewed Student-t distribution and show how the cumulative distribution function (cdf)and its inverse can be computed.They show how to simulate data distributed as a skewed Stu-dent-t distribution and discuss how to parametrize time-varying higher moments.We then consider two alternative copula functions which have different characteristics in terms of tail dependency:the Gaussian copula that does not allow any dependency in the tails of the distribution,and the Student-t copula that is able to capture such a tail dependency.Fi-nally,we propose several ways to condition the dependency parameter with past realizations.It is thus possible to test several hypotheses of the way in which dependency varies during turbu-lent periods.In the empirical part of the paper,we investigate the dependency structure between daily returns of major stock market indices over 20years.As a preliminary step,we provide evidence that the skewed Student-t distribution with time-varying higher moments fits very well the uni-variate behavior of the data.Then,we check that the Student-t copula is able to capture the de-pendency structure between market returns.Further scrutiny of the data reveals that the dependency between European markets increases,subsequent to movements in the same direc-tion,either positively or negatively.Furthermore,the strong persistence in the dynamics of the dependency structure is found to reflect a shift,over the sample period,in the dependency pa-rameter.This parameter has increased from about 0.3over the 1980s to about 0.6over the next decade.Such a pattern is not found to hold for the dependency structure between the U.S.mar-ket and the European markets.The remainder of the paper is organized as follows.In Section 2,we first introduce our uni-variate model which allows for volatility,skewness,and kurtosis,to vary over time.In Section 3,we introduce copula functions and describe the copulas used in the empirical application.We also describe how the dependency parameter may vary over time.In Section 4,we present the data and discuss our empirical results.Section 5summarizes our results and concludes.2.A model for the marginal distributionsIt is well known that the residuals obtained from a GARCH model are generally non-normal.This observation has led to the introduction of fat-tailed distributions for innovations.For in-stance,Nelson (1991)considered the generalized error distribution,while Bollerslev and Wool-dridge (1992)focused on Student-t innovations.Engle and Gonzalez-Rivera (1991)modeled residuals non-parametrically.Even though these contributions recognize the fact that errors have fat tails,they generally do not render higher moments time varying,i.e.,parameters of the error distribution are assumed to be constant over time.Our margin model builds on Hansen (1994).2.1.Hansen’s skewed Student-t distributionHansen (1994)was the first to propose a GARCH model,in which the first four mo-ments are conditional and time varying.For the conditional mean and volatility,he built on the usual GARCH model.To control higher moments,he constructed a new density with which he modeled the GARCH residuals.The new density is a generalization of the Student-t distribution while maintaining the assumption of a zero mean and unit vari-ance.The conditioning is obtained by defining parameters as functions of past realizations.829E.Jondeau,M.Rockinger /Journal of International Money and Finance 25(2006)827e 853Some extensions to this seminal contribution may be found in Theodossiou(1998)and Jondeau and Rockinger(2003a).3Hansen’s skewed Student-t distribution is defined bydðz;h;lÞ¼8>>><>>>:bc1þ1hÀ2bzþa1Àl2 À½ðhþ1Þ=2if z<Àa=bbc1þ1bzþa 2 À½ðhþ1Þ=2if z!Àa=b;ð1Þwherea h4l c hÀ2;b2h1þ3l2Àa2;c hGhþ12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipðhÀ2ÞpGh2;and h and l denote the degree-of-freedom parameter and the asymmetry parameter,respec-tively.If a random variable Z has the density d(z;h,l),we will write Z w ST(h,l).Additional useful results are provided in the Appendix A.In particular,we characterize the domain of definition for the distribution parameters h and l,give formulas relating higher moments to h and l,and we describe how the cdf of Hansen’s skewed Student-t distribution can be computed.This computation is necessary for the evaluation of the likelihood of the copula function.2.2.A GARCH model with conditional skewness and kurtosisLet the returns of a given asset be given by{r t},t¼1,.,T.Hansen’s margin model with time-varying volatility,skewness,and kurtosis,is defined byr t¼m tþ3t;ð2Þ3t¼s t z t;ð3Þs2 t ¼a0þbþÀ3þtÀ1Á2þbÀÀ3ÀtÀ1Á2þc0s2tÀ1;ð4Þz t w STðh t;l tÞ:ð5ÞEq.(2)decomposes the return of time t into a conditional mean,m t,and an innovation,3t. The conditional mean is modeled with10lags of r t and day-of-the-week dummies.Eq.(3) then defines this innovation as the product between conditional volatility,s t,and a residual, z t.Eq.(4)determines the dynamics of volatility.We use the notation3þt¼maxð3t;0Þand3Àt ¼maxðÀ3t;0Þ.For positivity and stationarity of the volatility process to be guaranteed,parameters are assumed to satisfy the following constraints:a0>0,bþ0;bÀ;c0!0,and3Harvey and Siddique(1999)have proposed an alternative specification,based on a non-central Student-t distribu-tion,in which higher moments also vary over time.This distribution is designed so that skewness depends on the non-centrality parameter and the degree-of-freedom parameter.Note also that the specification of the skewed Student-t distribution adopted by Lambert and Laurent(2002)corresponds to the distribution proposed by Hansen,but with asymmetry parametrized in a different way.830 E.Jondeau,M.Rockinger/Journal of International Money and Finance25(2006)827e853c 0þÀb þ0þb À0Á 2<1.Such a specification has been suggested by Glosten et al.(1993).Eq.(5)specifies that residuals follow a skewed Student-t distribution with time-varying parameters h t and l t .Many specifications could be used to describe the dynamics of h t and l t .To ensure that they remain within their authorized range,we consider an unrestricted dynamic that we constrain via a logistic map.4The type of functional specification that should be retained is discussed in Jon-deau and Rockinger (2003a).The general unrestricted model that we estimate is given by~h t ¼a 1þb þ13þt À1þb À13Àt À1þc 1~ht À1ð6Þ~l t ¼a 2þb þ23þt À1þb À23Àt À1þc 2~l t À1:ð7ÞWe map this dynamic into the authorized domain with h t ¼g L h ;U h ½ð~h t Þand l t ¼g L l ;U l ½ð~l t Þ.Several encompassing restrictions of this general specification are tested in the empirical section of the paper.In particular,we test,within the class of GARCH models of volatility,the following restrictions:a Gaussian conditional distribution,a standard Student-t distribution,and a skewed Student-t distribution with constant skewness and kurtosis.We will see that the most general model cannot be rejected for all the stock indices considered.3.Copula distribution functions3.1.The copula functionConsider two random variables X 1and X 2with marginal cdfs F i (x i )¼Pr[X i x i ],i ¼1,2.The joint cdf is denoted H (x 1,x 2)¼Pr[X 1 x 1,X 2 x 2].All cdfs F i ($)and H ($,$)range in the interval [0,1].In some cases,a multivariate distribution exists,so that the function H ($,$)has an explicit expression.One such case is the multivariate normal distribution.In many cases,however,the margins F i ($)are relatively easy to describe,while an explicit expression of the joint distribution H ($,$)may be difficult to obtain.5In such a context,copulas can be used to link margins into a multivariate distribution func-tion.The copula function extends the concept of multivariate distribution for random variables which are defined over [0,1].This property allows to define a multivariate distribution in terms of margins F i (x i )instead of realizations x i .Then,as highlighted in Sklar’s theorem (see the Appendix A ),one has the equality between the cdf H defined over realizations of the random variables x i and the copula function C defined over margins F i (x i ),so that H (x 1,x 2)¼C (F 1(x 1),F 2(x 2)).We now describe the two copula functions used in our empirical application.For notational convenience,set u i h F i (x i ).The Gaussian copula is defined by the cdfC ðu 1;u 2;r Þ¼F r ÀF À1ðu 1Þ;F À1ðu 2ÞÁ;and the density by4The logistic map,g L ;U ½ðx Þ¼L þðU ÀL Þð1þe Àx ÞÀ1maps R into the interval ]L ,U [.In practice,we use the bounds L h ¼2,U h ¼30for h and L l ¼À1,U l ¼1for l .5It may be argued that multivariate extensions of the skewed Student-t distribution exist (see,in particular,Bauwens and Laurent,2002).In fact,the difficulty comes in this case from the joint estimation of two or more distributions,each involving a large number of unknown parameters.831E.Jondeau,M.Rockinger /Journal of International Money and Finance 25(2006)827e 853cðu1;u2;rÞ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi1Àr2p expÀ12j0ÀRÀ1ÀI2Áj;where j¼ÀFÀ1ðu1Þ;FÀ1ðu2ÞÁ0.The matrix R is the(2,2)correlation matrix with r as depen-dency measure between X1and X2.F r is the bivariate standardized Gaussian cdf with correla-tion r,À1<r<1.The letter F represents the univariate standardized Gaussian cdf.Similarly,the Student-t copula is defined byCðu1;u2;r;nÞ¼T n;r ÀtÀ1nðu1Þ;tÀ1nðu2ÞÁ;and its associated density iscðu1;u2;r;nÞ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi1Àr2pGnþ22Gn2Gnþ1221þ1nj0UÀ1jÀ½ðnþ2Þ=2Q2i¼11þ1nj2iÀ½ðnþ1Þ=2 ;where j¼ÀtÀ1nðu1Þ;tÀ1nðu2ÞÁ0.T n,r is the bivariate Student-t cdf with degree-of-freedom param-eter n and correlation r˛[À1,1],while t n is the univariate Student-t cdf with degree-of-freedom parameter n.These two copula functions have different characteristics in terms of tail dependence.The Gaussian copula does not have tail dependence,while the Student-t copula has got it(see, for instance,Embrechts et al.,2003).Such a difference is likely to have important conse-quences on the modeling of the dependency parameter.Indeed,Longin and Solnik(2001) have shown,using an alternative methodology,that the correlation between market returns is higher in case of extreme events.6Finally,the two copula functions under study are symmetric. Therefore,when the dependency parameter is assumed to be constant,large joint positive re-alizations have the same probability of occurrence as large joint negative realizations.In Sec-tion3.2,we relax this assumption by allowing the dependency parameter to be conditional on past realizations.Both Gaussian and Student-t copulas belong to the elliptical-copula family.Thus,when mar-gins are elliptical distributions withfinite variances,r is just the usual linear correlation coef-ficient and can be estimated using a linear correlation estimator(see Embrechts et al.,1999).In the following,however,we provide evidence that margins can be well approximated by the skewed Student-t distribution,which does not belong to the elliptical-distribution family.It fol-lows that r is not the linear Pearson’s correlation and needs to be estimated via maximum-likelihood.3.2.Alternative specifications for conditional dependencyLet us consider a sample{z1t,z2t},t¼1,.,T.It is assumed that z it gets generated by a con-tinuous cdf F i($;q i),where q i represents the vector of unknown parameters pertaining to the marginal distribution of Z it,i¼1,2.In our context,z it is the residual of the univariate GARCH model presented in Section2.2.6Poon et al.(2004)have obtained that much of this increase in dependency may be explained by changes in volatility. 832 E.Jondeau,M.Rockinger/Journal of International Money and Finance25(2006)827e853The key observation is that the copula depends on parameters that can be easily conditioned.We define r t as the value taken by the dependency parameter at time t .For the Student-t copula,the degree-of-freedom parameter n may be conditioned as well.Several different specifications of the dependency parameter are possible in our context.Asa first approach,we follow Gourie´roux and Monfort (1992)and adopt a semi-parametric spec-ification in which r t depends on the position of past joint realizations in the unit square.This means that we decompose the unit square of joint past realizations into a grid,with parameter r t held constant for each element of the grid.More precisely,we consider an unrestricted specification:r t ¼X16j ¼1d j I Âðz 1t À1;z 2t À1Þ˛A j Ã;ð8Þwhere A j is the j th element of the unit square grid and d j ˛[À1,1].To each parameter d j ,an area A j is associated.For instance,A 1¼½0;p 1½Â½0;q 1½and A 2¼½p 1;p 2½Â½0;q 1½.7The choice of 16subintervals is somewhat arbitrary.It should be noticed,however,that it has the advantage of providing an easy testing of several conjectures concerning the impact of past joint returns on subsequent dependency while still allowing for a large number of ob-servations per area.In the empirical section,we test several hypotheses of interest on the parameters d j .It should be recognized that such a specification is not able to capture persistence in r t .Therefore,we first consider a time-varying correlation (TVC)approach,as proposed by Tse and Tsui (2002)in their modeling of the Pearson’s correlation in a GARCH context.The de-pendency parameter r t is assumed to be driven by the following model:r t ¼ð1Àa Àb Þr þax t À1þbr t À1;ð9Þwherex t ¼P m À1i ¼0z 1t Ài z 2t Ài ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP m À1i ¼0z 21t Ài P m À1i ¼0z 22t Àiq represents the correlation between the residuals over the recent period,with m !2.For statio-narity to be guaranteed,the following constraints are imposed:0 a ,b 1,a þb 1,and À1 r 1.Our empirical analysis,however,reveals a very large persistence in r t ,with a þb very close to 1in most cases.This result suggests that the TVC model may be inappropriate and that the long-memory feature may be in fact the consequence of a model with large but infrequentbreaks (see Lamoureux and Lastrapes,1990;Diebold and Inoue,1999;or Gourie ´roux and Ja-siak,2001).8This approach has been followed,among others,by Ramchand and Susmel7Fig.2illustrates the position of the areas d j s.In the figure,we have set equally spaced threshold levels,i.e.,p 1,p 2,and p 3take the values 0.25,0.5,and 0.75,respectively.The same for q 1,q 2,and q 3,respectively.In the empirical part of the paper,we use as thresholds the values 0.15,0.5,and 0.85.The reason for this choice is that we want to focus on rather large values.If we had used 0.25,0.5,and 0.75,the results would have been similar,however.8We are grateful to a referee for suggesting this interpretation.833E.Jondeau,M.Rockinger /Journal of International Money and Finance 25(2006)827e 853(1998),Chesnay and Jondeau (2001),and Ang and Bekaert (2002).These authors have ob-tained,for several market pairs,evidence of the presence of a high-volatility/high-correlation regime and a low-volatility/low-correlation regime.Thus,we also consider,as an alternative approach,a model in which parameters of the Student-t copula are driven by the following modelr t ¼r 0S t þr 1ð1ÀS t Þ;ð10Þn t ¼n 0S t þn 1ð1ÀS t Þ;ð11Þwhere S t denotes the unobserved regime of the system at time t .S t is assumed to follow a two-state Markov process,with transition probability matrix given byp 1Àp1Àq qwithp ¼Pr ½S t ¼0j S t À1¼0 ;q ¼Pr ½S t ¼1j S t À1¼1 :Note that,in this model,we do not assume that univariate characteristics of returns should also shift.Rather,only parameters pertaining to the dependence structure are driven by the Markov-switching model.Quasi maximum-likelihood estimation of this model can be easily obtained using the ap-proach developed by Hamilton (1989)and Gray (1995).For the degree-of-freedom parameter,we investigated several hypotheses.In particular,we tested whether it is regime independent (n 0¼n 1)or infinite,so that the Gaussian copula would prevail for a given regime.We also in-vestigated time-variation in transition probabilities,along the lines of Schaller and van Norden (1997).We tested specifications in which transition probabilities are allowed to depend on past volatilities (s 1t À1and s 2t À1)as well as on correlation between the residuals over the recent pe-riod (x t À1).We were unable,however,to obtain a significant time-variation in the probabilities using such variables.3.3.EstimationWe now assume that the copula function depends on a set of unknown parameters through the function Q (z 1t À1,z 2t À1;q c ).We have q c ¼(d 1,.,d 16,n )0for the semi-parametric specification,q c ¼(r ,a ,b ,n )0for the TVC specification,and q c ¼(r 0,r 1,n 0,n 1,p ,q )0for the Markov-switching model.We also denote f i as the marginal density of z it .In the context of a skewed Student-t marginal distribution,as presented in Section 2,this density is simply defined by f i (z it ;q i )¼d (z it ;q i ),i ¼1,2.We set q ¼Àq 01;q 02;q 0c Á0the vector of all parameters to be estimated.Conse-quently,the log-likelihood of a sample is given by[ðq Þ¼XT t ¼1[t ðq Þ;ð12Þ834 E.Jondeau,M.Rockinger /Journal of International Money and Finance 25(2006)827e 853where[t ðq Þ¼ln c ðF 1ðz 1t ;q 1Þ;F 2ðz 2t ;q 2Þ;Q ðz 1t À1;z 2t À1;q c ÞÞþX2i ¼1ln f i ðz it ;q i Þ:Maximum-likelihood estimation involves maximizing the log-likelihood function (12)si-multaneously over all parameters,yielding parameter estimates denoted b q ML ¼ðb q 01;b q 02;b q 0c Þ0,such thatb q ML ¼arg max [ðq Þ:In some applications,however,the ML estimation method may be difficult to implement,because of a large number of unknown parameters or of the complexity of the model.9In such a case,it may be necessary to adopt a two-step ML procedure,also called inference func-tions for margins.This approach,which has been introduced by Shih and Louis (1995)and Joe and Xu (1996),can be viewed as the ML estimation of the dependence structure given the es-timated margins.First,parameters pertaining to the marginal distributions are estimated separately:~q i ˛arg max XT t ¼1ln f i ðz it ;q i Þi ¼1;2:ð13ÞSecond,parameters pertaining to the copula function are estimated by solving the following equation:~q c ˛arg max XT t ¼1ln c ðF 1ðz 1t ;~q 1Þ;F 2ðz 2t ;~q 2Þ;Q ðz 1t À1;z 2t À1;q c ÞÞ:Patton (2006)has shown that this two-step estimation yields asymptotically efficient and normal parameter estimates.If q 0denotes the true value of the parameter vector,the asymptotic distribution of ~q TS ¼ð~q 01;~q 02;~q 0c Þ0is given byffiffiffiT p ð~q TS Àq 0Þ/N ð0;U Þ;where the asymptotic covariance matrix U may be estimated by the robust estimator b U¼b M À1b V b M À1,with b M¼ÀX T t ¼1v 2[t ð~q Þv q v q 0;b V ¼X T t ¼1v [t ð~q Þv q X T t ¼1v [t ð~q Þ0v q :9The dependency parameter of the copula function may be a convoluted expression of the parameters.In such a case,an analytical expression of the gradient of the likelihood might not exist.Therefore,only numerical gradients may be computable,implying a dramatic slowing down of the numerical procedure.835E.Jondeau,M.Rockinger /Journal of International Money and Finance 25(2006)827e 853836 E.Jondeau,M.Rockinger/Journal of International Money and Finance25(2006)827e8534.Empirical results4.1.The dataWe investigate the interactions between four major stock indices.The labels are SP for the S&P500,FTSE for the Financial Times100stock index,DAX for the Deutsche Aktien Index, and CAC for the French Cotation Automatique Continue index.Our sample covers the period from January1,1980to December29,2000.All the data are from Datastream,sampled at a daily frequency.To eliminate spurious correlation generated by holidays,we eliminated those observations when a holiday oc-curred at least for one country from the database.This reduced the sample from5479 observations to4578.Note that such an observation would not affect the dependency be-tween stock markets during extreme events.Yet,it would affect the estimation of the re-turn marginal distribution and,subsequently,the estimation of the distribution of the copula.In particular,the estimation of the copula would be distorted to account for the excessive occurrence of null returns in the distribution.To take into account the fact that international markets have different trading hours,we use once lagged U.S.returns, although this does not significantly affect the correlation with European markets(because trading times are partially overlapping).Preliminary estimations also revealed that the crash of October1987was of such importance that the standard errors of our model would be very much influenced by this event.For the S&P,on that date,the index drop-ped byÀ22%,while the second largest drop wasÀ9%only.For this reason,we elimi-nated the data between October17and24.This further reduces the sample to a total of 4572observations.Table1provides summary statistics on stock market returns.Returns(r t)are defined as 100Âln(P t/P tÀ1),where P t is the value of the index at time t.Statistics are computed after holidays have been removed from the time series.Therefore,the number of observations is the same one for all markets,and the series do not contain days when a market was closed. We begin with the serial dependency of returns.The LM(K)statistic tests whether the squared return is serially correlated up to lag K.This statistic clearly indicates that ARCH effects are likely to be found in all market returns.Also,when considering the Ljung e Box statistics, QW(K),after correction for heteroskedasticity,we obtain that returns are serially correlated for all the retained indices.We also consider the unconditional moments of the various series,with standard errors computed using a GMM-based procedure.We notice that for all series skewness,Sk,is neg-ative.Moreover,excess kurtosis,XKu,is significant for all return series.This indicates that the empirical distributions of returns display fatter tails than the Gaussian distribution.The Wald statistic of the joint test of significance of skewness and excess kurtosis corroborates thisfinding.10Finally,the unconditional correlation matrix indicates that a rather large dependency between market returns is expected.The correlation is the smallest between the SP and the CAC,and the largest between the DAX and the CAC.10When the1987crash is not removed,the SP distribution is characterized by a very strong asymmetry(with a skew-ness equal toÀ2.55)and fat tails(with an excess kurtosis as high as57).Yet,due to uncertainty around higher-moment point estimates,the Wald test would not reject normality.。

Discrete Applied Mathematics157(2009)2217–2220Contents lists available at ScienceDirectDiscrete Applied Mathematicsjournal homepage:/locate/damPreface$This special issue on Networks in Computational Biology is based on a workshop at Middle East Technical University in Ankara,Turkey,September10–12,2006(.tr/Networks_in_Computational_Biology/). Computational biology is one of the many currently emerging areas of applied mathematics and science.During the last century,cooperation between biology and chemistry,physics,mathematics,and other sciences increased dramatically,thus providing a solid foundation for,and initiating an enormous momentum in,many areas of the life sciences.This special issue focuses on networks,a topic that is equally important in biology and mathematics,and presents snapshots of current theoretical and methodological work in network analysis.Both discrete and continuous optimization,dynamical systems, graph theory,pertinent inverse problems,and data mining procedures are addressed.The principal goal of this special issue is to contribute to the mathematical foundation of computational biology by stressing its particular aspects relating to network theory.This special issue consists of25articles,written by65authors and rigorously reviewed by70referees.The guest editors express their cordial thanks to all of them,as well as to the Editors-in-Chief of Discrete Applied Mathematics,Prof.Dr.Endre Boros and his predecessor,Prof.Dr.Peter L.Hammer,who was one of the initiators of this special issue but left us in2006, and to Mrs.Katie D’Agosta who was at our side in each phase of preparation of this DAM special issue.The articles are ordered according to their contents.Let us briefly summarize them:In the paper of Jacek Błażewicz,Dorota Formanowicz,Piotr Formanowicz,Andrea Sackmann,and MichałSajkowski, entitled Modeling the process of human body iron homeostasis using a variant of timed Petri nets,the standard model of body iron homeostasis is enriched by including the durations of the pertinent biochemical reactions.A Petri-net variant in which, at each node,a time interval is specified is used in order to describe the time lag of the commencement of conditions that must be fulfilled before a biochemical reaction can start.Due to critical changes in the environment,switches can occur in metabolic networks that lead to systems exhibiting simultaneously discrete and continuous dynamics.Hybrid systems represent this accurately.The paper Modeling and simulation of metabolic networks for estimation of biomass-accumulation parameters by Uˇg ur Kaplan,Metin Türkay,Bülent Karasözen,and Lorenz Biegler develops a hybrid system to simulate cell-metabolism dynamics that includes the effects of extra-cellular stresses on metabolic responses.Path-finding approaches to metabolic-pathway analysis adopt a graph-theoretical approach to determine the reactions that an organism might use to transform a source compound into a target compound.In the contribution Path-finding approaches and metabolic pathways,Francisco J.Planes and John E.Beasley examine the effectiveness of using compound-node connectivities in a path-finding approach.An approach to path finding based on integer programming is also presented. Existing literature is reviewed.This paper is well illustrated and provides many examples as well as,as an extra service,some supplementary information.In A new constraint-based description of the steady-state flux cone of metabolic networks,Abdelhalim Larhlimi and Alexander Bockmayr present a new constraint-based approach to metabolic-pathway analysis.Based on sets of non-negativity constraints,it uses a description of the set of all possible flux distributions over a metabolic network at a steady state in terms of the steady-state flux cone.The constraints can be identified with irreversible reactions and,thus,allow a direct interpretation.The resulting description of the flux cone is minimal and unique.Furthermore,it satisfies a simplicity condition similar to the one for elementary flux modes.Most biological networks share some properties like being,e.g.,‘‘scale free’’.Etienne Birmeléproposes a new random-graph model in his contribution A scale-free graph model based on bipartite graphs that can be interpreted in terms of metabolic networks,and exhibits this specific feature.$Dedicated to our dear teacher and friend Prof.Dr.Peter Ladislaw Hammer(1936–2006).0166-218X/$–see front matter©2009Elsevier B.V.All rights reserved.doi:10.1016/j.dam.2009.01.0212218Preface/Discrete Applied Mathematics157(2009)2217–2220Differential equations have been established to quantitatively model the dynamic behaviour of regulatory networks representing interactions between cell components.In the paper Inference of an oscillating model for the yeast cell cycle, Nicole Radde and Lars Kaderali study differential equations within a Bayesian setting.First,an oscillating core network is learned that is to be extended,in a second step,using‘‘Bayesian’’methodology.A specifically designed hierarchical prior distribution over interaction strengths prevents overfitting and drives the solutions to sparse networks.An application to a real-world data set is provided,and its dynamical behaviour is reconstructed.The contribution An introduction to the perplex number system by Jerry L.R.Chandler derives from his approach to theoretical chemistry,and provides a universal source of diagrams.The perplex number system,a new logic for describing relationships between concrete objects and processes,provides in particular an exact notation for chemistry without invoking either chemical or‘‘alchemical’’symbols.Practical applications to concrete compounds(e.g.,isomers of ethanol and dimethyl ether)are given.In conjunction with the real number system,the relations between perplex numbers and scientific theories of concrete systems(e.g.,intermolecular dynamics,molecular biology,and individual medicine)are described.Since exact determination of haplotype blocks is usually impossible,a method is desired which can account for recombinations,especially,via phylogenetic networks or a simplified version.In their work Haplotype inferring via galled-tree networks using a hypergraph-covering problem for special genotype matrices,Arvind Gupta,Ján Maňuch,Ladislav Stacho, and Xiaohong Zhao reduce the problem via galled-tree networks to a hypergraph-covering problem for genotype matrices satisfying a certain combinatorial condition.Experiments on real data show that this condition is mostly satisfied when the minor alleles(per SNP)reach at least30%.Recently the Quartet-Net or,for short,‘‘QNet’’method was introduced by Stefan Grünewald et al.as a method for computing phylogenetic split networks from a collection of weighted quartet trees.Here,Stefan Grünewald,Vincent Moulton,and Andreas Spillner show that QNet is a‘‘consistent’’method.This key property of QNet does not only guarantee to produce a tree if the input corresponds to a tree—and an outer-labeled planar split network if the input corresponds to such a network;the proof given in their contribution Consistency of the QNet algorithm for generating planar split networks from weighted quartets also provides the main guiding principle for the design of the method.Kangal and Akbash dogs are the two well-known shepherd dog breeds in Turkey.In the article The genetic relationship between Kangal,Akbash,and other dog populations,Evren Koban,Çigdem Gökçek Saraç,Sinan Can Açan,Peter Savolainen, andİnci Togan present a comparative examination by mitochondrial DNA control region,using a consensus neighbour-joining tree with bootstrapping which is constructed from pairwise FST values between populations.This study indicates that Kangal and Akbash dogs belong to different branches of the tree,i.e.,they might have descended maternally from rather different origins created by an early branching event in the history of the domestic dogs of Eurasia.In their paper The Asian contribution to the Turkish population with respect to the Balkans:Y-chromosome perspective,Ceren Caner Berkman and inci Togan investigate historical migrations from Asia using computational approaches.The admixture method of Chikhi et al.was used to estimate the male genetic contribution of Central Asia to hybrids.The authors observed that the male contribution from Central Asia to the Turkish population with reference to the Balkans was13%.Comparison of the admixture estimate for Turkey with those of neighboring populations indicated that the Central Asian contribution was lowest in Turkey.Split-decomposition theory deals with relations between real-valued split systems and metrics.In his work Split decomposition over an Abelian group Part2:Group-valued split systems with weakly compatible support,Andreas Dress uses a general conceptual framework to study these relations from an essentially algebraic point of view.He establishes the principal results of split-decomposition theory regarding split systems with weakly compatible support within this new algebraic framework.This study contributes to computational biology by analyzing the conceptual mathematical foundations of a tool widely used in phylogenetic analysis and studies of bio-diversity.The contribution Phylogenetic graph models beyond trees of Ulrik Brandes and Sabine Cornelsen deals with methods for phylogenetic analysis,i.e.,the study of kinship relationships between species.The authors demonstrate that the phylogenetic tree model can be generalized to a cactus(i.e.,a tree all of whose2-connected components are cycles)without losing computational efficiency.A cactus can represent a quadratic rather than a linear number of splits in linear space.They show how to decide in linear time whether a set of splits can be accommodated by a cactus model and,in that case,how to construct it within the same time bounds.Finally,the authors briefly discuss further generalizations of tree models.In their paper Whole-genome prokaryotic clustering based on gene lengths,Alexander Bolshoy and Zeev Volkovich present a novel method of taxonomic analysis constructed on the basis of gene content and lengths of orthologous genes of 66completely sequenced genomes of unicellular organisms.They cluster given input data using an application of the information-bottleneck method for unsupervised clustering.This approach is not a regular distance-based method and, thus,differs from other recently published whole-genome-based clustering techniques.The results correlate well with the standard‘‘tree of life’’.For characterization of prokaryotic genomes we used clustering methods based on mean DNA curvature distributions in coding and noncoding regions.In their article Prokaryote clustering based on DNA curvature distributions,due to the extensive amount of data Limor Kozobay-Avraham,Sergey Hosida,Zeev Volkovich,and Alexander Bolshoy were able to define the external and internal factors influencing the curvature distribution in promoter and terminator regions.Prokaryotes grow in the wide temperature range from4◦C to100◦C.Each type of bacteria has an optimal temperature for growth.They found very strong correlation between arrangements of prokaryotes according to the growth temperature and clustering based on curvature excess in promoter and terminator regions.They found also that the main internal factors influencingPreface/Discrete Applied Mathematics157(2009)2217–22202219 the curvature excess are genome size and A+T composition.Two clustering methods,K-means and PAM,were applied and produced very similar clusterings that reflect the aforementioned genomic attributes and environmental conditions of the species’habitat.The paper Pattern analysis for the prediction of fungal pro-peptide cleavage sites by SüreyyaÖzöˇgür Ayzüz,John Shawe-Taylor,Gerhard-Wilhelm Weber,and Zümrüt B.Ögel applies support-vector machines to predict the pro-peptide cleavage site of fungal extra-cellular proteins displaying mostly a monobasic or dibasic processing site.A specific kernel is expressed as an application of the Gaussian kernel via feature spaces.The novel approach simultaneously performs model selection, tests the accuracy,and computes confidence levels.The results are found to be accurate and compared with the ones provided by a server.Preetam Ghosh,Samik Ghosh,Kalyan Basu,and Sajal Das adopt an‘‘in silico’’stochastic-event-based simulation methodology to determine the temporal dynamics of different molecules.In their paper Parametric modeling of protein–DNA binding kinetics:A discrete event-based simulation approach,they present a parametric model for predicting the execution time of protein–DNA binding.It considers the actual binding mechanism along with some approximated protein-and DNA-structural information using a collision-theory-based approach incorporating important biological parameters and functions into the consideration.Murat Ali Bayır,Tacettin Doˇg acan Güney,and Tolga Can propose a novel technique in their paper Integration of topological measures for eliminating non-specific interactions in protein interaction networks for removing non-specific interactions in a large-scale protein–protein interaction network.After transforming the interaction network into a line graph,they compute betweenness and other clustering coefficients for all the edges in the network.The authors use confidence estimates and validate their method by comparing the results of a test case relating to the detection of a molecular complex with reality.The article Graph spectra as a systematic tool in computational biology by Anirban Banarjee and Jürgen Jost deals with the obviously important question of how biological content can be extracted from the graphs to which biological data are often reduced.From the spectrum of the graph’s Laplacian that yields an essentially complete qualitative characterization of a graph,a spectral density plot is derived that can easily be represented graphically and,therefore,analyzed visually and compared for different classes of networks.The authors apply this method to the study of protein–protein interaction and other biological and infrastructural networks.It is detected that specific such classes of networks exhibit common features in their spectral plots that readily distinguish them from other classes.This represents a valuable complement to the currently fashionable search for universal properties that hold across networks emanating from many different contexts.Konstantin Klemm and Peter F.Stadler’s Note on fundamental,nonfundamental,and robust cycle bases investigates the mutual relationships between various classes of cycle bases in a network that have been studied in the literature.The authors show for instance that strictly fundamental bases are not necessarily cyclically robust;and that,conversely, cyclically robust bases are not necessarily fundamental.The contribution focuses on cyclically robust cycle bases whose existence for arbitrary graphs remains open despite their practical use for generating all cycles of a given2-connected graph. It presents also a class of cubic graphs for which cyclically robust bases can be constructed explicitly.Understanding the interplay and function of a system’s components also requires the study of the system’s functional response to controlled experimental perturbations.For biological systems,it is problematic with an experimental design to aim at a complete identification of the system’s mechanisms.In his contribution A refinement of the common-cause principle,Nihat Ay employs graph theory and studies the interplay between stochastic dependence and causal relations within Bayesian networks and information theory.Applying a causal information-flow measure,he provides a quantitative refinement of Reichenbach’s common-cause principle.Based on observing an appropriate collection of nodes of the network, this refinement allows one to infer a hitherto unknown lower bound for information flows within the network.In their article Discovering cis-regulatory modules by optimizing barbecues,Axel Mosig,Türker Bıyıkoˇg lu,Sonja J.Prohaska, and Peter F.Stadler ask for simultaneously stabbing a maximum number of differently coloured intervals from K arrangements of coloured intervals.A decision version of this best barbecue problem is shown to be NP-complete.Because of the relevance for complex regulatory networks on gene expression in eukaryotic cells,they propose algorithmic variations that are suitable for the analysis of real data sets comprising either many sequences or many binding sites.The optimization problem studied generalizes frequent itemset mining.The contribution A mathematical program to refine gene regulatory networks by Guglielmo Lulli and Martin Romauch proposes a methodology for making sense of large,multiple time-series data sets arising in expression analysis.It introduces a mathematical model for producing a reduced and coherent regulatory system,provided a putative regulatory network is given.Two equivalent formulations of the problem are given,and NP-completeness is established.For solving large-scale instances,the authors implemented an ant-colony optimization procedure.The proposed algorithm is validated by a computational analysis on randomly generated test instances.The practicability of the proposed methodology is also shown using real data for Saccharomyces cerevisiae.Jutta Gebert,Nicole Radde,Ulrich Faigle,Julia Strösser,and Andreas Burkovski aim in their paper Modelling and simulation of nitrogen regulation in Corynebacterium glutamicum at understanding and predicting the interactions of macromolecules inside the cell.It sets up a theoretical model for biochemical networks,and introduces a general method for parameter estimation,applicable in the case of very short time series.This approach is applied to a special system concerning nitrogen uptake.The equations are set up for its main components,the corresponding optimization problem is formulated and solved, and simulations are carried out.2220Preface/Discrete Applied Mathematics157(2009)2217–2220Gerhard-Wilhelm Weber,Ömür Uˇg ur,Pakize Taylan,and Aysun Tezel model and predict gene-expression patterns incorporating a rigorous treatment of environmental aspects,and aspects of errors and uncertainty.For this purpose,they employ Chebyshev approximation and generalized semi-infinite optimization in their paper On optimization,dynamics and uncertainty:A tutorial for gene–environment networks.Then,time-discretized dynamical systems are studied,the region of parametric stability is detected by a combinatorial algorithm and,then,the topological landscape of gene–environment networks is analyzed in terms of its‘‘structural stability’’.We are convinced that all papers selected for this special issue constitute valuable contributions to many different areas in computational biology,employing methods from discrete mathematics and related fields.We again thank all colleagues who have participated in this exciting endeavor with care,foresight,and vision,for their highly appreciated help.Guest editorsAndreas DressBülent KarasözenPeter F.StadlerGerhard-Wilhelm Weber125July2008Available online29March2009 1Assistant to the guest editors:Mrs.Cand.MSc.Bengisen Pekmen(Institute of Applied Mathematics,METU,Ankara).。

2017年第36卷第5期 CHEMICAL INDUSTRY AND ENGINEERING PROGRESS·1581·化 工 进展基于复杂网络理论的大型换热网络节点重要性评价王政1，孙锦程1，刘晓强1，姜英1，贾小平2，王芳2（1青岛科技大学化工学院，山东 青岛 266042；2青岛科技大学环境与安全工程学院，山东 青岛 266042） 摘要：鉴于换热网络大型化和流股间复杂关系，使得换热网络换热器节点重要性的研究显得越来越重要，对其控制和安全运行的工程实践方面具有指导意义。

本文以大型换热网络为研究对象，将换热器抽象为节点，换热器之间的干扰传递抽象为边，构造网络拓扑结构。

在复杂网络理论的基础上，提出了评价大型换热网络节点重要性的策略和模型。

首先，从网络的点度中心性、中间中心性、接近中心性和特征向量中心性等网络拓扑结构属性出发，依据多属性决策方法对网络节点重要性进行综合评价；其次，考虑换热网络的方向性，基于PageRank 算法对该网络进行节点重要性评价研究。

综合两个算法的计算结果得出最终结论。

案例分析表明：该研究方法是有效的，可从不同的角度全面评价换热网络的节点重要性，丰富了换热器节点重要性评价的相关理论。

关键词：换热网络；复杂网络；节点重要性；多属性决策；PageRank 算法中图分类号：X92 文献标志码：A 文章编号：1000–6613（2017）05–1581–08 DOI ：10.16085/j.issn.1000-6613.2017.05.004Evaluation of the node importance for large heat exchanger networkbased on complex network theoryWANG Zheng 1，SUN Jincheng 1，LIU Xiaoqiang 1，JIANG Ying 1，JIA Xiaoping 2，WANG Fang 2（1College of Chemical Engineering ，Qingdao University of Science and Technology ，Qingdao 266042，Shandong ，China ；2College of Environment and Safety Engineering ，Qingdao University of Science and Technology ，Qingdao266042，Shandong ，China ）Abstract ：Because of the complexity of large-scale heat exchanger network ，it is important to investigate the importance of heat exchanger nodes in heat exchanger network. It can provide guidance for the control and safe operation of heat exchanger networks ，as well as engineering practices. In this paper ，the network topology structure of large-scale heat exchanger network was constructed by treating heat exchangers as nodes and treating the transfer of interference between heat exchangers as edges. Based on the complex network theory ，the strategies and models for evaluating the node importance of the heat exchanger network were proposed. Firstly ，the importance of nodes were evaluated by the multi-attribute decision method based on the degree centrality, betweenness ，closeness and eigenvector centralities. Next ，considering the direction of case heat exchanger network ，PageRank algorithm was used to evaluate the importance of nodes. Considering the results from these two algorithms ，the final results were obtained. The case analysis showed that the strategy is effective and it can evaluate the node importance from different views ，which will enrich the node importance evaluation theory for heat exchanger network.Key words ：heat exchanger network ；complex network ；node importance ；multi-attribute decision ；PageRank algorithm第一作者及联系人：王政（1968—），男，博士，副教授，硕士生导师，主要研究过程系统工程。

RFS Advance Access published August 27, 2007 Informed and Strategic Order Flow in theBond MarketsPaolo PasquarielloRoss School of Business,University of MichiganClara VegaSimon School of Business,University of Rochester and FRBG We study the role played by private and public information in the process of price formation in the U.S.Treasury bond market.To guide our analysis,we develop a parsimonious model of speculative trading in the presence of two realistic market frictions—information heterogeneity and imperfect competition among informed traders—and a public signal.We test its equilibrium implications by analyzing the response of two-year,five-year,and ten-year U.S.bond yields to orderflow and real-time U.S.macroeconomic news.Wefind strong evidence of informational effects in the U.S.Treasury bond market:unanticipated orderflow has a significant and permanent impact on daily bond yield changes during both announcement and nonannouncement days.Our analysis further shows that,consistent with our stylized model,the contemporaneous correlation between orderflow and yield changes is higher when the dispersion of beliefs among market participants is high and public announcements are noisy.(JEL E44;G14)Identifying the causes of daily asset price movements remains a puzzling issue infinance.In a frictionless market,asset prices should immediately adjust to public news surprises.Hence,we should observe price jumps only during announcement times.However,asset pricesfluctuate significantly during nonannouncement days as well.This fact has motivated the introduction of various market frictions to better explain the behavior The authors are affiliated with the Department of Finance at the Ross School of Business,University of Michigan(Pasquariello)and the University of Rochester,Simon School of Business and the Federal Reserve Board of Governors(Vega).We are grateful to the Q Group forfinancial support.We benefited from the comments of an anonymous referee,Sreedhar Bharath,Michael Brandt,Michael Fleming, Michael Goldstein,Clifton Green,Joel Hasbrouck(the editor),Nejat Seyhun,Guojun Wu,Kathy Yuan, and other participants in seminars at the2005European Finance Association meetings in Moscow, the2006Bank of Canada Fixed Income Markets conference in Ottawa,the2007American Finance Association meetings in Chicago,Federal Reserve Board of Governors,George Washington University, the University of Maryland,the University of Michigan,the University of Rochester,and the University of Utah.Any remaining errors are our own.Address correspondence to Paolo Pasquariello,Ross School of Business,University of Michigan,701Tappan Street,Room E7602,Ann Arbor,MI48109-1234,or e-mail:ppasquar@ and Clara.Vega@.©The Author2007.Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved.For Permissions,please email:journals.permissions@.doi:10.1093/rfs/hhm034The Review of Financial Studies/v20n52007of asset prices.One possible friction is asymmetric information.1When sophisticated agents trade,their private information is(partially)revealed to the market,via orderflow,causing revisions in asset prices even in the absence of public announcements.The goal of this article is to theoretically identify and empirically measure the effect of these two complementary mechanisms responsible for daily price changes:aggregation of public news and aggregation of orderflow. In particular,we assess the relevance of each mechanism conditional on the dispersion of beliefs among traders and the public signals’noise.To guide our analysis,we develop a parsimonious model of speculative trading in the spirit of Kyle(1985).The model builds upon two realistic market frictions:information heterogeneity and imperfect competition among informed traders(henceforth,speculators).In this setting,more diverse information among speculators leads to lower equilibrium market liquidity,since their trading activity is more cautious than if they were homogeneously informed,thus making the market makers(MMs)more vulnerable to adverse selection.We then introduce a public signal and derive equilibrium prices and trading strategies on announcement and nonannouncement days.The contribution of the model is two-fold.To our knowledge,it provides a novel theoretical analysis of the relationship between the trading activity of heterogeneously informed,imperfectly competitive speculators,the availability and quality of public information, and market liquidity.Furthermore,its analytically tractable closed-form solution,in terms of elementary functions,generates several explicit and empirically testable implications on the nature of that relationship.2In particular,we show that the availability of a public signal improves market liquidity(the more so the lower that signal’s volatility)since its presence reduces the adverse selection risk for the MMs and mitigates the quasimonopolistic behavior of the speculators.1According to Goodhart and O’Hara(1997,p.102),‘‘one puzzle in the study of asset markets,either nationally or internationally,is that so little of the movements in such markets can be ascribed to identified public‘news.’In domestic(equity)markets thisfinding is often attributed to private information being revealed.’’This friction has been recently studied by Fleming and Remolona(1997,1999),Brandt and Kavajecz(2004)and Green(2004)in the US Treasury bond market,by Andersen and Bollerslev(1998) and Evans and Lyons(2002,2003,2004)in the foreign exchange market,by Berry and Howe(1994)in the US stock market,and by Brenner et al.(2005)in the US corporate bond market,among others.2Foster and Viswanathan(1996)and Back et al.(2000)extend Kyle(1985)to analyze the impact of competition among heterogeneously informed traders on market liquidity and price volatility in discrete-time and continuous-time models of intraday trading,respectively.Foster and Viswanathan(1993)show that,when the beliefs of perfectly informed traders are represented by elliptically contoured distributions, price volatility and trading volume depend on the surprise component of public information.Yet,neither model’s equilibrium is in closed-form,except the(analytically intractable)inverse incomplete gamma function in Back et al.(2000).Hence,their implications are sensitive to the chosen calibration parameters. Further,neither model,by its dynamic nature,generates unambiguous comparative statics for the impact of information heterogeneity or the availability of public information on market liquidity.Finally,neither model can be easily generalized to allow for both a public signal of the traded asset’s payoff and less than perfectly correlated private information.2Informed and Strategic Order Flow in the Bond MarketsThis model is not asset-specific,that is,it applies to stock,bond,and foreign exchange markets.In this study,we test its implications for the US government bond market for three reasons.First,Treasury market data contains signed trades;thus,we do not need to rely on algorithms[e.g.Lee and Ready(1991)]that add measurement error to our estimates of order flow.Second,government bond markets represent the simplest trading environment to analyze price changes while avoiding omitted variable biases.For example,most theories predict an unambiguous link between macroeconomic fundamentals and bond yield changes,with unexpected increases in real activity and inflation raising bond yields[e.g.Fleming and Remolona(1997)and Balduzzi et al.(2001),among others].In contrast, the link between macroeconomic fundamentals and the stock market is less clear[e.g.Andersen et al.(2004)and Boyd et al.(2005)].Third,the market for Treasury securities is interesting in itself since it is among the largest,most liquid USfinancial markets.Our empirical results strongly support the main implications of our model.During nonannouncement days,adverse selection costs of unanticipated orderflow are higher when the dispersion of beliefs—measured by the standard deviation of professional forecasts of macroeconomic news releases—is high.For instance,we estimate that a one standard deviation shock to abnormal orderflow decreases two-year,five-year,and ten-year bond yields by7.19,10.04,and6.84basis points, respectively,on high dispersion days compared to4.08,4.07,and2.86 basis points on low dispersion days.These differences are economically and statistically significant.Consistently,these higher adverse selection costs translate into higher contemporaneous correlation between order flow changes and bond yield changes.For example,the adjusted R2of regressing dailyfive-year Treasury bond yield changes on unanticipated orderflow is41.38%on high dispersion days compared to9.65%on low dispersion days.Intuitively,when information heterogeneity is high,the speculators’quasimonopolistic trading behavior leads to a‘‘cautious’’equilibrium where changes in unanticipated orderflow have a greater impact on bond yields.The release of a public signal,a trade-free source of information about fundamentals,induces the speculators to trade more aggressively on their private information.Accordingly,wefind that the correlation between unanticipated orderflow and day-to-day bond yield changes is lower during announcement days.For example,comparing nonannouncement days with Nonfarm Payroll Employment release dates,the explanatory power of orderflow decreases from15.31%to6.47%,21.03%to19.61%,and6.74% to3.59%for the two-year,five-year,and ten-year bonds,respectively.Yet, when both the dispersion of beliefs and the noise of the public signal —measured as the absolute difference between the actual announcement and its last revision—are high,the importance of orderflow in setting3The Review of Financial Studies/v20n52007bond prices increases.All of the above results are robust to alternative measures of the dispersion of beliefs among market participants,as well as to different regression specifications and the inclusion of different control stly,our evidence cannot be attributed to transient inventory or portfolio rebalancing considerations,since the unanticipated government bond orderflow has a permanent impact on yield changes during both announcement and nonannouncement days in the sample.Our article is most closely related to two recent studies of orderflow in the U.S.Treasury market.Brandt and Kavajecz(2004)find that orderflow accounts for up to26%of the variation in yields on days without major macroeconomic announcements.Green(2004)examines the effect of order flow on intraday bond price changes surrounding U.S.macroeconomic news announcements.We extend both studies by identifying a theoretical and empirical link between the price discovery role of orderflow and the degree of information heterogeneity among investors and the quality of macroeconomic data releases.In particular,we document important effects of both dispersion of beliefs and public signal noise on the correlation between daily bond yield changes and orderflow during announcement and nonannouncement days.This evidence complements the weak effects reported by Green(2004)over30-minute intervals around news releases. Since the econometrician does not observe the precise arrival time of private information signals,narrowing the estimation window may lead to underestimating the effect of dispersion of beliefs on market liquidity.3 Our work also belongs to the literature bridging the gap between asset pricing and market microstructure.Evans and Lyons(2003)find that signed orderflow is a good predictor of subsequent exchange rate movements;Brandt and Kavajecz(2004)show that this is true for bond market movements;and Easley et al.(2002)argue that the probability of informed trading(PIN),a function of orderflow,is a pricedfirm characteristic in stock returns.These studies enhance our understanding of the determinants of asset price movements,but do not provide any evidence on the determinants of orderflow.Evans and Lyons(2004)address this issue by showing that foreign exchange order flow predicts future macroeconomic surprises(i.e.it conveys information about fundamentals).We go a step further in linking the impact of order flow on bond prices to macroeconomic uncertainty(public signal noise) and the heterogeneity of beliefs about real shocks.We proceed as follows.In Section1,we construct a stylized model of trading to guide our empirical analysis.In Section2,we describe the data. In Section3,we present the empirical results.We conclude in Section4. 3For instance,heterogeneously informed investors may not trade immediately after public news releases but instead wait to preserve(and exploit)their informational advantage as long(and as much)as possible, as in Foster and Viswanathan(1996)4Informed and Strategic Order Flow in the Bond Markets1.Theoretical ModelIn this section we motivate our investigation of the impact of the disper-sion of beliefs among sophisticated market participants and the release of macroeconomic news on the informational role of trading.We first describe a one-shot version of the multiperiod model of trading of Foster and Viswanathan (1996)and derive closed-form solutions for the equi-librium market depth and trading volume.Then,we enrich the model by introducing a public signal and consider its implications for the equilib-rium price and trading strategies.All proofs are in the Appendix unless otherwise noted.1.1Benchmark:no public signalThe basic model is a two-date,one-period economy in which a single risky asset is exchanged.Trading occurs only at the end of the period (t =1),after which the asset payoff,a normally distributed random variable v with mean zero and variance σ2v ,is realized.The economy is populated by three types of risk-neutral traders:a discrete number (M )of informed traders (that we label speculators),liquidity traders,and perfectly competitive MMs.All traders know the structure of the economy and the decision process leading to order flow and prices.At time t =0there is neither information asymmetry about v nor trading.Sometime between t =0and t =1,each speculator k receives a private and noisy signal of v ,S vk .We assume that the resulting signal vector S v is drawn from a multivariate normal distribution (MND)with mean zero and covariance matrix s such that var (S vk )=σ2s and cov S vk ,S vj =σss .We also impose that the speculators together know the liquidation value of the risky asset: M k =1S vk =v ;therefore,cov (v,S vk )=σ2v M .This specification makes the total amount of information available to the speculators independent from the correlation of theirprivate signals,albeit still implying the most general information structure up to rescaling by a constant [see Foster and Viswanathan (1996)].These assumptions imply that δk ≡E (v |S vk )=σ2v Mσ2s S vk and E δj |δk =γδk ,where γ=σss σ2s is the correlation between any two private information endowments δk and δj .As in Foster and Viswanathan (1996),we parametrize the degree of diversity among speculators’signals by requiring that σ2s −σss =χ≥0.This restriction ensures that s is positive definite.If χ=0,then speculators’private information is homogeneous :Allspeculators receive the same signal S vk =v M such that σ2s =σss =σ2v M 2and γ=1.If χ=σ2v M ,then speculators’information is heterogeneous :σ2s =χ,σss =0,and γ=0.Otherwise,speculators’signals are only 5The Review of Financial Studies /v 20n 52007partially correlated:Indeed,γ∈(0,1)if χ∈ 0,σ2v M and γ∈ −1M −1,0 if χ>σ2v M .4At time t =1,both speculators and liquidity traders submit their orders to the MMs,before the equilibrium price p 1has been set.We define the market order of the k -th speculator to be x k .Thus,that speculator’s profit is given by πk (x k ,p 1)=(v −p 1)x k .Liquidity traders generate a random,normally distributed demand u ,with mean zero and variance σ2u .For simplicity,we assume that u is independent from all other random variables.MMs do not receive any information,but observe the aggregate order flow ω1= M k =1x k +u from all market participants and set the market-clearing price p 1=p 1(ω1).1.1.1Equilibrium.Consistently with Kyle (1985),we define a Bayesian Nash equilibrium as a set of M +1functions x 1(·),...,x M (·),and p 1(·)such that the following two conditions hold:1.Profit maximization :x k (S vk )=arg max E (πk |S vk );and2.Semistrong market efficiency :p 1(ω1)=E (v |ω1).We restrict our attention to linear equilibria.We first conjecture general linear functions for the pricing rule and speculators’demands.We then solve for their parameters satisfying conditions 1and 2.Finally,we show that these parameters and those functions represent a rational expectations equilibrium.The following proposition accomplishes this task.Proposition 1.There exists a unique linear equilibrium given by the price functionp 1=λω1(1)and by the k -th speculator’s demand strategyx k =λ−12+(M −1)γδk ,(2)where λ=σ2vσu σs √M [2+(M −1)γ]>0.The optimal trading strategy of each speculator depends on the information received about the asset payoff (v )and on the depth of the 4The assumption that the total amount of information available to speculators is fixed ( Mk =1S vk =v )implies that σ2s =σ2v +M(M −1)χM 2and σss =σ2v −MχM 2,hence γ=σ2v −Mχσ2v +M(M −1)χ.Further,the absolute bound to the largest negative private signal correlation γcompatible with a positive definite s , −1M −1 ,is a decreasing function of M .6Informed and Strategic Order Flow in the Bond Marketsmarket (λ−1).If M =1,Equations (1)and (2)reduce to the well-known equilibrium of Kyle (1985).The speculators,albeit risk-neutral,exploit their private information cautiously (|x k |<∞),to avoid dissipating their informational advantage with their trades.Thus,the equilibrium market liquidity in p 1reflects MMs’attempt to be compensated for the losses they anticipate from trading with speculators,as it affects their profits from liquidity trading.1.1.2Testable implications.The intuition behind the parsimonious equilibrium of Equations (1)and (2)is similar to that in the multiperiod models of Foster and Viswanathan (1996)and Back et al.(2000).Yet,its closed-form solution (in Proposition 1)translates that intuition into unambiguous predictions on the impact of information heterogeneity on market depth.5The optimal market orders x k depend on the number of speculators (M )and the correlation among their information endowments (γ).The intensity of competition among speculators affects their ability to maintain the informativeness of the order flow as low as possible.A greater number of speculators trade more aggressively —that is,their aggregate amount of trading is higher—since (imperfect)competition among them precludes any collusive trading strategy.For instance,when M >1speculators are homogeneously informed (γ=1),then x k =σu σv √M v ,which implies that the finite difference Mx k =(M +1)x k −Mx k =σu σv √M +1−√M v >0.This behavior reduces the adverse selection problem for the MMs,thus leading to greater market liquidity (lower λ).The heterogeneity of speculators’signals attenuates their trading aggressiveness.When information is less correlated (γcloser to zero),each speculator has some monopolistic power on the private signal,because at least part of the information is exclusively known.Hence,as a group,they trade more cautiously—that is,their aggregate amount of trading is lower—to reveal less of their own information endowments δk .For example,when M >1speculators are heterogeneously informed (γ=0),then x k =σu σv S vk ,which implies that M k =1x k =σu σv v <σu M σv √M v ,that is,lower than the aggregate amount of trading by M >1homogeneously informed speculators (γ=1)but identical to the trade of a monopolistic speculator (M =1).This ‘‘quasimonopolistic’’behavior makes the MMs more vulnerable to adverse selection,thus the market less liquid (higher λ).The following corollary summarizes the first set of empirical implications of our model.5This contrasts with the numerical examples of the dynamics of market depth reported in Foster and Viswanathan (1996,Figure 1C)and Back et al.(2000,Figure 3A).7The Review of Financial Studies /v 20n 52007Corollary 1.Equilibrium market liquidity is increasing in the number of speculators and decreasing in the heterogeneity of their information endowments.To gain further insight on this result,we construct a simple numerical example by setting σv =σu =1.We then vary the parameter χto study the liquidity of this market with respect to a broad range of signal correlations γ(from very highly negative to very highly positive)when M =1,2,and4.By construction,both the private signals’variance (σ2s )and covariance (σss )change with χand M ,yet the total amount of information available to the speculators is unchanged.We plot the resulting λin Figure 1.Multiple,perfectly heterogeneously informed speculators (γ=0)collectively trade as cautiously as a monopolist speculator.Under these circumstances,adverse selection is at its highest,and market liquidity atits lowest (λ=σv 2σu ).A greater number of competing speculators improvesmarket depth,but significantly so only if accompanied by more correlated private signals.However,ceteris paribus ,the improvement in market liquidity is more pronounced (and informed trading less cautious)when speculators’private signals are negatively correlated.When γ<0,each speculator expects her competitors’trades to be negatively correlated toFigure 1Equilibrium without a public signalIn this figure we plot the market liquidity parameter defined in Proposition 1,λ=σ2v σu σs √M [2+(M −1)γ],as a function of the degree of correlation of the speculators’signals,γ,in the presence of M =1,2,or 4speculators,when σ2v =σ2u =1.Since σ2s =σ2v +M(M −1)χM 2,σss =σ2v −MχM 2,and γ=σ2v −Mχσ2v +M(M −1)χ,the range of correlations compatible with a positive definite s is obtained by varying the parameterχ=σ2s −σss within the interval [0,10]when M =2,and the interval [0,5]when M =4.8Informed and Strategic Order Flow in the Bond Marketsher own (pushing p 1against her signal),hence trading on it to be more profitable.1.2Extension:a public signalWe now extend the basic model of Section 1.1by providing each player with an additional,common source of information about the risky asset before trading takes place.According to Kim and Verrecchia (1994,p.43),‘‘public disclosure has received little explicit attention in theoretical models whose major focus is understanding market liquidity.’’6More specifically,we assume that,sometime between t =0and t =1,both the speculators and the MMs also observe a public and noisy signal,S p ,of the asset payoffv .This signal is normally distributed with mean zero and variance σ2p >σ2v .We can think of S p as any surprise public announcement (e.g.macroeco-nomic news)released simultaneouslyto all market participants.We further impose that cov S p ,v =σ2v ,so that the parameter σ2p controls for the quality of the public signal and cov S p ,S vk =σ2v M .The private informa-tion endowment of each speculator is then given by δ∗k≡E v |S vk ,S p −E v |S p =αS vk +βS p ,where α=Mσ2v σ2p −σ2v σ2p [σ2v +M(M −1)χ]−σ4v >0and β=−σ4v σ2p −σ2v σ2p {[σ2v +M(M −1)χ]−σ4v}<0.Thus,E δ∗j |δ∗k =E δ∗j |S vk ,S p =γp δ∗k ,where γp =Mα2σss +2αβσ2v +Mβ2σ2pMα2σ2s +2αβσ2v +Mβ2σ2p ≤γ.1.2.1Equilibrium.Again we search for linear equilibria.The following proposition summarizes our results.Proposition 2.There exists a unique linear equilibrium given by the price functionp 1=λp ω1+λs S p(3)and by the k -th speculator’s demand strategyx k =λ−1p 2+(M −1)γp δ∗k ,(4)6Admati and Pfleiderer (1988)and Foster and Viswanathan (1990)consider dynamic models of intraday trading in which the private information of either perfectly competitive insiders or a monopolistic insider is either fully or partially revealed by the end of the trading period.McKelvey and Page (1990)provide experimental evidence that individuals make inferences from publicly available information using Bayesian updating.Diamond and Verrecchia (1991)argue that the disclosure of public information may reduce the volatility of the order flow,leading some MMs to exit.Kim and Verrecchia (1994)show that,in the absence of better informed agents but in the presence of better information processors with homogeneous priors,the arrival of a public signal leads to greater information asymmetry and lower market liquidity.9The Review of Financial Studies /v 20n 52007where λp =α12σv σ2p −σ2v 12σu σp [2+(M −1)γp ]>0and λs =σ2v σ2p.The optimal trading strategy of each speculator in Equation (4)mirrors that of Proposition 1[Equation (2)],yet it now depends only on δ∗k ,the truly private—hence less correlated (γp ≤γ)—component of speculator k ’s original private signal (S vk )in the presence of a public signal of v .Hence,the MMs’belief update about v stemming from S p makes speculators’private information less valuable.The resulting equilibrium price p 1in Equation (3)can be rewritten as:p 1=α2+(M −1)γp v +λp u + λs +Mβ2+(M −1)γp S p .(5)According to Equation (5),the public signal impacts p 1through two channels that [in the spirit of Evans and Lyons (2003)]we call direct ,related to MMs’belief updating process (λs >0),and indirect ,via the speculators’trading activity (β<0).Since λs 2+(M −1)γp >−Mβ>0,the former always dominates the latter.Therefore,public news always enter the equilibrium price with the ‘‘right’’sign.1.2.2Additional testable implications.Foster and Viswanathan (1993)generalize the trading model of Kyle (1985)to distributions of the elliptically contoured class (ECC)and show that,in the presence of a discrete number of identically informed traders,the unexpected realization of a public signal has no impact on market liquidity regardless of the ECC used.This is the case for the equilibrium of Proposition 2as well.7Nonetheless,Proposition 2allows us to study the impact of the availability of noisy public information on equilibrium market depth in the presence of imperfectly competitive and heterogeneously informed speculators.To our knowledge,this analysis is novel to the financial literature.We start with the following result.Corollary 2.The availability of a public signal of v increases equilibrium market liquidity.The availability of the public signal S p reduces the adverse selection risk for the MMs,thus increasing the depth of this stylized market,for two reasons.First,the public signal represents an additional,trade-free source of information about v .Second,speculators have to trade more aggressively to extract rents from their private information.In Figure 27Specifically,it can be shown that the one-shot equilibrium in Foster and Viswanathan (1993,Proposition1)is a special case of our Proposition 2when private signal correlations γ=1for any ECC.10Informed and Strategic Order Flow in the Bond MarketsFigure 2Equilibrium with a public signalIn this figure we plot the difference between the sensitivity of the equilibrium price to the order flow in the absence and in the presence of a public signal S p ,λ−λp ,as a function of the degree of correlation ofthe speculators’signals,γ,in the presence of M =1,2,or 4speculators,when σ2v =σ2u =1.Accordingto Proposition 1,λ=σ2v σu σs √M [2+(M −1)γ],while λp =α12σv σ2p −σ2v 12σu σp 2+(M −1)γpin Proposition 2,where γp =σ2p σ2v −Mχ −σ4v σ2p σ2v +M(M −1)χ −σ4v and α=Mσ2v σ2p −σ2v σ2p σ2v +M(M −1)χ −σ4v .Since σ2s =σ2v +M(M −1)χM 2,σss =σ2v −MχM 2,and γ=σ2v−Mχσ2v +M(M −1)χ,the range of correlations compatible with a positive definite s is obtained by varying the parameter χ=σ2s −σss within the interval [0,10]when M =2,and the interval [0,5]whenM =4.we plot the ensuing gain in liquidity,λ−λp ,as a function of private signalcorrelations γwhen the public signal’s noise σp =1.25,that is,by varying χand M (so σ2s and σss as well,but not the total amount of information available to the speculators)as in Figure 1.The increase in market depth is greater when γis negative and the number of speculators (M )is high.In those circumstances,the availability of a public signal reinforces speculators’existing incentives to place market orders on their private signals,S vk ,more aggressively.However,greater σ2p ,ceteris paribus ,increases λp ,since the poorer quality of S p (lower information-to-noise ratioσ2vσ2p)induces the MMs to rely more heavilyon ω1to set market-clearing prices,hence the speculators to trade less aggressively.Remark 1.(The increase in)market liquidity is decreasing in the volatility of the public signal.11。

Spread of epidemic disease on networksM.E.J.NewmanCenter for the Study of Complex Systems,University of Michigan,Ann Arbor,Michigan48109-1120 Santa Fe Institute,1399Hyde Park Road,Santa Fe,New Mexico87501͑Received4December2001;published26July2002͒The study of social networks,and in particular the spread of disease on networks,has attracted considerable recent attention in the physics community.In this paper,we show that a large class of standard epidemiological models,the so-called susceptible/infective/removed͑SIR͒models can be solved exactly on a wide variety of networks.In addition to the standard but unrealistic case offixed infectiveness time andfixed and uncorrelated probability of transmission between all pairs of individuals,we solve cases in which times and probabilities are nonuniform and correlated.We also consider one simple case of an epidemic in a structured population,that of a sexually transmitted disease in a population divided into men and women.We confirm the correctness of our exact solutions with numerical simulations of SIR epidemics on networks.DOI:10.1103/PhysRevE.66.016128PACS number͑s͒:89.75.Hc,87.23.Ge,05.70.Fh,64.60.AkI.INTRODUCTIONMany diseases spread through human populations by con-tact between infective individuals͑those carrying the dis-ease͒and susceptible individuals͑those who do not have the disease yet,but can catch it͒.The pattern of these disease-causing contacts forms a network.In this paper we investi-gate the effect of network topology on the rate and pattern of disease spread.Most mathematical studies of disease propagation make the assumption that populations are‘‘fully mixed,’’meaning that an infective individual is equally likely to spread the disease to any other member of the population or subpopu-lation to which they belong͓1–3͔.In the limit of large popu-lation size this assumption allows one to write down nonlin-ear differential equations governing,for example,numbers of infective individuals as a function of time,from which solutions for quantities of interest can be derived,such as typical sizes of outbreaks and whether or not epidemics oc-cur.͑Epidemics are defined as outbreaks that affect a non-zero fraction of the population in the limit of large system size.͒Epidemic behavior usually shows a phase transition with the parameters of the model—a sudden transition from a regime without epidemics to one with.This transition hap-pens as the‘‘reproductive ratio’’R0of the disease,which is the fractional increase per unit time in the number of infec-tive individuals,passes though one.Within the class of fully mixed models much elaboration is possible,particularly concerning the effects of age struc-ture in the population,and population turnover.The crucial element however that all such models lack is network topol-ogy.It is obvious that a given infective individual does not have equal probability of infecting all others;in the real world each individual only has contact with a small fraction of the total population,although the number of contacts that people have can vary greatly from one person to another.The fully mixed approximation is made primarily in order to al-low the modeler to write down differential equations.For most diseases it is not an accurate representation of real con-tact patterns.In recent years a large body of research,particularly within the statistical physics community,has addressed thetopological properties of networks of various kinds,fromboth theoretical and empirical points of view,and studied theeffects of topology on processes taking place on those net-works͓4,5͔.Social networks͓6–9͔,technological networks ͓10–13͔,and biological networks͓14–18͔have all been ex-amined and modeled in some detail.Building on insightsgained from this work,a number of authors have pursued amathematical theory of the spread of disease on networks ͓19–24͔.This is also the topic of the present paper,in which we show that a large class of standard epidemiological mod-els can be solved exactly on networks using ideas drawn from percolation theory.The outline of the paper is as follows.In Sec.II we intro-duce the models studied.In Sec.III we show how percola-tion ideas and generating function methods can be used toprovide exact solutions of these models on simple networkswith uncorrelated transmission probabilities.In Sec.IV weextend these solutions to cases in which probabilities oftransmission are correlated,and in Sec.V to networks repre-senting some types of structured populations.In Sec.VI wegive our conclusions.II.EPIDEMIC MODELS AND PERCOLATION The mostly widely studied class of epidemic models,and the one on which we focus in this paper,is the class of susceptible/infective/removed or SIR models.The original and simplest SIR model,first formulated͑though never pub-lished͒by Lowell Reed and Wade Hampton Frost in the 1920s,is as follows.A population of N individuals is divided into three states:susceptible͑S͒,infective͑I͒,and removed ͑R͒.In this context‘‘removed’’means individuals who are either recovered from the disease and immune to further in-fection,or dead.͑Some researchers consider the R to stand for‘‘recovered’’or‘‘refractory.’’Either way,the meaning is the same.͒Infective individuals have contacts with randomly chosen individuals of all states at an average rate␤per unit time,and recover and acquire immunity͑or die͒at an aver-age rate␥per unit time.If those with whom infective indi-viduals have contact are themselves in the susceptible state,PHYSICAL REVIEW E66,016128͑2002͒then they become infected.In the limit of large N this model is governed by the coupled nonlinear differential equations ͓1͔:ds dt ϭϪ␤is,didtϭ␤isϪ␥i,drdtϭ␥i,͑1͒where s(t),i(t),and r(t)are the fractions of the population in each of the three states,and the last equation is redundant, since sϩiϩrϭ1necessarily at all times.This model is ap-propriate for a rapidly spreading disease that confers immu-nity on its survivors,such as influenza.In this paper we will consider only diseases of this type.Diseases that are endemic because they propagate on time scales comparable to or slower than the rate of turnover of the population,or because they confer only temporary immunity,are not well repre-sented by this model;other models have been developed for these cases͓3͔.The model described above assumes that the population is fully mixed,meaning that the individuals with whom a sus-ceptible individual has contact are chosen at random from the whole population.It also assumes that all individuals have approximately the same number of contacts in the same time,and that all contacts transmit the disease with the same probability.In real life none of these assumptions is correct, and they are all grossly inaccurate in at least some cases.In the work presented here we remove these assumptions by a series of modifications of the model.First,as many others have done,we replace the‘‘fully mixed’’aspect with a network of connections between indi-viduals͓19–28͔.Individuals have disease-causing contacts only along the links in this network.We distinguish here between‘‘connections’’and actual contacts.Connections be-tween pairs of individuals predispose those individuals to disease-causing contact,but do not guarantee it.An individu-al’s connections are the set of people with whom the indi-vidual may have contact during the time he or she is infective—people that the individual lives with,works with, sits next to on the bus,and so forth.We can vary the number of connections each person has with others by choosing a particular degree distribution for the network.͑Recall that the degree of a vertex in a network is the number of other vertices to which it is attached.͒For example,in the case of sexual contacts,which can commu-nicate sexually transmitted diseases,the degree distribution has been found to follow a power-law form͓8͔.By placing the model on a network with a power-law degree distribution we can emulate this effect in our model.Our second modification of the model is to allow the probability of disease-causing contact between pairs of indi-viduals who have a connection to vary,so that some pairs have higher probability of disease transmission than others.Consider a pair of individuals who are connected,one of whom i is infective and the other j susceptible.Suppose that the average rate of disease-causing contacts between them is r i j,and that the infective individual remains infective for a time␶i.Then the probability1ϪT i j that the disease will not be transmitted from i to j is1ϪT i jϭlim␦t→0͑1Ϫr i j␦t͒␶i/␦tϭeϪr i j␶i,͑2͒and the probability of transmission isT i jϭ1ϪeϪr i j␶i.͑3͒Some models,particularly computer simulations,use dis-crete time steps rather than continuous time,in which case instead of taking the limit in Eq.͑2͒we simply set␦tϭ1, givingT i jϭ1Ϫ͑1Ϫr i j͒␶i,͑4͒where␶is measured in time steps.In general r i j and␶i will vary between individuals,so that the probability of transmission also varies.Let us assume initially that these two quantities are independent identically distributed͑iid͒random variables chosen from some appro-priate distributions P(r)and P(␶).͑We will relax this as-sumption later.͒The rate r i j need not be symmetric—the probability of transmission in either direction might not be the same.In any case,T i j is in general not symmetric be-cause of the appearance of␶i in Eqs.͑3͒and͑4͒.Now here is the trick:because r i j and␶i are idd random variables,so is T i j,and hence the a priori probability of transmission of the disease between two individuals is sim-ply the average T of T i j over the distributions P(r)and P(␶),which isTϭ͗T i j͘ϭ1Ϫ͵0ϱdr d␶P͑r͒P͑␶͒eϪr␶͑5͒for the continuous time case orTϭ1Ϫ͵0ϱdr͚␶ϭ0ϱP͑r͒P͑␶͒͑1Ϫr͒␶͑6͒for the discrete case͓23͔.We call T the‘‘transmissibility’’of the disease.It is necessarily always in the range0рTр1.Thus the fact that individual transmission probabilities vary makes no difference whatsoever;in the population as a whole the disease will propagate as if all transmission prob-abilities were equal to T.We demonstrate the truth of this result by explicit simulation in Sec.III E.It is this result that makes our models solvable.Cases in which the variables r and␶are not idd are trickier,but,as we will show,these are sometimes solvable as well.We note further that more complex disease transmission models,such as SEIR models in which there is an infected-but-not-infective period͑E͒,are also covered by this formal-ism.The transmissibility T i j is essentially just the integrated probability of transmission of the disease between two indi-viduals.The precise temporal behavior of infectivity and other variables is unimportant.Indeed the model can be gen-eralized to include any temporal variation in infectivity of the infective individuals,and transmission can still be repre-sented correctly by a simple transmissibility variable T,as above.M.E.J.NEWMAN PHYSICAL REVIEW E66,016128͑2002͒Now imagine watching an outbreak of the disease,which starts with a single infective individual,spreading across our network.If we were to mark or‘‘occupy’’each edge in the graph across which the disease is transmitted,which happens with probability T,the ultimate size of the outbreak would be precisely the size of the cluster of vertices that can be reached from the initial vertex by traversing only occupied edges.Thus,the model is precisely equivalent to a bond percolation model with bond occupation probability T on the graph representing the community.The connection between the spread of disease and percolation was in fact one of the original motivations for the percolation model itself͓29͔,but seems to have been formulated in the manner presented here first by Grassberger͓30͔for the case of uniform r and␶,and by Warren et al.͓23,24͔for the nonuniform case.In the following section we show how the percolation problem can be solved on random graphs with arbitrary de-gree distributions,giving exact solutions for the typical size of outbreaks,presence of an epidemic,size of the epidemic ͑if there is one͒,and a number of other quantities of interest.III.EXACT SOLUTIONS ON NETWORKS WITH ARBITRARY DEGREE DISTRIBUTIONS One of the most important results to come out of empiri-cal work on networks is thefinding that the degree distribu-tions of many networks are highly right skewed.In other words,most vertices have only a low degree,but there are a small number whose degree is very high͓5,7,11,31͔.The network of sexual contacts discussed above provides one ex-ample of such a distribution͓8͔.It is known that the presence of highly connected vertices can have a disproportionate ef-fect on certain properties of the network.Recent work sug-gests that the same may be true for disease propagation on networks͓21,32͔,and so it will be important that we incor-porate nontrivial degree distributions in our models.As a first illustration of our method therefore,we look at a simple class of unipartite graphs studied previously by a variety of authors͓33–42͔,in which the degree distribution is speci-fied,but the graph is in other respects random.Our graphs are simply defined.One specifies the degree distribution by giving the properly normalized probabilities p k that a randomly chosen vertex has degree k.A set of N degrees͕k i͖,also called a degree sequence,is then drawn from this distribution and each of the N vertices in the graph is given the appropriate number k i of‘‘stubs’’—ends of edges emerging from it.Pairs of these stubs are then chosen at random and connected together to form complete edges. Pairing of stubs continues until none are left.͑If an odd number of stubs is by chance generated,complete pairing is not possible,in which case we discard one k i and draw an-other until an even number is achieved.͒This technique guar-antees that the graph generated is chosen uniformly at ran-dom from the set of all graphs with the selected degree sequence.All the results given in this section are averaged over the ensemble of possible graphs generated in this way,in the limit of large graph size.A.Generating functionsWe wish then to solve for the average behavior of graphs of this type under bond percolation with bond occupation probability T.We will do this using generating function tech-niques͓43͔.Following Newman et al.͓36͔,we define a gen-erating function for the degree distribution thusG0͑x͒ϭ͚kϭ0ϱp k x k.͑7͒Note that G0(1)ϭ͚k p kϭ1if p k is a properly normalized probability distribution.This function encapsulates all of the information about the degree distribution.Given it,we can easily reconstruct the distribution by repeated differentiationp kϭ1k!d k G0dx k͉xϭ0.͑8͒We say that the generating function G0‘‘generates’’the dis-tribution p k.The generating function is easier to work with than the degree distribution itself because of two crucial properties.Powers.If the distribution of a property k of an object is generated by a given generating function,then the distribu-tion of the sum of k over m independent realizations of the object is generated by the m th power of that generating func-tion.For example,if we choose m vertices at random from a large graph,then the distribution of the sum of the degrees of those vertices is generated by͓G0(x)͔m.Moments.The mean of the probability distribution gener-ated by a generating function is given by thefirst derivative of the generating function,evaluated at1.For instance,the mean degree z of a vertex in our network is given byzϭ͗k͘ϭ͚k kp kϭG0Ј͑1͒.͑9͒Higher moments of the distribution can be calculated from higher derivatives also.In general,we have͗k n͘ϭ͚k k n p kϭͫͩx d dxͪn G0͑x͒ͬxϭ1.͑10͒A further observation that will also prove crucial is the following.While G0above correctly generates the distribu-tion of degrees of randomly chosen vertices in our graph,a different generating function is needed for the distribution of the degrees of vertices reached by following a randomly cho-sen edge.If we follow an edge to the vertex at one of its ends,then that vertex is more likely to be of higher degree than is a randomly chosen vertex,since high-degree vertices have more edges attached to them than low-degree ones.The distribution of degrees of the vertices reached by following edges is proportional to kp k,and hence the generating func-tion for those degrees isSPREAD OF EPIDEMIC DISEASE ON NETWORKS PHYSICAL REVIEW E66,016128͑2002͚͒k kp k x k͚kkp k ϭxG 0Ј͑x ͒G 0Ј͑1͒.͑11͒In general we will be concerned with the number of ways ofleaving such a vertex excluding the edge we arrived along,which is the degree minus 1.To allow for this,we simply divide the function above by one power of x ,thus arriving at a new generating functionG 1͑x ͒ϭG 0Ј͑x ͒G 0Ј͑1͒ϭ1z G 0Ј͑x ͒,͑12͒where z is the average vertex degree,as before.In order to solve the percolation problem,we will also need generating functions G 0(x ;T )and G 1(x ;T )for the dis-tribution of the number of occupied edges attached to a ver-tex,as a function of the transmissibility T .These are simple to derive.The probability of a vertex having exactly m of the k edges emerging from it occupied is given by the binomialdistribution (mk)T m (1ϪT )k Ϫm ,and hence the probability dis-tribution of m is generated byG 0͑x ;T ͒ϭ͚m ϭ0ϱ͚k ϭm ϱp kͩk mͪT m͑1ϪT ͒k Ϫm x m ϭ͚k ϭ0ϱp k͚m ϭ0kͩkmͪ͑xT ͒m ͑1ϪT ͒k Ϫm ϭ͚k ϭ0ϱp k ͑1ϪT ϩxT ͒k ϭG 0…1ϩ͑x Ϫ1͒T ….͑13͒Similarly,the probability distribution of occupied edges leav-ing a vertex arrived at by following a randomly chosen edgeis generated byG 1͑x ;T ͒ϭG 1…1ϩ͑x Ϫ1͒T ….͑14͒Note that,in our notationG 0͑x ;1͒ϭG 0͑x ͒,͑15a ͒G 0͑1;T ͒ϭG 0͑1͒,͑15b ͒G 0Ј͑1;T ͒ϭTG 0Ј͑1͒,͑15c ͒and similarly for G 1.͓G 0Ј(x ;T )here represents the derivativeof G 0(x ;T )with respect to its first argument.͔B.Outbreak size distributionThe first quantity we will work out is the distribution P s (T )of the sizes s of outbreaks of the disease on our net-work,which is also the distribution of sizes of clusters ofvertices connected together by occupied edges in the corre-sponding percolation model.Let H 0(x ;T )be the generatingfunction for this distribution,H 0͑x ;T ͒ϭ͚s ϭ0ϱP s ͑T ͒x s .͑16͒By analogy with the preceding section we also define H 1(x ;T )to be the generating function for the cluster of con-nected vertices we reach by following a randomly chosen edge.Now,following Ref.͓36͔,we observe that H 1can be broken down into an additive set of contributions as follows.The cluster reached by following an edge may be:͑1͒a single vertex with no occupied edges attached to it,other than the one along which we passed in order to reach it;͑2͒a single vertex attached to any number m у1of occupied edges other than the one we reached it by,each leading to another cluster whose size distribution is also generated by H 1.We further note that the chance that any two finite clus-ters that are attached to the same vertex will have an edge connecting them together directly goes as N Ϫ1with the size N of the graph,and hence is zero in the limit N →ϱ.In other words,there are no loops in our clusters;their structure is entirely treelike.Using these results,we can express H 1(x ;T )in a Dyson-equation-like self-consistent form thusH 1͑x ;T ͒ϭxG 1…H 1͑x ;T ͒;T ….͑17͒Then the size of the cluster reachable from a randomly cho-sen starting vertex is distributed according toH 0͑x ;T ͒ϭxG 0…H 1͑x ;T ͒;T ….͑18͒It is straightforward to verify that for the special case T ϭ1of 100%transmissibility,these equations reduce to those given in Ref.͓36͔for component size in random graphs with arbitrary degree distributions.Equations ͑17͒and ͑18͒pro-vide the solution for the more general case of finite transmis-sibility which applies to SIR models.Once we have H 0(x ;T ),we can extract the probability distribution of clus-ters P s (T )by differentiation using Eq.͑8͒on H 0.In most cases however it is not possible to find arbitrary derivatives of H 0in closed form.Instead we typically evaluate them numerically.Since direct evaluation of numerical derivatives is prone to machine precision problems,we recommend evaluating the derivatives by numerical contour integration using the Cauchy formulaP s ͑T ͒ϭ1s !d s H 0dx sͯx ϭ0ϭ12␲iͶH 0͑␨;T ͒␨s ϩ1d ␨,͑19͒where the integral is over the unit circle ͓44͔.It is possible to find the first thousand derivatives of a function without dif-ficulty using this method ͓36͔.By this method then,we can find the exact probability P s that a particular outbreak of our disease will infect s people in total,as a function of the transmissibility T .M.E.J.NEWMAN PHYSICAL REVIEW E 66,016128͑2002͒C.Outbreak sizes and the epidemic transitionAlthough in general we must use numerical methods to find the complete distribution P s of outbreak sizes from Eq.͑19͒,we canfind the mean outbreak size in closed form.Using Eq.͑9͒,we have͗s͘ϭH0Ј͑1;T͒ϭ1ϩG0Ј͑1;T͒H1Ј͑1;T͒,͑20͒where we have made use of the fact that the generating func-tions are1at xϭ1if the distributions that they generate are properly normalized.Differentiating Eq.͑17͒,we haveH1Ј͑1;T͒ϭ1ϩG1Ј͑1;T͒H1Ј͑1;T͒ϭ11ϪG1Ј͑1;T͒,͑21͒and hence͗s͘ϭ1ϩG0Ј͑1;T͒1ϪG1Ј͑1;T͒ϭ1ϩTG0Ј͑1͒1ϪTG1Ј͑1͒.͑22͒Given Eqs.͑7͒,͑12͒,͑13͒,and͑14͒,we can then evaluate this expression to get the mean outbreak size for any value of T and degree distribution.We note that Eq.͑22͒diverges when TG1Ј(1)ϭ1.This point marks the onset of an epidemic;it is the point at which the typical outbreak ceases to be confined to afinite number of individuals,and expands tofill an extensive fraction of the graph.The transition takes place when T is equal to the criti-cal transmissibility T c,given byT cϭ1G1Ј͑1͒ϭG0Ј͑1͒G0Љ͑1͒ϭ͚kkp k͚kk͑kϪ1͒p k.͑23͒For TϾT c,we have an epidemic,or‘‘giant component’’in the language of percolation.We can calculate the size of this epidemic as follows.Above the epidemic threshold Eq.͑17͒is no longer valid because the giant component is ex-tensive and therefore can contain loops,which destroys the assumptions on which Eq.͑17͒was based.The equation is valid however if we redefine H0to be the generating func-tion only for outbreaks other than epidemic outbreaks,i.e., isolated clusters of vertices that are not connected to the giant component.These however do notfill the entire graph, but only the portion of it not affected by the epidemic.Thus, above the epidemic transition,we haveH0͑1;T͒ϭ͚s P sϭ1ϪS͑T͒,͑24͒where S(T)is the fraction of the population affected by the epidemic.Rearranging Eq.͑24͒for S and making use of Eq.͑18͒,wefind that the size of the epidemic isS͑T͒ϭ1ϪG0͑u;T͒,͑25͒where uϵH1(1;T)is the solution of the self-consistencyrelationuϭG1͑u;T͒.͑26͒Results equivalent to Eqs.͑22͒–͑26͒were given previouslyin a different context in Ref.͓40͔.Note that it is not the case,even above T c,that all out-breaks give rise to epidemics of the disease.There are stillfinite outbreaks even in the epidemic regime.While this ap-pears very natural,it stands nonetheless in contrast to thestandard fully mixed models,for which all outbreaks giverise to epidemics above the epidemic transition point.In thepresent case,the probability of an outbreak becoming an epidemic at a given T is simply equal to S(T).D.Degree of infected individualsThe quantity u defined in Eq.͑26͒has a simple interpre-tation:it is the probability that the vertex at the end of arandomly chosen edge remains uninfected during an epi-demic͑i.e.,that it belongs to one of thefinite components͒.The probability that a vertex does not become infected via one of its edges is thus vϭ1ϪTϩTu,which is the sum of the probability1ϪT that the edge is unoccupied,and the probability Tu that it is occupied but connects to an unin-fected vertex.The total probability of being uninfected if a vertex has degree k is v k,and the probability of having de-gree k given that a vertex is uninfected is p k v k/͚k p k v k ϭp k v k/G0(v),which distribution is generated by the func-tion G0(v x)/G0(v).Differentiating and setting xϭ1,we thenfind that the average degree z out of vertices outside thegiant component isz outϭv G0Ј͑v͒G0͑v͒ϭv G1͑v͒G0͑v͒zϭu͓1ϪTϩTu͔1ϪSz.͑27͒Similarly the degree distribution for an infected vertex is generated by͓G0(x)ϪG0(v x)͔/͓1ϪG0(v)͔,which gives a mean degree z in for vertices in the giant component ofz inϭ1Ϫv G1͑v͒1ϪG0͑v͒zϭ1Ϫu͓1ϪTϩTu͔Sz.͑28͒Note that1ϪSϭG0(u;T)рu,since all coefficients of G0(x;T)are by definition positive͑because they form a probability distribution͒and hence G0(x;T)has only posi-tive derivatives,meaning that it is convex everywhere on the positive real line within its domain of convergence.Thus, from Eq.͑27͒,z outрz.Similarly,z inуz,and hence,as we would expect,the mean degree of infected individuals is al-ways greater than or equal to the mean degree of uninfected ones.Indeed,the probability of a vertex being infected, given that it has degree k,goes as1Ϫv kϭ1ϪeϪk ln(1/v),i.e., tends exponentially to unity as degree becomes large.E.An exampleLet us now look at an application of these results to a specific example of disease spreading.First of all we need to define our network of connections between individuals,SPREAD OF EPIDEMIC DISEASE ON NETWORKS PHYSICAL REVIEW E66,016128͑2002͒which means choosing a degree distribution.Here we will consider graphs with the degree distributionp kϭͭ0for kϭ0CkϪ␣eϪk/␬for kу1,͑29͒where C,␣,and␬are constants.In other words,the distri-bution is a power-law of exponent␣with an exponential cutoff around degree␬.This distribution has been studied before by various authors͓7,36,37,40͔.It makes a good ex-ample for a number of reasons:͑1͒distributions of this form are seen in a variety of real-world networks͓7,45͔;͑2͒it includes pure power-law and pure exponential distributions, both of which are also seen in various networks͓7,11,12,31͔, as special cases when␬→ϱor␣→0;͑3͒it is normalizable and has all momentsfinite for anyfinite␬.The constant C isfixed by the requirement of normaliza-tion,which gives Cϭ͓Li␣(eϪ1/␬)͔Ϫ1and hencep kϭkϪ␣eϪk/␬Li␣͑eϪ1/␬͒for kу1,͑30͒where Li n(x)is the n th polylogarithm of x.We also need to choose the distributions P(r)and P(␶) for the transmission rate and the time spent in the infective state.For the sake of easier comparison with computer simu-lations we use discrete time and choose both distributions to be uniform,with r real in the range0рrϽr max and␶integer in the range1р␶р␶max.The transmissibility T is then given by Eq.͑6͒.From Eq.͑30͒,we haveG0͑x͒ϭLi␣͑xeϪ1/␬͒Li␣͑eϪ1/␬͒͑31͒andG1͑x͒ϭLi␣Ϫ1͑xeϪ1/␬͒x Li␣Ϫ1͑eϪ1/␬͒.͑32͒Thus the epidemic transition in this model occurs atT cϭLi␣Ϫ1͑eϪ1/␬͒Li␣Ϫ2͑eϪ1/␬͒ϪLi␣Ϫ1͑eϪ1/␬͒.͑33͒Below this value of T there are only small͑nonepidemic͒outbreaks,which have mean size͗s͘ϭ1ϩT͓Li␣Ϫ1͑eϪ1/␬͔͒2Li␣͑eϪ1/␬͓͒͑Tϩ1͒Li␣Ϫ1͑eϪ1/␬͒ϪT Li␣Ϫ2͑eϪ1/␬͔͒.͑34͒Above it,we are in the region in which epidemics can occur, and they affect a fraction S of the population in the limit of large graph size.We cannot solve for S in closed form,but we can solve Eqs.͑25͒and͑26͒by numerical iteration and hencefind S.In Fig.1we show the results of calculations of the aver-age outbreak size and the size of epidemics from the exactformulas,compared with explicit simulations of the SIRmodel on networks with the degree distribution͑30͒.Simu-lations were performed on graphs of Nϭ100000vertices, with␣ϭ2,a typical value for networks seen in the real world,and␬ϭ5,10,and20͑the three curves in each panel of thefigure͒.For each pair of the parameters␣and␬for the network,we simulated10000disease outbreaks each for (r,␶)pairs with r max from0.1to1.0in steps of0.1,and␶max from1to10in steps of1.Figure1shows all of these resultson one plot as a function of the transmissibility T,calculatedfrom Eq.͑6͒.Thefigure shows two important things.First,the points corresponding to different values of r max and␶max but the same value of T fall in the same place and the two-parameter set of results for r and␶collapses onto a single curve.This indicates that the arguments leading to Eqs.͑5͒and͑6͒arecorrect͑as also demonstrated by Warren et al.͓23,24͔͒andthat the statistical properties of the disease outbreaks reallydo depend only on the transmissibility T,and not on theindividual rates and times of infection.Second,the dataclearly agree well with our analytic results for average out-break size and epidemic size,confirming the correctness ofour exact solution.The small disagreement between simula-tions and exact solution for͗s͘close to the epidemic transi-tion in the lower panel of thefigure appears to be afinite sizeeffect,due to the relatively small system sizes used in thesimulations.To emphasize the difference between our results and thosefor the equivalent fully mixed model,we compare the posi-tion of the epidemic threshold in the two cases.In the case ␣ϭ2,␬ϭ10͑the middle curve in each frame of Fig.1͒,our analytic solution predicts that the epidemic threshold occurs at T cϭ0.329.The simulations agree well with this predic-tion,giving T cϭ0.32(2).By contrast,a fully mixedSIR FIG.1.Epidemic size͑top͒and average outbreak size͑bottom͒for the SIR model on networks with degree distributions of the form ͑30͒as a function of transmissibility.Solid lines are the exact solu-tions,Eqs.͑25͒and͑22͒,for␣ϭ2and͑left to right in each panel͒␬ϭ20,10,and5.Each of the points is an average result for10000 simulations on graphs of100000vertices each with distributions of r and␶as described in the text.M.E.J.NEWMAN PHYSICAL REVIEW E66,016128͑2002͒。