Strong tunable coupling between a superconducting charge and phase qubit

couplingCoupling: An Essential Concept in Systems Design and AnalysisIntroductionIn the field of systems design and analysis, the concept of coupling plays a critical role in understanding and evaluating the complexity and efficiency of a software system. Coupling refers to the degree of interdependence between different modules or components within a system. It measures how closely one module is connected with another, and how much one module relies on another to perform its functionality. This document aims to provide a comprehensive overview of coupling, its types, impacts, and strategies to reduce coupling in software systems.Types of CouplingThere are several types of coupling that can exist in a software system. Each type represents a different level of interdependence between modules. Understanding thesetypes is crucial for analyzing software systems and identifying potential areas for improvement.1. Content Coupling: Content coupling refers to the highest level of interdependence between modules. In this type of coupling, one module directly accesses the internal data or methods of another module. This close dependency can make the system fragile and less maintainable, as any changes in one module may require modifications in other closely coupled modules.2. Common Coupling: Common coupling occurs when multiple modules share a global data item. Any changes in the shared data can impact the behavior of multiple modules. This type of coupling can make it difficult to maintain and reason about the system, as modifications in one module can have unintended consequences on others.3. Control Coupling: Control coupling happens when one module controls the flow of execution of another module by passing parameters, flags, or function calls. This type of coupling can lead to code complexity and reduce modularity, as changes in one module can have ripple effects on other modules that rely on the control flow.4. Stamp Coupling: Stamp coupling occurs when modules share a structure with multiple related elements, but only a subset of the elements is actually used by each module. This can lead to inefficiency and code bloat, as modules may require unnecessary data or functionality.5. Data Coupling: Data coupling represents a lower level of interdependence, where modules share data through parameters or return values. However, they do not rely on each other's internal implementation details. This is considered a desirable type of coupling, as it promotes modular and independent development of components.Impacts of CouplingHigh coupling in a software system can have various negative impacts on its maintainability, extensibility, and overall performance. Some of the key impacts of coupling include:1. Difficult Maintenance: When modules are tightly coupled, any changes or bug fixes in one module can have unintended consequences on other modules. This makes it more difficult to identify and fix issues, increasing the risk of introducing new bugs and errors.2. Limited Extensibility: In a system with high coupling, making changes or adding new features becomes a challenging task. The interdependence between modules can restrict the flexibility and modularity needed for easy extensibility. This can limit the ability to adapt to evolving requirements or integrate with new technologies.3. Testing Complexity: Closely coupled modules often require more extensive testing efforts. The high interdependence between modules makes it harder to isolate and test individual components. Any changes to one module may require retesting of multiple other modules, increasing the time and effort required for testing.Strategies to Reduce CouplingReducing coupling is crucial for developing and maintaining robust, flexible, and efficient software systems. Here are some strategies to minimize coupling and improve the overall quality of the system:1. Encapsulate Functionality: Encapsulating the functionality of modules and exposing only necessary interfaces canreduce the level of interdependence. This promotes modularity and allows changes to be localized within individual modules, minimizing the impact on other parts of the system.2. Use Abstraction and Interfaces: Instead of tightly coupling modules with concrete implementations, use abstractions and interfaces to define contracts between components. This allows for loose coupling, as modules can interact through well-defined interfaces without relying on the concrete implementation details.3. Apply Design Patterns: Design patterns like the Observer pattern, Dependency Injection, or Strategy pattern can help reduce coupling by decoupling modules and promoting more flexible interactions. These patterns provide standardized approaches for achieving loose coupling and modular designs.4. Follow Separation of Concerns: Ensure that each module or component has a clear and single responsibility. This helps to minimize the dependencies between modules and allows for better separation of concerns. By separating the functionality, changes or updates in one aspect of the system will have minimal impact on other components.5. Continuous Refactoring: Regularly review and refactor the codebase to identify and reduce coupling. Refactoring involves restructuring the code without changing its external functionality. By continuously refactoring, developers can identify opportunities to decouple modules and improve the overall code quality.ConclusionCoupling is a fundamental concept in systems design and analysis, influencing the complexity and maintainability of software systems. Understanding the different types of coupling and their impacts is essential for evaluating and improving the quality of a system. By adopting strategies like encapsulation, abstraction, design patterns, separation of concerns, and continuous refactoring, developers can minimize coupling and build more flexible, modular, and maintainable software systems.。

《软件工程》习题汇锦一、单项选择题提示：在每小题列出的四个备选项中只有一个是符合题目要求的，请将其代码填写在下表中。

错选、多选或未选均无分.1. ( ）If a system is being developed where the customers are not sure of what theywant, the requirements are often poorly defined。

Which of the following would be an appropriate process model for this type of development?（A）prototyping（B）waterfall（C)V-model（D）spiral2. （)The project team developing a new system is experienced in the domain.Although the new project is fairly large, it is not expected to vary much from applications that have been developed by this team in the past. Which process model would be appropriate for this type of development？（A）prototyping（B）waterfall（C）V-model（D）spiral3. （）Which of the items listed below is not one of the software engineering layers?(A）Process（B）Manufacturing（C）Methods（D)T ools4. （)Which of these are the 5 generic software engineering framework activities？（A）communication，planning，modeling，construction，deployment（B) communication, risk management, measurement，production, reviewing(C）analysis，designing，programming, debugging, maintenance（D）analysis, planning，designing，programming，testing5. （）The incremental model of software development is（A）A reasonable approach when requirements are well defined.(B）A good approach when a working core product is required quickly。

Liang Guo Stephen L.Hodson Timothy S.FisherXianfan Xu1e-mail:xxu@ School of Mechanical Engineering and Birck Nanotechnology Center,Purdue University,West Lafayette,IN47907Heat Transfer AcrossMetal-Dielectric Interfaces During Ultrafast-Laser Heating Heat transfer across metal-dielectric interfaces involves transport of electrons and pho-nons accomplished either by coupling between phonons in metal and dielectric or by cou-pling between electrons in metal and phonons in dielectric.In this work,we investigate heat transfer across metal-dielectric interfaces during ultrafast-laser heating of thin metalfilms coated on dielectric substrates.By employing ultrafast-laser heating that cre-ates strong thermal nonequilibrium between electrons and phonons in metal,it is possible to isolate the effect of the direct electron–phonon coupling across the interface and thus facilitate its study.Transient thermo-reflectance measurements using femtosecond laser pulses are performed on Au–Si samples while the simulation results based on a two-temperature model are compared with the measured data.A contact resistance between electrons in Au and phonons in Si represents the coupling strength of the direct electron–phonon interactions at the interface.Our results reveal that this contact resist-ance can be sufficiently small to indicate strong direct coupling between electrons in metal and phonons in dielectric.[DOI:10.1115/1.4005255]Keywords:interface thermal resistance,ultrafast laser,thermo-reflectance,two-temper-ature model,electron–phonon coupling1IntroductionInterface heat transfer is one of the major concerns in the design of microscale and nanoscale devices.In metal,electrons,and pho-nons are both energy carriers while in dielectric phonons are the main energy carrier.Therefore,for metal-dielectric composite structures,heat can transfer across the interface by coupling between phonons in metal and dielectric or by coupling between electrons in metal and phonons in dielectric through electron-interface scattering.Phonon–phonon coupling has been simulated mainly by the acoustic mismatch model and the diffuse mismatch model[1].As for electron–phonon coupling,there are different viewpoints.Some studies have assumed that electron–phonon coupling across a metal-dielectric interface is negligible and heat transfer occurs as electron–phonon coupling within metal and then phonon–phonon coupling across the interface[2].Electron–phonon coupling between metal(Cr,Ti,Al,Ni,and Pt)and SiO2 has exhibited negligible apparent thermal resistance using a parallel-strip technique[3].On the other hand,comparison between simulations and transient thermal reflectance(TTR) measurements for Au-dielectric interfaces reveals that energy could be lost to the substrate by electron-interface scattering dur-ing ultrafast-laser heating,and this effect depends on electron temperature and substrate thermal properties[4–6].In this study,we employ TTR techniques to investigate inter-face heat transfer for thin goldfilms of varying thicknesses on sili-con substrates.(Here,we consider silicon as a dielectric since heat is carried by phonons in silicon.)Similar work has been reported [5].In our model,we consider two temperatures in metal and also the temperature in the dielectric substrate.This allows us to inves-tigate the effect of both the coupling between electrons in metal and phonons in the dielectric substrate,and the coupling between phonons in metal and phonons in the dielectric substrate,and allows us to isolate the effect of the electron–phonon coupling across the interface that can be determined from the TTR mea-surement.Experimentally,we employ pulse stretching to mini-mize the effect of nonequilibrium among the electrons.As a result,the experimental data can be well-explained using the com-putational model.The thermal resistance between electrons in Au and phonons in Si,which quantifies the direct electron–phonon coupling strength,is calculated from the measured data.The results reveal that in the thermal nonequilibrium state,this electron–phonon coupling at the interface is strong enough to dominate the overall interface heat transfer.2TTR MeasurementAu–Si samples of varying Au thicknesses were prepared by thermal evaporation at a pressure of the order of10À7Torr.The thicknesses of the goldfilms are39,46,60,77,and250nm,meas-ured using an atomic force microscope.The pump-and-probe technique is used in a collinear scheme to measure the thermo-reflectance signal.The laser pulses are generated by a Spectra Physics Ti:Sapphire amplified femtosecond system with a central wavelength of800nm and a repetition rate of5kHz.The wave-length of the pump beam is then converted to400nm with a sec-ond harmonic crystal.The pump pulse has a pulse width(full width at half maximum-FWHM)of390fs measured by the sum-frequency cross-correlation method and is focused onto the sam-ple with a spot radius of20.3l m.The probe beam has a central wavelength of800nm and a pulse width of205fs measured by autocorrelation and is focused with a spot radius of16.9l m.This pump pulse width is intentionally stretched from the original pulse width of50fs to minimize the influence of thermal nonequili-brium among electrons since the electron thermalization time in Au can be of the order of100fs[7].This thermalization time is pump wavelength and pumpfluence dependent,and can be of the order of10fs if higher laserfluence is used[8,9].Our experiments did show the importance of pulse stretching.Figure1shows the TTR measurement results for the sample of thickness77nm with different pumpfluences before and after stretching the pulse.The plots show the normalized relative reflectance change(ÀD R/R)1Corresponding author.Contributed by the Heat Transfer Division of ASME for publication in the J OURNAL OF H EAT T RANSFER.Manuscript received May18,2011;final manuscript received September30,2011;published online February13,2012.Assoc.Editor: Robert D.Tzou.with the delay time between the pump and the probe pulses to show the contrast in cooling rates.With a shorter pulse (Fig.1(a )),a steep initial drop is seen in the signal,which is attributed to the behavior of nonequilibrium among electrons.Since the TTM to be used for simulation assumes a well-defined tempera-ture for electrons,i.e.,the electrons in gold have reached thermal equilibrium (not necessarily a uniform temperature),the model cannot predict the fast initial drop in the signals in Fig.1(a ).As will be shown later,the signals obtained by stretching the pulse can be predicted well using the TTM.3Two-Temperature Model for Thermal Reflectance MeasurementsUltrafast-laser heating induces thermal nonequilibrium between electrons and phonons in metal,which can be described by the TTM [10–13].We note that the heterogeneous interface consid-ered here involves three primary temperature variables (two in the metal and one in the dielectric).The “two-temperature”model is applied to the metal side.For investigating electron–phonon and phonon–phonon coupling at the interface,two thermal resistances are defined:R es (its reciprocal)indicates the coupling strength between electrons in metal and phonons in dielectric,while R ps indicates the coupling strength between phonons in metal and phonons in dielectric.(Large thermal resistance corresponds to weak coupling.)The resulting governing equations,initial,and interface conditions areC e @T e @t ¼k e @2T e@x2ÀG ðT e ÀT p ÞþS (1a )C p @T p @t ¼k p @2T p @x 2þG ðT e ÀT p Þ(1b )C s @T s @t ¼k s @2T s@x(1c )T e ðt ¼0Þ¼T p ðt ¼0Þ¼T s ðt ¼0Þ¼T 0(2)Àk e@T e @xx ¼L ¼T e ÀT s R es x ¼L(3a )Àk p @T px ¼L ¼T p ÀT s ps x ¼L(3b )Àk s@T sx ¼L ¼T e ÀT s es x ¼L þT p ÀT s ps x ¼L(3c )The subscripts e ,p ,and s denote electrons in metal,phonons in metal,and phonons in the dielectric substrate,respectively.C is the volumetric heat capacity,k is the thermal conductivity,G is the electron–phonon coupling factor governing the rate of energy transfer from electrons to phonons in metal,and L is the thickness of the metal layer.At the front surface of the metal layer insula-tion boundary condition is used due to the much larger heat flux caused by laser heating relative to the heat loss to air.At the rear surface of the substrate,since the thickness of the substrate used is large enough (1l m)so that there is no temperature rise during the time period of consideration,the insulation boundary condition is also applied.Thermal properties of phonons in both metal and dielectric are taken as temperature-independent due to the weak temperature dependence.The thermal conductivity of phonons in metal is much smaller than that of the electrons and is taken in this work as 0.001times the bulk thermal conductivity of gold (311W/(mK)).The volumetric heat capacity of the metal phonon is taken as that of the bulk gold.C e is taken as proportional to T e [14]with the proportion coefficient being 70J/(m 3K 2)[15],and k e is calculated by the model and the data used in Ref.[13]which is valid from the room temperature to the Fermi temperature (6.39Â104K in Au,[14]).G can be obtained using the model derived in Ref.[16].In this work,the value of G at the room tem-perature is taken as 4.6Â1016W/(m 3K)[17],and its dependence on electron and phonon temperatures follows [16].The laser heat-ing source term S is represented by the model used in [13]asS ¼0:94ð1ÀR ÞJ t p ðd þd b Þ1Àexp ÀL d þd bexp Àx d þd b À2:77t t p2"#(4)which assumes all the absorbed laser energy is deposited in the metal layer.J is the fluence of the pump laser,R is the surface re-flectance to the pump,t p is the pulse width (FWHM),d is the opti-cal penetration depth,and d b is the electron ballistic length (around 100nm in Au,[18]).R es and R ps are treated as free pa-rameters for fitting the experimental data.The wavelength of the probe laser in the experiment is centered at 800nm.For this wavelength,the incident photon energy is below the interband transition threshold in Au,which is around 2.47eV [18],and the Drude model can be used to relate the tem-peratures of electrons and phonons to the dielectric function and then the index of refraction,which is expressed as [19]e ¼e 1Àx 2px ðx þi x s Þ(5)x is the frequency of the probe laser and x p is the plasma fre-quency (1.37Â1016rad/s in Au evaluated using the data in Ref.[14]).x s is the electron collisional frequency,the inverse of the electron relaxation time.The temperature dependence ofelectricalFig.1TTR measurement results for the Au–Si sample of Authickness 77nm with different fluences.(a )Results before pulse stretching;(b )results after pulse stretching.resistivity indicates that x s is approximately proportional to pho-non temperature at high temperature [14]and from the Fermi liq-uid theory,its variation with electron temperature is quadratic (T e 2)[20].Therefore,x s is related to T e and T p approximately asx s ¼A ee T 2e þB ep T p(6)A ee is estimated from the low-temperature measurement [21]andB ep is usually estimated from the thermal or electrical resistivity near the room temperature [14].In this work,A ee is taken as the lit-erature value 1.2Â107s À1K À2[6]while e 1and B ep are evaluated by fitting the room-temperature value of the complex dielectric con-stant at 800nm wavelength provided in Ref.[22],which are found to be 9.7and 3.6Â1011s À1K À1,respectively.The complex index of refraction n 0þin 00is the square root of the dielectric ing Eqs.(5)and (6),n 0and n 00are evaluated as 0.16and 4.90,respectively,which agree with the empirical values [23].The re-flectance is then calculated from n 0and n 00by the method of transfer matrix [24],which considers multiple reflections in thin films.4Results and DiscussionThe results of TTR measurements with a pump fluence of 147J/m 2are plotted in Fig.2.The fast decrease of the reflectance indicates that energy transfer between electrons and phonons in metal,followed by a relatively slow decrease after several ps which indicates electrons and phonons have reached thermal equi-librium.The initial cooling rates are smaller for samples with thicknesses less than the electron ballistic length since the electron temperature is almost uniform across the thin film,and coupling with phonons within the metal film and the dielectric substrate is the only cooling mechanism.For a thicker sample of thickness 250nm,the initial decrease is much faster due to thermal diffu-sion in the gold film caused by a gradient of the electron tempera-ture in the film.We investigate the effect of R es and R ps using the thermo-reflectance signal.Two values of R ps ,1Â10À10m 2K/W and 1Â10À7m 2K/W,are used,each with a parameterized range of values for R es .Figure 3shows the calculated results for the sample with a 39nm-thick gold film.Little difference can be seen between Figs.3(a )and 3(b )while different cooling rates are obtained with varying R es in either plot,indicating that the cooling rate is not sensitive to the coupling strength between phonons in metal and dielectric.Note that an interface resistance of 1Â10À10m 2K/W is lower than any reported value,indicating a very high coupling strength between the phonons in metal and dielectric.Conversely,the results vary greatly with the coupling strength between electrons in metal and phonons in dielectric at the interface.This is because the lattice (phonon)temperature rise in metal is much smaller than the elec-tron temperature that the interface coupling between phonons in metal and dielectric does not influence the surface temperature,which directly determines the measured reflectance.On the other hand,the temperature rise of electrons is much higher,and conse-quently,the cooling rate is sensitive to R es .The relatively high sensitivity of R es to that of R ps demonstrates that the former can be isolated for the study of the coupling between electrons in metal and phonons in dielectric.We now use the measured TTR data to estimate R es ,the thermal resistance between electrons in metal and phonons in dielectric.R es is adjusted by the least square method to fit the simulation results with the measured data,and the results are shown in Fig.4.We note that it is impossible to fit the measured results using insu-lation interface condition (i.e.,no coupling or extremely large thermal resistance between electrons in metal and phonons in the dielectric substrate),which will significantly underestimate the cooling rate.For thin samples,we find that the value of R es is of the order of 10À10to 10À9m 2K/W.This value is below the ther-mal resistances of representative solid–solid interfaces measured in thermal equilibrium [25].This indicates that the direct coupling between electrons in metal and phonons in dielectric is strong.It is also noted that the resistance values increases with the thickness of the gold film,indicating a decrease in the coupling strength between electrons in metal and the dielectric substrate.This could be due to the lower electron temperature obtained in thicker films,and a decrease of the coupling strength with a decrease in the electron temperature [5].For the sample of thickness 250nm,R es has little effect on the simulation result since the interface is too far from the absorbing surface to influence the surface tempera-ture,and therefore it is not presented here.The agreement between the fitted results and the measured data is generally good.The small discrepancy between the measured and the fitted results can result from inaccuracy in computingtheFig.2TTR measurement results on Au–Si samples of varying AuthicknessesFig.3Simulation results with varying R es for the Au–Si sample of Au thickness 39nm.(a )R ps 51310210m 2K/W;(b )R ps 5131027m 2K/W.absorption or the temperature.Figure 1(b)shows the normalized TTR measurement results on the sample of thickness 77nm with three laser fluences.It is seen that small variations in the shape of the TTR signals can be caused by different laser fluences and thus the maximum temperature reached in the film.Absorption in metal,multiple reflections between the metal surface and the Au–Si inter-face,and possible deviations of the properties of thin films from those of bulk can all contribute to uncertainties in the temperature simulation;therefore affecting the calculated reflectance.With the values of R es shown in Fig.4,the calculation shows that the highest electron temperature,which is at the surface of 39nm–thick gold film,is about 6700K.The highest temperature of electrons is roughly inversely proportional to the thickness of the films for the four thinner films.The highest temperature of elec-trons is much less than the Fermi temperature and thus ensures the validity of the linear dependence of C e on T e [14].The highest temperature for the lattice in metal is about 780K,also in the 39nm-thick gold film.This large temperature difference between electrons and lattice indicates that the interface heat transfer is dominated by the coupling between electrons in metal and the phonons in the dielectric substrate.As shown in Fig.4,the meas-ured R es is very low,of the order of 10À10to 10À9m 2K/W.Even if R ps ,which is not determined in this study,is also that low (note that 10À10to 10À9m 2K/W is lower than any reported values),because of the large difference in temperatures between electrons and the phonons in metal,the interface heat transfer rate (Eqs.(3a )–(3c ))due to the coupling between electrons in metal and the substrate is much larger than that due to the coupling between phonons in metal and the substrate.5ConclusionsIn conclusion,TTR measurements using femtosecond laser pulses are performed on Au–Si samples and the results are analyzed using the TTM model.It is shown that due to the strong nonequilibrium between electrons and phonons during ultrafast-laser heating,it is possible to isolate the effect of the direct electron–phonon coupling across the interface,allowing investiga-tion of its ing stretched femtosecond pulses is shown to be able to minimize the nonequilibrium effect among electrons,and is thus more suitable for this study.The TTR measurement data can be well-represented using the TTM parison between the TTR data and the TTM results indicates that the direct coupling due to electron-interface scattering dominates the interface heat transfer during ultrafast-laser heating of thin films.AcknowledgmentThis paper is based upon work supported by the Defense Advanced Research Projects Agency and SPAWAR Systems Cen-ter,Pacific under Contract No.N66001-09-C-2013.The authors also thank C.Liebig,Y.Wang,and W.Wu for helpful discussions.NomenclatureA ee ¼coefficient in Eq.(6),s À1K À2B ep ¼coefficient in Eq.(6),s À1K À1C ¼volumetric heat capacity,J/(m 3K)G ¼electron–phonon coupling factor,W/(m 3K)i ¼unit of the imaginary number J ¼fluence of the pump,J/m 2k ¼thermal conductivity,W/(mK)L ¼metal film thickness,mn 0¼real part of the complex index of refractionn 00¼imaginary part of the complex index of refraction R ¼interface thermal resistance,m 2K/W;reflectance S ¼laser source term,W/m3Fig.4Comparison between the measurement and the simulation results for Au–Si samples of different Au thicknesses.The open circle represents the meas-ured data and the solid line represents the simulation results.(a )39nm fitted by R es 55310210m 2K/W;(b )46nm fitted by R es 56310210m 2K/W;(c )60nm fitted by R es 51.231029m 2K/W;and (d )77nm fitted by R es 51.831029m 2K/W.T¼temperature,Kt¼time,st p¼pulse width of the pump(FWHM),sx¼spatial coordinate,me¼complex dielectric constante1¼constant in the Drude modeld¼radiation penetration depth,md b¼electron ballistic depth,mx¼angular frequency of the probe,rad/sx p¼plasma frequency,rad/sx s¼electron collisional frequency,rad/sSubscripts0¼initial statee¼electron in metales¼electron in metal and phonon in dielectricp¼phonon in metalps¼phonon in metal and phonon in dielectrics¼phonon in 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a r X i v :h e p -t h /0703131v 1 14 M a r 2007Preprint typeset in JHEP style -HYPER VERSIONMatteo Beccaria Dipartimento di Fisica,Universita’del Salento,Via Arnesano,I-73100Lecce INFN,Sezione di Lecce E-mail:matteo.beccaria@le.infn.it Gian Fabrizio De Angelis Dipartimento di Fisica,Universita’del Salento,Via Arnesano,I-73100Lecce INFN,Sezione di Lecce E-mail:deangelis@le.infn.it Valentina Forini Dipartimento di Fisica,Universita’di Perugia,Via A.Pascoli,I-06123Perugia INFN,Sezione di Perugia and Humboldt-Universit¨a t zu Berlin,Institut f¨ur Physik,Newtonstraße 15,D-12489Berlin E-mail:forini@pg.infn.it,forini@physik.hu-berlin.de A BSTRACT :We study at strong coupling the scaling function describing the large spin anomalous dimension of twist two operators in super Yang-Mills theory.In the spirit of AdS/CFT duality,it is possible to extract it from the string Bethe Ansatz equa-tions in the slsector of the superstring.To this aim,we present a detailed analysis of the Bethe equations by numerical and analytical methods.We recover several short string semiclassical results as a check.In the more difficult case of the long string limit providing the scaling function,we analyze the strong coupling version of the Eden-Staudacher equation,including the Arutyunov-Frolov-Staudacher phase.We prove that it admits a unique solution,at least in perturbation theory,leading to the correct prediction consistent with semiclassical string calculations.K EYWORDS :integrable quantum field theory,integrable spin chains (vertex models),quantum integrability (Bethe ansatz).1.IntroductionThe-gluon maximally helicity violating(MHV)amplitudes in planar SYM obey very remarkable iterative relations[1,2,3]suggesting solvability or even integrability of the maximally supersymmetric gauge theory.The main ingredient of the construction is the so-called scaling function defined in terms of the large spin anomalous dimension of leading twist operators in the gauge theory[4].The scaling function can be obtained by considering operators in the sl sector of the formTr permutations(1.1) The classical dimension is,so is the twist,with minimal value.The minimal anomalous dimension in this sector is predicted to scale at large spin as(1.2) where the planar’t Hooft coupling is defined as usual byThe one and two loops explicit perturbative calculation of is described in[5,6,7] and[8,9].Based on the QCD calculation[10],the three-loop calculation is per-formed in[11,12]by exploiting the so-called trascendentality principle(KLOV).In principle,one would like to evaluate the scaling function,possibly at all loop order by Bethe Ansatz methods exploiting the conjectured integrability of SYM.This strategy has been started in[13].In that paper,an integral equation providing is proposed by taking the large spin limit of the Bethe equations[14].Its weak coupling expansion disagrees with the four loop contribution.The reason of this discrepancy is well under-stood.The Bethe Ansatz equations contain a scalar phase,the dressing factor,which is not constrained by the superconformal symmetry of the model.Its effects at weak-coupling show up precisely at the fourth loop order.A major advance was done by Beisert,Eden and Staudacher(BES)in[15].In the spirit of AdS/CFT duality,they considered the dressing factor at strong coupling.In that regime,it has been conjectured a complete asymptotic series for the dressing phase[16]. This has been achieved by combining the tight constrains from integrability,explicit1-loop-model calculations[17,18,19,20]and crossing symmetry[21].By an impressive insight,BES proposed a weak-coupling all-order continuation of the dressing.Including it in the ES integral equation they obtained a new(BES)equation with a rather complicated kernel.The predicted analytic four-loop result agrees with the KLOV principle.Very re-markably,an explicit and independent perturbative4-loop calculation of the scaling func-tion appeared in[22].In thefinal stage,the4-loop contribution is evaluated numerically with full agreement with the BES prediction.This important result is one of the main checks of AdS/CFT duality.Indeed,a non trivial perturbative quantity is evaluated in the gauge theory by using in an essential way input data taken from the string side.As a further check,one would like to recover at strong coupling the asymptotic behav-ior of the scaling function,as predicted by the usual semiclassical expansions of spinning string solutions[23,24,25].Actually,the BES equation passes this check,partly numer-ically[26]and partly by analytical means[27].One could say that this is a check that nothing goes wrong if one performs the analytic continuation of the dressing phase from strong to weak coupling.From a different perspective,one would like to close this logical circle and check that the same result is obtained in the framework of the quantum string Bethe equations pro-posed originally in[17].Indeed,it would be very nice to show that these equations repro-duce the scaling function in the suitable long string limit.Also,one expects tofind some simplifications due to the fact that only thefirst terms in the strong coupling expression of the dressing must be dealt with.On the other side,the BES equation certainly requires all the weak-coupling terms if it has to be extrapolated at large coupling.In this paper,we pursue this approach.As afirst step,we study numerically the quantum Bethe Ansatz equations in the sl sector and check various results not directly related to the scaling function.Then,we work out the long string limit which is relevant to the calculation of.From our encouraging numerical results,we move to an analytical study of a new version of the BES equation suitable for the string coupling region.Thisequation has beenfirst derived by Eden and Staudacher in[13]as a minor result.Indeed,it has been left over because the main interest was focusing on matching the weak-coupling 4-loop prediction.However,we believe that it is a quite comfortable tool if the purpose is that of reproducing the strong coupling behavior of the scaling function.We indeed prove that the solution described in[27]is the unique solution of the strong coupling ES equation.The plan of the paper is the following.In Sec.(2)we recall the Bethe Ansatz equations valid in the sl sector of SYM without and with dressing corrections.In Sec.(3)we present various limits obtained in the semiclassical treatment of the superstring. We present our results for short and long string configurations.In Sec.(4)we analyze the strong coupling ES equation building explicitly its solution and checking that it agrees with the result of[27].We also investigate numerically the equation without making any strong coupling expansion to show that the equation is well-defined.Sec.(5)is devoted to a summary of the presented results.2.Gauge Bethe Ansatz predictions for the scaling function without dressing In the seminal paper[13],Eden and Staudacher(ES)proposed to study the scaling func-tion in the framework of the Bethe Ansatz for the sl sector of SYM.The states to be considered in this rank-1perturbatively closed sector take the general formTr(2.1) They are associated to the states of an integrable spin chain.The anomalous dimension is related to the chain energies by(2.2) The all-loop conjectured Bethe Ansatz equations valid for up to wrapping terms are fully described in[28,14].Some explicit tests are can be found in[29,30].The Bethe Ansatz equations for the roots are(2.3)where we have defined the maps(2.4) The solutions of Eq.(2.3)must obey the following constraint to properly represent single trace operatorsThe quantum part of the anomalous dimension,i.e.the chain spectrum,is obtained from(2.6) Taking the large limit of the Bethe Ansatz equations,ES obtain the following represen-tation of the scaling function(2.9) Notice that given we can simply write[31].These equations are independent on the twist which drops in the large limit.This is important since the scaling function is expected to be universal[32,13]and therefore can be computed at large twist.Unfortunately,the perturbative expansion of disagrees at4loops with the ex-plicit calculation in the gauge theory.This is well known to be due to the missing contri-bution of the dressing phase.2.1Input from string theory:Dressing correctionsThe effect of dressing is discussed in[15]to which we defer the reader for general dis-cussions about its origin and necessity.The Bethe equations are corrected by a universal dressing phase according to1This general formula holds unchanged in various deformations of the SYM theory[35,36],see for ex.[37].Thefirst non trivial constant is.Indeed,consistently with the3-loops agreement with explicit perturbation theory.The proposed coefficients for the all-order weak-coupling expansion of the dressing phase[16]are given in[15](see also[38]and[39]).They read(2.15) with(2.17) The modified integral equation can be exploited to compute the perturbative expansion of.Now,there is agreement with the4loop explicit calculation.As we stressed in the Introduction,it is very remarkable that this weak coupling agreement is found with various inputs from string theory.In this sense,this is a powerful check of AdS/CFT duality.3.Strong coupling regime and the string Bethe equationsAs we explained,the BES equation is obtained by including in the ES equation an all-order weak-coupling expansion of the dressing phase.This expansion comes from a clever combination of string theory inputs and constraints from integrability.In our opinion,this is the essence of the integrability approach to AdS/CFT duality.As a consistency check, one would like to recover from the BES equation the known semiclassical predictions valid in the superstring at large coupling.There are indeed several limits that can be computed.The semiclassical limit is eval-uated in terms of the BMN-like[40]scaled variables which are keptfixed as(3.1) where are the semiclassical energy of a string rotating in with angular momen-tum and spinning in with spin.The classical solution,and thefirst quantum corrections as well,are described in[23,24,25].The simplest limits that can be considered are those describing short strings that do not probe regions with large curvature.We call themshort-GKP(3.2)short-BMNfixed(3.3) The scaling function is instead reproduced in the simplest long string limit which readslong string(3.4) In this limit,one can read the strong coupling behavior of the scaling function which iswhich at means.One can ask if it is possible to explore numerically the Bethe equations in the gauge theory up to and with to extract the scaling function.Actually,this is a hard task.At the regime is not perturbative.The complete dressing should be resummed and it is not easy to do that,although some very interesting results have been presented in[15].An alternative hybrid approach would be that of taking the string Bethe equations with leading dressing.This should be enough to study the leading terms of at strong coupling.Of course,the problem is now that must be large and then must be unre-alistically large to deal with the numerical solution.However,not much is known about the properties of the strong coupling dressing expansion.It is divergent,but possibly asymptotic.Therefore,it would be difficult to estimate its accuracy at.In the next Section,we illustrate the detailed exploration of the above three limits.3.1The String Bethe equations and their numerical solutionFor the subsequent analysis it is convenient to pass to the variables defining(3.7)(3.8) The loop corrections to the energy are now(3.9) where are obtained by solving the Bethe equations with dressing Eq.(2.10).At strong coupling,we use the leading dressing phase and write the string Bethe equations in loga-rithmic form as(3.10)(3.11) where and we have utilized the symmetry of the solution for the ground state as well as its known mode numbers[13].The numerical solution of the String Bethe equations is perfectly feasible.The tech-niques have already been illustrated in the two compact rank-1subsectors su and su as discussed in[41,42].First,we solve the equations at.This is the one-loop contribution.It is known exactly at.The Bethe roots are obtained,for an even spin,as the roots of the resolvent polynomial[43,13]At,we use the solution at as the starting point for the numerical root finder.Then,we increase.At each step,we use a linear extrapolation of the previous solutions to improve the guess of the new solution.This procedure is quite stable and allows to explore a wide range of values.Notice that changing the twist is trivial since the complexity of the equations does not change.3.2Short string in the GKP limitAs afirst numerical experiment wefix and and increase up to large values where the equations are reliable.This is the short string limit where the geometry is approx-imatelyflat.One expects to recover the Gubser-Klebanov-Polyakov law[44]. Solving the equations,we indeed verify that the Bethe momenta have the asymptotic scaling.This is clearly illustrated in Fig.(1).Despite the non trivial distribution of the Bethe roots,it is straightforward to compute the anomalous dimension at large.To do that,wefirst write the Bethe equations in the form(3.14)At large we can write(3.15)The Bethe momenta can be divided into a set with and a symmetric set obtained byflipping.The expansion of the energy is(3.16)(3.17) From the Bethe equations we have(3.18) where we exploited.Each Bethe momentum with can be written as with.Hence we can write(3.19) At large(3.21)The next-to-leading terms are(3.22)(3.24) The subleading term cancels the classical contribution leaving a pure behavior.This can be compared with theflat limit in the semiclassical approximation that reads for(3.25) with full agreement.3.3Short string in the BMN limitIf we keepfixed and increase withfixed,we can reach the BMN limit[40]. This is numerically very easy because enters trivially the equations.Fig.(2)shows the convergence to the BMN limit when is increased from10to100and isfixed at. The various curves clearly approach a limiting one.This is very nice since it is an explicit show of how the BMN regimes sets up.Fig.(3)shows the limiting curves forat very large.The three curves are perfectlyfit by the expected law(3.26)3.4Long string limit and the scaling functionThe previous pair of tests in the(easy)short string limits is a clear illustration that the numerical solution of the Bethe equations is reliable.The slow string limit is much more difficult.We begin with a plot of the energy at fixed twist and increasing spin from10to60.It is shown in Fig.(4).Each curve bends downward as increases,since it ultimately must obey the law.However,at fixed,when increases the energy increases slowly eventually following the law. We attempted an extrapolation at at each.In Fig.(5)we show our estimate for the derivative of the scaling function,byfitting the data with.We also show the analytical prediction.It seems to be roughly reproduced as soon as .The above”dirty”numerical procedure shows that it is reasonable to expect that the quantum string Bethe equations are able to capture the correct strong coupling behavior of the scaling function.However,the above extrapolation has a high degree of arbitrariness, especially concerning thefitting function employed to estimate the limit.Also, one would like to go to quite larger requiring a huge number of Bethe roots,equalindeed to the spin.In practice,as it stands,the numerical investigation could hardly be significantly improved.For all these reasons,in thefinal part of this paper we explore an equation analogous to the BES equation,but derived for the string Bethe equation,at least with the leading order dressing phase.4.The strong coupling ES equationThe inclusion of the AFS phase[17]in the ES equation(4.1)(4.3)(4.4) where the AFS kernel is(4.6) For the AFS kernel we have2(4.8) We change variables to put the equation in a somewhat simpler form and define(4.9)(4.7) The result Eq.(4.8)follows immediately from the expansions reported in the appendices of[15].The strong coupling ES equation for is then(4.12) Taking the terms with the leading power of wefind that satisfies the remarkably simple equation(4.15) and now the equation reads(4.16) Now,the following question arises:Is this equation a constraint or does it determine a unique solution?As afirst step,we prove that the solution of[27]is indeed a solution to the above equation.This solution reads(4.17) Now,the detailed values of the matrix elements areand(4.19) The linear equations Eq.(4.16)are then(stands for even Bessel)(4.20) whereThese equations are indeed satisfied by the solution defined through Eq.(4.17).This can be checked by evaluating the infinite sum in closed form by the Sommerfeld-Watson trans-formation methods.For instance thefive terms in Eq.(4.21)read at(4.22)(4.23)(4.24) However,this is not the unique solution of Eq.(4.13).As is quite usual,the straightfor-ward strong coupling limit of Bethe equations does not determine completely the solution which isfixed by the tower of subleading corrections.A similar difficulty is explained in details in[27].For instance,a second solution of Eq.(4.16)is(4.25) Indeed,this produces the remarkably simple solution(4.26) Notice that,before summing the series,this second solution is precisely of the general class Eq.(4.15).As a check,we have indeedfor(4.27) In practice,we still need the equal-weight condition on the even/odd Bessel functions contained in the solution[27].Tofind a unique solution,we must examine the next orders in the strong coupling expansion.Indeed,the next orders provide both equations for the various subleading corrections to the solution and constraints on the previous contributions.This is due to the fact that the AFS kernel is a function of expressed as a Neumann series of purely even Bessel functions.The odd Bessel functions provide the above mentioned constraints as we now illustrate.4.2NLO order at strong couplingLet us work out the constraints from the subleading correction.If we take into account the next terms in the expansion and write(4.28)(4.29) wefind the following equation for(4.30)(4.34) Hence,the equation can be written(4.35) where,explicitly,the constants are given by(4.36)(4.38)otherwiseandEquating to zero the coefficients of the odd Bessel functions we obtain the con-straint on(4.40) where(4.42) So this constraint adds nothing new and,in particular,is satisfied by the Alday’s solu-tion[27].Looking at the even Bessel functions,we obtain the homogeneous equation(4.44)(4.47)and(4.48) One obtains immediately the crucial relationThis relation permits to write all odd coefficients in terms of the even ones.Substi-tuting this relation in the truncated versions of the basic conditions(4.50) one obtains a well-posed problem converging rapidly to the solution[27]without any a priori condition on the solution.For instance,by truncating the problem dropping all with,wefind the table Tab.(1)of values for.A simple polynomial extrapolation to provides the correct limit.Hence,the strong coupling expansion is well defined and the leading solution is unique.Of course,it is the one described in[27].1020304050607080Table1:Coefficient from the truncated full rank linear problem.4.4Numerical integration of the strong coupling ES equationTo summarize,we have shown that the strong coupling ES equation is consistent with the results of[27].In that paper,it was crucial tofix the relative weights of the even/odd Bessel functions appearing in the general solution.These weights were shown to be more than an Ansatz.They are encoded in the full equation before expanding at strong cou-pling.Alternatively,they can be derived by analyzing the next-to-leading and next-to-leading corrections.As afinal calculation and check,we provide the results from a numerical investigation without any strong coupling expansion to see how the correct strong coupling solution arises.This can be done along the lines illustrated in[26,27].We start again from the Neumann expansion5.ConclusionsIn this paper,we have considered several properties of the quantum string Bethe equa-tions in the sl sector with the leading strong coupling dressing,i.e.the AFS phase.We have performed a numerical investigation of the equations showing that their analysis is quite feasible.As an interesting result,we have repeated the calculation of the GKP limit of the anomalous dimensions as for the highest excited states in the compact rank-1 subsectors su and su.Also,we have been able to observe the setting of the BMN scaling regime by reproducing the plane wave energy formula atfixed spin and large twist.In the case of the long string regime,we have been able to provide numerical ev-idences for a scaling function exhibiting an early strong coupling behavior as expected from the numerical solution of the BES equation.Motivated by these results,we have analyzed analytically and perturbatively at strong coupling an almost trivially modified version of the BES equation with the very simple strong coupling dressing[17].In particular,we have proved that this equation admits,as it should,a unique solution for the asymptotic Bethe root(Fourier transformed)density in full agreement with existing results.While this work was under completion,the paper[45]presented an analysis partially overlapping with our results.That paper derives an integral equation for the Bethe root density taking into account the dressing at strong coupling and is based on a novel integral representation of the dressing kernel.We hope that the two alternative approaches will turn out to be useful in computing the one-loop string correction to the large scaling function.Indeed,this interesting contribution has been checked numerically in[26]but it still evades an analytical confirmation.Hopefully,these various efforts might give insight on the general structure of the dressing phase as well as on the role of the asymptotic Bethe equations in an exact de-scription of the planar spectrum[46].Significative studies aboutfinite size effects[47] and corrections that arise in afinite volume to the magnon dispersion relation at strong coupling[48],see also[49],as well as the recent observation[50]that the dressing phase could originate from the elimination of”novel”Bethe roots,strongly demand a deeper understanding.AcknowledgmentsWe thank D.Serban and M.Staudacher for very useful discussions and comments.M.B.also thanks G.Marchesini for conversations about the properties of twist-2anomalous dimensions atfinite spin.The work of V.F.is supported in part by the PRIN project 2005-24045”Symmetries of the Universe and of the Fundamental Interactions”and by DFG 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强互相作用力物质的英文Strong Interaction Materials: A Brief Overview.Strong interaction materials, often referred to as hadronic matter, are composed of particles that interact through the strong nuclear force. This force, one of the four fundamental forces of nature, is responsible for binding protons and neutrons within atomic nuclei and for the existence of stable matter as we know it. The strong force is characterized by its short-range nature and its ability to bind particles into composite particles known as hadrons.Hadrons are subatomic particles that participate in the strong interaction. They include protons, neutrons, pions, kaons, and many others. These particles are classified into two broad categories: baryons, which have an odd number of valence quarks, and mesons, which have an even number of valence quarks. Baryons, such as protons and neutrons, are composed of three quarks, while mesons, such as pions andkaons, are composed of a quark and an antiquark.The strong force is transmitted by gluons, which are massless particles that carry the color charge. This charge is analogous to the electric charge but operates within the context of quantum chromodynamics (QCD), the theory that describes the strong interaction. Gluons interact with quarks and antiquarks, binding them into hadrons throughthe exchange of color charge.The strength of the strong force increases with decreasing distance between particles, reaching a maximumat very short distances. This behavior is known as asymptotic freedom, a prediction of QCD that has been confirmed through experiments. However, at larger distances, the strong force becomes weaker, allowing hadrons to exist as separate particles.The study of strong interaction materials is crucial to understanding the inner workings of atomic nuclei and the stability of matter. It also plays a pivotal role inparticle physics experiments, which aim to probe thefundamental nature of matter and energy. Experiments such as the Large Hadron Collider (LHC) at CERN are designed to create and study hadrons under extreme conditions, revealing insights into the behavior of the strong force.In addition to its fundamental importance, the strong force has applications in fields such as nuclear physics, nuclear engineering, and even medicine. For example, nuclear reactors use the strong force to maintain the stability of nuclear fuel, while radiation therapy uses radioactive particles to treat cancer by exploiting the strong force's ability to bind particles together.In conclusion, strong interaction materials are an integral part of our understanding of the fundamental forces of nature and the structure of matter. Their study continues to reveal new insights into the inner workings of the universe and holds promise for future technological applications. As research in this field continues to progress, we can expect to gain even deeper understanding of the role of the strong force in shaping our world.。

a rX iv:112.206v1[physi cs .data-an]1D ec21Catastrophic Cascade of Failures in Interdependent Net-works S.Havlin Minerva Center and Department of Physics,Bar-Ilan University,52900Ramat-Gan,Israel N.A.M.Ara ´u jo Computational Physics for Engineering Materials,IfB,ETH Z¨u rich,Schafmattstr.6,8093Z¨u rich,Switzerland S.V.Buldyrev Department of Physics,Yeshiva University,500West 185th Street,New York 10033USA Center for Polymer Studies and Department of Physics,Boston University,Boston Massachusetts 02215,USA C.S.Dias GCEP-Centro de F´ısica da Universidade do Minho,4710-057Braga,Portugal R.Parshani Minerva Center and Department of Physics,Bar-Ilan University,52900Ramat-Gan,Israel G.PaulCenter for Polymer Studies and Department of Physics,Boston University,Boston Massachusetts 02215,USAH.E.StanleyCenter for Polymer Studies and Department of Physics,Boston University,Boston Massachusetts 02215,USAc Societ`a Italiana di Fisica 12H.E.Stanley Summary.—Modern network-like systems are usually coupled in such a waythat failures in one network can affect the entire system.In infrastructures,biology,sociology,and economy,systems are interconnected and events taking place in onesystem can propagate to any other coupled system.Recent studies on such coupledsystems show that the coupling increases their vulnerability to random failure.Prop-erties for interdependent networks differ significantly from those of single-networksystems.In this article,these results are reviewed and the main properties discussed.1.–IntroductionThe last decade witnessed an intensive study of complex networks[1,2,3,4,5, 6]boosted by several real-world data revealing complex structures in the topology of their network like,Internet,airport connections,and power grids[7].Recently,special emphasis has been focused on the robustness of such systems to random failures or malicious attacks[8,9,10,11,12,13,14].Most of these works have been focused on single,isolated networks where no interaction with other networks is considered,i.e., the behavior of the system is independent of any other,coupled with it.Such conditions rarely occur in nature nor in technology.Typically,systems are interconnected and events taking place in one are likely to affect the others.Only recently[15,16],the effect of coupled networks has been considered,where a failure of one node in a network may lead to a cascade of failures in the entire system.In this manuscript we review these results, obtained through analytical and numerical approaches,based on percolation principles, which are rather surprising.In Fig.1is a scheme,from Peerenboom et al.[17],showing the interdependence of the relevant infrastructures for daily life,namely,oil,transportation,electric power, natural gas,water,and telecommunications.Not only all infrastructures depend on the electrical-power network,as one would expect,but also electrical power depends on the others,i.e.,a bidirectional coupling exists between all networks.This strong coupling can lead to catastrophic effects when,for some reason,a failure occurs in one of the networks.In September28th,2003,Italy was affected by a country-wide blackout[18], which was understood due to the coupling between the electrical power and communi-cation networks.In Fig.2is the map of Italy with the electrical power network on top. Each node is a power station and the edges represent the connection between stations. Slightly shifted to the top,is a scheme of the communication network that controls the power distribution,where the nodes are servers and the edges are links between them.In both cases,nodes are positioned based on their real geographical coordinates and com-munication servers are connected to their geographically nearest power station.When, for some reason,a failure occurs in one power station(red node in Fig.2.a),the node is removed from the network,and consequently four servers are turned offdue to the lack ofCatastrophic Cascade of F ailures in Interdependent Networks3Fig.1.–Interdependency of infrastructures(After Peerenboom et al.[17]).power supply.When these servers go down,three other servers(in green)become inac-tive since they are disconnected from the giant communication cluster.Then,a sequence of events takes place,in a cascade fashion,where the power stations connected to these servers are shut down(see Fig.2.b)and a set of other power stations(in green)become disconnected from the power grid giant component and,therefore,become inactive(see Fig.2.c).This example shows how a fail in a single power station can lead to a cascade of events ending in a blackout spanning over more than half of the system.Not only in infrastructures onefinds interdependent networks.From economy to biology there exist many examples of coupled systems.The network of banks is inter-dependent to the network of insurance companies,as well as to the network offirms in differentfields.These interconnections played a major role in the recentfinancial crisis. The human body networks are also interdependent,e.g.,the cardio-vascular,the respi-ratory,and the nervous systems depend on each other in order to function.Examples of interdependent networks can also be found in social sciences.Therefore,the study of the properties of coupled networks is a matter of paramount importance in multidisci-plinary science.In this article,we review the recent analytical and numerical results on their percolation properties.We start by recalling the main features of single networks, in Sec.2.In Secs.3and4the case of fully and partially interdependent networks are discussed.We presentfinal remarks in Sec.5.2.–Single Network RobustnessFor more than a decade the critical properties of isolated(single)networks have been studied extensively,see e.g.,[5,6].One relevant question is related with percolation,4H.E.StanleyFig.2.–Cartoon of a typical cascade obtained by implementing the described model on the real coupled system in Italy.Over the map is the network of the Italian power network and,slightly shifted to the top,is the communication network.Every server was considered to be connected to the geographically nearest power station.(After Buldyrev et al.[15])i.e.,the emergence of a giant cluster when nodes(or links)are sequentially added to the network[19,20,21,22,23].Due to symmetry reasons,for a single network,the problem can also be formulated in the inverse way.Let us consider an initial configuration of a network made of nodes and links connecting them.How the fraction of nodes in the giant cluster(largest one)is changed when a fraction1−p of nodes(or links)is removed?The fraction of the giant component is called the order parameter in the language of critical phenomena[19,20].Several models of networks have been proposed.Their description and critical prop-erties can be found elsewhere[5,6].In this article we focus on two types of model networks:a random graph(Erd˝o s-R´e nyi)[21,22,23]and a scale-free network(Barab´a si-Albert)[24].The Erd˝o s-R´e nyi(ER)network is a random graph obtained by randomly distributing M links between N nodes,being a statistical ensemble with equal probabil-ity for any generated configuration.The scheme in Fig.3.a is one possible configuration. On the other hand,the Barab´a si-Albert(BA)is a network which was grown under the preferential attachment rule,i.e.,at each iteration a new node is added to the network and connected to m already existing nodes with a probability of linking to a certain node proportional to the actual degree(number of links)of that node.A typical configuration obtained with this model is shown in Fig.3.b.These two models of generating networks lead to different topologies and statistical properties.For the ER network,since links are distributed in an uncorrelated way,theCatastrophic Cascade of F ailures in Interdependent Networks5Fig.3.–Single networks.a)Classical Erd˝o s-R´e nyi model and b)Barab´a si-Albert model. degree distribution is Poissonian,i.e.,the frequency of nodes with k links is[21],λkP(k)=exp(−λ)6H.E.Stanley The power-law nature of the degree distribution in a scale-free graph leads to richer percolation properties in this type of networks.In fact,such properties are dependent on the value of the degree exponentγ.For2≤γ≤3,the giant component only vanish for p c=0and no percolation transition occurs[26].However,for values ofγabove3,alike ER a second-order transition is found but with different critical exponents[5,25].Yet, the network is very robust to random failure which explains,for example,the stability of the Internet to random failures and the high longevity of viruses in the Internet,despite the large number of anti-virus softwares in the market.For both types of networks,when the percolation transition occurs,it is of second-order nature,meaning that a smooth decrease of the order parameter with the increasing fraction of removed nodes is observed.Besides,this fraction,at the critical point,is rather large(p c is low)being a sign of robustness in the system.As discussed below, this is not the case,when interdependent systems are considered.Recently,several studies have been published,discussing how to change the stochastic rule of percolation to obtain an abrupt transition[27,28,29,30,31,32]in a single network.When coupled interdependent systems are considered the order of the transition becomes naturally of first order nature[15,16].3.–Interdependent Networks RobustnessDespite the technological and fundamental relevance of coupled networks,only re-cently they have been considered and their percolation properties have been studied [15,16].The results of Buldyrev et al.[15]disclose an emergency of novel percolation properties under coupling not predicted from the behavior of single networks.To account for the coupling,let us consider two different networks,hereafter referred as A and B.Every A-node depends on a B-node,and vice versa,i.e.,a bidirectional,one-to-one coupling exists such that if node A i depends on node B i then node B i depends on node A i.The dependency is such that coupled nodes are only active if both are connected to the giant component of their work A and network B have degree distributions P A(k)and P B(k)respectively.The dependency links between the networks are achieved by randomly connecting(A i,B i)pairs of nodes in both networks under the constraint of having only one inter-network link per node.Percolation properties are studied by randomly removing nodes in the network A, mimicking a failure or malicious attack.When a node A i is removed,its A-links and the coupled node B i are also removed.As discussed above,in the context of the2003 blackout in Italy,removing the node A i can ignite failures in other nodes.All A-nodes which become disconnected from the giant cluster through A i become inactive and are removed together with their correspondent B-nodes.Analogously,all B-nodes bridged to the giant component through node B i are also removed.The same procedure is, recurrently,followed for all removed nodes and their counterparts in the other network leading to cascading failures between the two networks.This procedure reveals,as in the case of single networks,discussed in Sec.2,that when a fraction of nodes1−p is removed,a percolation transition occurs at a certain threshold,p=p c.Only for valuesCatastrophic Cascade of F ailures in Interdependent Networks7Fig.4.–Fraction of nodes in the giant component(p n/p)after n stages of cascading failures for different realizations of two coupled ER networks with the same degree distribution and128000 nodes.Initial removal is just below the percolation threshold with p=2.4554/<k>=p c. (After Buldyrev et al.[15]).of p above this threshold a giant mutually connected cluster exists.Below it,the entire system becomes completely fragmented.For two coupled ER networks the problem can be solved explicitly using generating functions[15,33,34,35].When the two networks have the same degree,i.e.,<k>A=< k>B=<k>,the value of p c is,1p c=8H.E.StanleyFig.5.–Scaled distribution of the number of stages in the cascade failures for two Erd˝o s-R´e nyi graphs,with the same degree distribution,at criticality,for different values of N.(After Buldyrev et al.[15]).At p c,the average number number of stages in the cascade scales with N1/4(see Fig.5).In Fig.6we plot the threshold,p c,and the fraction of nodes in the mutual giant component,µ∞,for different ratios between the average degrees of the networks,<k>A /<k>B.When the average degree is the same for both networks,<k>A=<k>B,as discussed above,the threshold is given by eq.(4)andµ∞at p c is nonzero,i.e.,the coupled system is more vulnerable than the single network case and the percolation transition is first order.As the ratio between the average degree of the networks decreases,both the threshold and the jump at the transition diminish.Therefore,the smaller the ratio the more resilient is the system to failures.In the limit where the ratio approach zero,the single-network features are recovered,i.e.,the transition becomes of second-order nature, with p c=1/<k>.When coupled scale-free networks are considered,a percolation transition,at nonzero p c,is obtained even for values of the degree exponent2<γ≤3.This is in contrast to the single network case,where a giant cluster is observed for any positive fraction p of nodes(see Sec.2).Analogously to ER networks,the coupling between scale-free networks significantly increases the vulnerability of the system,with a larger p c compared to the case of a single network.Since hubs can have a low-degree counterpart node,their vulnerability evinces with the coupling.In contrast to single networks,the broader the degree distribution the larger is p c,i.e.,the smaller the fraction of nodes that needs to be removed to fully fragment the system.In general,the coupling between networks,increase the vulnerability of the system due to the cascade of failures that can be activated by removing a small fraction of nodes.Catastrophic Cascade of F ailures in Interdependent Networks 9Fig.6.–For two coupled Erd˝o s-R´e nyi networks,the threshold p c and the fraction of nodes in the mutual giant cluster at the transition,µ∞,are plotted as a function of the ratio between the average degrees of networks A and B,<k >A /<k >B .Findings are summarized in Fig.7.While for a single network,a second-order percolation transition is observed where the order parameter vanishes smoothly at criticality,for coupled systems,the transition occurs for larger p c (lower fraction of removed nodes),and is rather abrupt,characterized by a discontinuity in the order parameter.In the next section,the case of partially dependent networks is reviewed.SingleCascades,SuddenbreakdownCoupled 1st order1012nd order p c p p c P ∞Fig.7.–The order parameter (fraction of sites in the giant component)as a function of the fraction of left nodes in single and coupled networks.For single networks,above the percolation threshold a smooth increase of the order parameter is observed,with a critical exponent β.However,for strongly coupled networks,an abrupt transition is observed,characteristic of first-order transitions,due to the cascade phenomenon discussed in the text.10H.E.Stanleyp P 8Fig.8.–Order parameter P ∞as a function of the fraction of nodes left p for Erd˝o s-R´e nyi and scale-free network (γ=2.7)with strong and weak coupling.Both systems contain 5×104nodes.For both network types,first-order transitions occurs for strong coupling in contrast to second order transition in weak coupling.(After Parshani et al.[16]).4.–Partially Dependent NetworksFrequently,in real-world systems not all the nodes in one network are interdependent on the other network and vice versa,via bidirectional links.Instead,some nodes may be autonomous and independent on nodes from the other networks.For example,in the communication/power-grid system,a fraction of servers may be protected by emergency power supplies which are activated when the local power station is shut down.Parshani et al.[16]have studied the behavior of partially interdependent networks under random failure and found quantitatively how reducing the coupling improves the robustness of the system against random attack and how percolation transition changes from first to second order.The model introduced in Sec.3,can now be generalized to account for partially coupling between networks.Two coupled networks are then considered,A and B ,with degree distributions P A (k )and P B (k ),respectively.In network A ,a fraction of nodes q A depends on nodes from network B .Likewise,in network B ,a fraction of nodes q B depends on nodes in A .In the limit of q A =q B =1the fully interdependent system,discussed above,is recovered.Following the procedure described before,1−p nodes are randomly removed and the percolation properties of the system studied.If a removed node has a dependent in the other network,this node is also removed.All nodes linked to the correspondent giant component solely through the removed nodes are also considered inactive.In Fig.8we show the curves of the fraction of sites belonging to the mutually connected giant component,P ∞,as a function of p .ER and scale-free networks (with γ=2.7)have beenCatastrophic Cascade of F ailures in Interdependent Networks11010203040n 00.10.20.30.40.5P n (a)051015n00.050.10.150.20.25P n (b)Fig.9.–The giant component as a function of the number of cascade failures in two coupled Erd˝o s-R´e nyi networks,with N A =N B =8×105and <k >A =<k >B =2.5,for two different coupling strengths.(a)p =0.7455,q A =0.7,and q B =0.6(first order).(b)p =0.605,q A =0.2,and q B =0.75(second order).Solid lines correspond to results obtained from theory based on generating functions.(After Parshani et al.[16]).considered,with strong (q A =q B =q =0.8)and weak (q =0.1)coupling.For both systems in the strong coupling case,the robustness of the coupled networks systems is similar to that observed in the limit of q =1(see Sec.3).Reducing the coupling leads to a second order phase transition similar to single networks (the case of q =0).Increasing the coupling leads to a percolation transition at larger p c and to a change from second order,under weak coupling,to first order for strong coupling.The giant component as a function of the number of iterations in the cascade of failures,close to the transition point,for a strong and weak coupled system is shown in Fig.9.Two coupled ER networks have been considered,with N A =N B =8×105and the same average degree of nodes,with strong and weak coupling.In the strong coupling regime,a plateau is obtained followed by an abrupt decrease of the order parameter,similar to the case q =1of the a first-order transition.While,in the weak coupling regime,the order parameter smoothly vanish with the number of iterations of the cascades.Results for partially dependent networks are summarized in the two-parameter phase diagram of Fig.10.In the horizontal axis is the fraction of removed nodes in network A ,1−p ,while in the vertical one is the fraction of independent nodes in the same network,1−q A .The value of q B is chosen to be q B =1.Two different regimes can be identified in the diagram.In the right side of the diagram (below the curve),no finite giant component exists in network B at p c .That is,the system is below p c .In the left side,a giant component exists in the system.This two different regimes are separated by a phase transition line.When one crosses this line,a percolation transition is observed.The solid line corresponds to a first-order transition,characterized by a jump in the order parameter at the transition point.The dashed line stands for a second-order transition,where a continuous change of the order parameter is obtained.For a fully coupled system,12H.E.StanleyFig.10.–Diagram for percolation in two coupled networks in the two-parameter space,namely, the fraction of independent nodes in one network and the fraction of removed nodes in the same network.For weak coupling(large fraction of independent nodes)the transition is continuous (dashed line).Below a certain fraction of independent nodes,due to the coupling strength,the transition becomes discontinuous(first order).The vulnerability of the system increases with increasing in the coupling strength.(After Parshani et al.[16]).1−q A=0,only a small fraction of nodes needs to be removed to observe a fragmentation of the whole system in a discontinuous way.As one increases the fraction of independent nodes in the system,the fraction of removed nodes at the transition increases,meaning that the vulnerability of the system increases.Thisfirst-order line of the transition ends at a critical point above which the transition is no longer abrupt,being smooth and of second-order nature.5.–Final remarksIn this review,the recent work by Buldyrev et al.[15]and Parshani et al.[16]on the percolation properties of interdependent networks have been revisited.The review is based on a Lecture given by S.Havlin in the June2010Varenna School.Modern systems tend to be more coupled together.Infrastructures,biology,sociology and economy sys-tems are interconnected such that events taking place in one system may propagate and influence other coupled networks.Recent studies[15,16]show that coupling between systems increases their vulnerability to random failure.Properties of interdependent net-works significantly differ from the ones of a single network.In this article,these results are reviewed and the main properties are discussed.When a system of two interdepen-Catastrophic Cascade of F ailures in Interdependent Networks13 dent networks is considered,where nodes in one network have a bidirectional coupling with nodes in the other,the percolation properties are significantly affected.Due to cou-pling,not only the transition threshold is increased but also the order of the transition changes.The presence of interdependency between nodes in different networks,such that if one of the nodes is inactive the other can not function as well,leads to catastrophic effects when some nodes are removed from the system.A cascade of events is then ignited leading to an abrupt decomposition of the mutually connected giant component. 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