Robustness of complex networks with the local protection strategy against cascading failures

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a r X i v :c o n d -m a t /0008064v 1 [c o n d -m a t .d i s -n n ] 3 A u g 2000Error and attack tolerance of complex networksR´e ka Albert,Hawoong Jeong,Albert-L´a szl´o Barab´a siDepartment of Physics,University of Notre Dame,Notre Dame,IN 46556Many complex systems display a surprising degree of tolerance against er-rors.For example,relatively simple organisms grow,persist and reproduce de-spite drastic pharmaceutical or environmental interventions,an error tolerance attributed to the robustness of the underlying metabolic network [1].Complex communication networks [2]display a surprising degree of robustness:while key components regularly malfunction,local failures rarely lead to the loss of the global information-carrying ability of the network.The stability of these and other complex systems is often attributed to the redundant wiring of the func-tional web defined by the systems’components.In this paper we demonstrate that error tolerance is not shared by all redundant systems,but it is displayed only by a class of inhomogeneously wired networks,called scale-free networks.We find that scale-free networks,describing a number of systems,such as the World Wide Web (www)[3–5],Internet [6],social networks [7]or a cell [8],display an unexpected degree of robustness,the ability of their nodes to com-municate being unaffected by even unrealistically high failure rates.However,error tolerance comes at a high price:these networks are extremely vulnerable to attacks,i.e.to the selection and removal of a few nodes that play the mostimportant role in assuring the network’s connectivity.Such error tolerance and attack vulnerability are generic properties of communication networks,such as the Internet or the www,with complex implications on assuring information readiness.The increasing availability of topological data on large networks,aided by the computer-ization of data acquisition,has lead to major advances in our understanding of the generic aspects of network structure and development[9–16].The existing empirical and theoretical results indicate that complex networks can be divided into two major classes based on their connectivity distribution P(k),giving the probability that a node in the network is connected to k other nodes.Thefirst class of networks is characterized by a P(k)that is peaked at an average k and decays exponentially for large k.The most investigated examples of such exponential networks are the random graph model of Erd˝o s and R´e nyi[9,10]and the small-world model of Watts and Strogatz[11],both leading to a fairly homogeneous network,in which each node has approximately the same number of links,k≃ k .In contrast,results on the world-wide web(www)[3–5],Internet[6]and other large networks[17–19]indicate that many systems belong to a class of inhomogeneous networks,referred to as scale-free networks,for which P(k)decays as a power-law,i.e.P(k)∼k−γ,free of a characteristic scale.While the probability that a node has a very large number of connections(k>> k ) is practically prohibited in exponential networks,highly connected nodes are statistically significant in scale-free networks(see Fig.1).We start by investigating the robustness of the two basic network models,the Erd˝o s-R´e nyi(ER)model[9,10]that produces a network with an exponential tail,and the scale-free model[17]with a power-law tail.In the ER model wefirst define the N nodes,and then connect each pair of nodes with probability p.This algorithm generates a homogeneous net-work(Fig.1),whose connectivity follows a Poisson distribution peaked at k and decaying exponentially for k>> k .The inhomogeneous connectivity distribution of many real networks is reproduced by the scale-free model[17,18]that incorporates two ingredients common to real networks: growth and preferential attachment.The model starts with m0nodes.At every timestep t a new node is introduced,which is connected to m of the the already existing nodes.The probabilityΠi that the new node is connected to node i depends on the connectivity k i ofthat node,such thatΠi=k i/jk j.For large t the connectivity distribution is a power-lawfollowing P(k)=2m2/k3.The interconnectedness of a network is described by its diameter d,defined as the av-erage length of the shortest paths between any two nodes in the network.The diameter characterizes the ability of two nodes to communicate with each other:the smaller d is,the shorter is the expected path between works with a very large number of nodes can have a rather small diameter;for example the diameter of the www,with over800million nodes[20],is around19[3],while social networks with over six billion individuals are be-lieved to have a diameter of around six[21].To properly compare the two network models we generated networks that have the same number of nodes and links such that P(k)follows a Poisson distribution for the exponential,and a power-law for the scale-free network.Error tolerance—To address the networks’error tolerance,we study the changes in the diameter when a small fraction f of the nodes is removed.The malfunctioning(absence)of a node in general increases the distance between the remaining nodes,since it can eliminate some paths that contribute to the system’s interconnectedness.Indeed,for the exponential network the diameter increases monotonically with f(Fig.2a),thus,despite its redundant wiring(Fig.1),it is increasingly difficult for the remaining nodes to communicate with each other.This behavior is rooted in the homogeneity of the network:since all nodes have approximately the same number of links,they all contribute equally to the network’s diameter,thus the removal of each node causes the same amount of damage.In contrast,we observe a drastically different and surprising behavior for the scale-free network(Fig.2a): the diameter remains unchanged under an increasing level of errors.Thus even when as high as5%of the nodes fail,the communication between the remaining nodes in the network is unaffected.This robustness of scale-free networks is rooted in their extremely inhomogeneous connectivity distribution:since the power-law distribution implies that the majority of nodes have only a few links,nodes with small connectivity will be selected with much higher probability,and the removal of these”small”nodes does not alter the path structure of the remaining nodes,thus has no impact on the overall network topology.Attack survivability—An informed agent that attempts to deliberately damage a net-work,such as designing a drug to kill a bacterium,will not eliminate the nodes randomly,but will rather target the most connected nodes.To simulate an attack wefirst remove the most connected node,and continue selecting and removing nodes in the decreasing order of their connectivity k.Measuring the diameter of an exponential network under attack,wefind that,due to the homogeneity of the network,there is no substantial difference whether the nodes are selected randomly or in decreasing order of connectivity(Fig.2a).On the other hand,a drastically different behavior is observed for scale-free networks:when the most connected nodes are eliminated,the diameter of the scale-free network increases rapidly, doubling its original value if5%of the nodes are removed.This vulnerability to attacks is rooted in the inhomogeneity of the connectivity distribution:the connectivity is ensured by a few highly connected nodes(Fig.1b),whose removal drastically alters the network’s topology,and decreases the ability of the remaining nodes to communicate with each other.Network fragmentation—When nodes are removed from a network,clusters of nodes, whose links to the system disappear,can get cut offfrom the main cluster.To better understand the impact of failures and attacks on the network structure,we next investigate this fragmentation process.We measure the size of the largest cluster,S,shown as a fraction of the total system size,when a fraction f of the nodes are removed either randomly or in an attack mode.Wefind that for the exponential network,as we increase f,S displays a threshold-like behavior such that for f>f c≃0.28we have S≃0.A similar behavior is observed when we monitor the average size s of the isolated clusters(i.e.all the clusters except the largest one),finding that s increases rapidly until s ≃2at f c,after which it decreases to s =1.These results indicate the following breakdown scenario(Fig.4):For small f,only single nodes break apart, s ≃1,but as f increases,the size of the fragments that fall offthe main cluster increases,displaying a singular behavior at f c.At f c the system practically falls apart,the main cluster breaking into small pieces,leading to S≃0,and the size of the fragments, s ,peaks.As we continue to remove nodes(f>f c),we fragment these isolated clusters,leading to a decreasing s .Since the ER model is equivalent to theinfinite dimensional percolation[22],the observed threshold behavior is qualitatively similar to the percolation critical point.However,the response of a scale-free network to attacks and failures is rather different (Fig.3b).For random failures no threshold for fragmentation is observed,rather the size of the largest cluster slowly decreases.The fact that s ≃1for most f indicates that the network is deflated by nodes breaking offone by one,the increasing error level leading to the isolation of single nodes only,not clusters of nodes.Thus,in contrast with the catastrophic fragmentation of the exponential network at f c,the scale-free network stays together as a large cluster for very high values of f,providing additional evidence of the topological stability of these networks under random failures.This behavior is consistent with the existence of an extremely delayed critical point(Fig.3),the network falling apart only after the main cluster has been completely deflated.On the other hand,the response to attack of the scale-free network is similar(but swifter)to the response to attack and failure of the exponential network(Fig.3b):at a critical threshold f sf c≃0.18,smaller than the value f e c≃0.28observed for the exponential network,the system breaks apart,forming many isolated clusters(Fig.4).While great efforts are being made to design error tolerant and low yield components for communication systems,little is known about the effect of the errors and attacks on the large-scale connectivity of the network.To demonstrate the impact of our model based studies to these systems,next we investigate the error and attack tolerance of two networks of increasing economic and strategic importance:the Internet and the www.Recently Faloutsos et al.[6]investigated the topological properties of the Internet at the router and inter-domain level,finding that the connectivity distribution follows a power-law, P(k)∼k−2.48.Consequently,we expect that it should display the error tolerance and attack vulnerability predicted by our study.To test this,we used the latest survey of the Internet topology,giving the network at the inter-domain(autonomous system)level.Indeed,wefind that the diameter of the Internet is unaffected by the random removal of as high as2.5%of the nodes(an order of magnitude larger than the failure rate(0.33%)of the Internet routers[23]),while if the same percentage of the most connected nodes are eliminated(attack),d more than triples(Fig.2b).Similarly,the large connected cluster persists for high rates of random node removal,but if nodes are removed in the attack mode,the size of the fragments that break offincreases rapidly,the critical point appearing at f I c≃0.03(Fig.3b).The www forms a huge directed graph whose nodes are documents and edges are the URL hyperlinks that point from one document to another,its topology determining the search engines’ability to locate information on it.The www is also a scale-free network:the probabilities P out(k)and P in(k)that a document has k outgoing and incoming links follow a power-law over several orders of magnitude,i.e.P(k)∼k−γ,withγin=2.1andγout=2.45 [3,4,24].Since no complete topological map of the www is available,we limited our study to a subset of the web containing325,729nodes and1,469,680links( k =4.59)[3].Despite the directedness of the links,the response of the system is similar to the undirected networks we investigated earlier:after a slight initial increase,d remains constant in the case of random failures,while it increases for attacks(see Fig.2c).The network survives as a large cluster under high rates of failure,but the behavior of s indicates that under attack the system abruptly falls apart at f w c=0.067(Fig.3c).In summary,wefind that scale-free networks display a surprisingly high degree of toler-ance against random failures,a property not shared by their exponential counterparts.This robustness is probably the basis of the error tolerance of many complex systems,ranging from cells[8]to distributed communication systems.It also explains why,despite frequent router problems[23],we rarely experience global network outages or,despite the temporary unavailability of many webpages,our ability to surf and locate information on the web is unaffected.However,the error tolerance comes at the expense of attack survivability:the diameter of these networks increases rapidly and they break into many isolated fragments when the most connected nodes are targeted.Such decreased attack survivability is useful for drug design[8],but it is less encouraging for communication systems,such as the Internet or the www.While the general wisdom is that attacks on networks with distributed resource management are less successful,our results indicate that the topological weaknesses of thecurrent communication networks,rooted in their inhomogeneous connectivity distribution, have serious effects on their attack survivability,that could be exploited by those seeking to damage these 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The exponential network a is rather homogeneous,i.e.most nodes have approximately the same number of links.In contrast,the scale-free network b is extremely inhomogeneous:while the ma-jority of the nodes have one or two links,a few nodes have a large number of links,guaranteeing that the system is fully connected.We colored with red thefive nodes with the highest number of links, and with green theirfirst neighbors.While in the exponential network only27%of the nodes are reached by thefive most connected nodes,in the scale-free network more than60%are,demonstrat-ing the key role the connected nodes play in the scale-free network.Note that both networks contain 130nodes and215links( k =3.3).The network visualization was done using the Pajek program for large network analysis<http://vlado.fmf.uni-lj.si/pub/networks/pajek/pajekman.htm>.0.000.010.021015200.000.010.020510150.000.020.044681012abcfdInternetwwwAttackFailureAttackFailureSFE AttackFailure FIG.2.Changes in the diameter of the network as a function of the fraction of the removed nodes.a ,Comparison between the exponential (E)and scale-free (SF)network models,each containing N =10,000nodes and 20,000links (i.e. k =4).The blue symbols correspond to the diameter of the exponential (triangles)and the scale-free (squares)network when a fraction f of the nodes are removed randomly (error tolerance).Red symbols show the response of the exponential (diamonds)and the scale-free (circles)networks to attacks,when the most connected nodes are removed.We determined the f dependence of the diameter for different system sizes (N =1,000,5,000,20,000)and found that the obtained curves,apart from a logarithmic size correction,overlap with those shown in a ,indicating that the results are independent of the size of the system.Note that the diameter of the unperturbed (f =0)scale-free network is smaller than that of the exponential network,indicating that scale-free networks use more efficiently the links available to them,generating a more interconnected web.b ,The changes in the diameter of the Internet under random failures (squares)or attacks (circles).We used the topological map of the Internet,containing 6,209nodes and 12,200links ( k =3.4),collected by the National Laboratory for Applied Network Research </Routing/rawdata/>.c ,Error (squares)and attack (circles)survivability of the world-wide web,measured on a sample containing 325,729nodes and 1,498,353links [3],such that k =4.59.012301<s > a n d S0.00.20.40120.00.20.401210-110101102f<work fragmentation under random failures and attacks.The relative size of the largest cluster S (open symbols)and the average size of the isolated clusters s (filled symbols)in function of the fraction of removed nodes f for the same systems as in Fig.2.The size S is defined as the fraction of nodes contained in the largest cluster (i.e.S =1for f =0).a ,Fragmentation of the exponential network under random failures (squares)and attacks (circles).b ,Fragmentation of the scale-free network under random failures (blue squares)and attacks (red circles).The inset shows the error tolerance curves for the whole range of f ,indicating that the main cluster falls apart only after it has been completely deflated.Note that the behavior of the scale-free network under errors is consistent with an extremely delayed percolation transition:at unrealistically high error rates (f max ≃0.75)we do observe a very small peak in s ( s max ≃1.06)even in the case of random failures,indicating the existence of a critical point.For a and b we repeated the analysis for systems of sizes N =1,000,5,000,and 20,000,finding that the obtained S and s curves overlap with the one shown here,indicating that the overall clustering scenario and the value of the critical point is independent of the size of the system.Fragmentation of the Internet (c )and www (d ),using the topological data described in Fig.2.The symbols are the same as in b .Note that s in d in the case of attack is shown on a different scale,drawn in the right side of the frame.While for small f we have s ≃1.5,at f w c=0.067the average fragment size abruptly increases,peaking at s max ≃60,then decays rapidly.For the attack curve in d we ordered the nodes in function of the number of outgoing links,k out .Note that while the three studied networks,the scale-free model,the Internet and the www have different γ, k and clustering coefficient [11],their response to attacks and errors is identical.Indeed,we find that the difference between these quantities changes only f c and the magnitude of d ,S and s ,but not the nature of the response of these networks to perturbations.f≈0.05f≈0.18f≈0.45 FIG.4.Summary of the response of a network to failures or attacks.The insets show the cluster size distribution for various values of f when a scale-free network of parameters given in Fig.3b is subject to random failures(a-c)or attacks(d-f).Upper panel:Exponential networks under random failures and attacks and scale-free networks under attacks behave similarly:for small f clusters of different sizes break down,while there is still a large cluster.This is supported by the cluster size distribution:while we see a few fragments of sizes between1and16,there is a large cluster of size9,000(the size of the original system being10,000).At a critical f c(see Fig.3)the network breaks into small fragments between sizes1and100(b)and the large cluster disappears. At even higher f(c)the clusters are further fragmented into single nodes or clusters of size two. Lower panel:Scale-free networks follow a different scenario under random failures:The size of the largest cluster decreases slowly asfirst single nodes,then small clusters break off.Indeed,at f=0.05only single and double nodes break off(d).At f=0.18,when under attack the network is fragmented(b),under failures the large cluster of size8,000coexists with isolated clusters of size1through5(e).Even for unrealistically high error rate of f=0.45the large cluster persists, the size of the broken-offfragments not exceeding11(f).。

电力线的点云聚类方法Point cloud clustering is an important and challenging task in various fields, including the analysis of power lines. 电力线的点云聚类是各种领域中重要且具有挑战性的任务。

One perspective to consider in addressing the point cloud clustering problem for power lines is the use of advanced machine learning and deep learning techniques. This involves leveraging algorithms such as k-means clustering, DBSCAN, or hierarchical clustering to extract meaningful patterns and structures from the point cloud data. Additionally, deep learning methods like convolutional neural networks (CNNs) and recurrent neural networks (RNNs) can be employed to further improve the accuracy and efficiency of power line point cloud clustering. 一种应对电力线点云聚类问题的角度是使用先进的机器学习和深度学习技术。

这涉及利用k均值聚类、DBSCAN或分层聚类等算法从点云数据中提取有意义的模式和结构。

此外，还可以采用卷积神经网络（CNN）和循环神经网络（RNN）等深度学习方法，以进一步提高电力线点云聚类的准确性和效率。

第45卷第3期2019年6月兰州理工大学学报Journal of Lanzhou University of TechnologyVol.45No.3Jun.2019文章编号：1673-5196(2019)03-0101-07聚类系数指标对复杂网络鲁棒性的影响分析卢鹏丽1,董瑚1,曹乐2（1.兰州理工大学计算机与通信学院，甘肃兰州730050； 2.天水师范学院电子信息与电气工程学院，甘肃兰州741000）摘要：分析了采用度分布相同且聚类系数不同的三种类型网络（中性网络、同配网络和异配网络）在遇到随机故障或者蓄意攻击时，网络的初始聚类系数变化对网络鲁棒性的影响.实验分析表明，网络的初始聚类系数越大，网络在受到随机故障或蓄意攻击时网络中最大连通子图的直径和网络中最大连通子图的平均路径长度的起伏也就越大.初始聚类系数的变化在异配网中对网络鲁棒性的作用最明显，中性网次之，对同配网的鲁棒性不明显.关键词：复杂网络；鲁棒性；聚类系数中图分类号：TP202文献标志码：AAnalysis of influence of clustering coefficient as itsindex on robustness of complex networkLU Peng-li1，DONG Men1，CAO Le2(1.College of Computer and Communication,Lanzhou Univ,of Tech.,Lanzhou730050,China； 2.School of Electronic Information and Electrical Engineering,Tianshui Normal University,Tianshui741000,China)Abstract:The complex networks can be divided into three types according to chaining rules of their nodes,namely neutral network,assortative network,and hetero-assortative network.In this paper,three types of network with identical degree distribution and different clustering coefficient are used to analyze the influence of their initial clustering coefficient on their robustness when they are subjected to random failure and deliberate attack.Experimental analysis shows that the larger the initial clustering coefficient of the network is，the larger the fluctuation of the diameter and average path length of the maximum connect­ed subgraph in network will be when the network is subj ected to random failures or deliberate attacks. And the effect of initial clustering coefficient in hetero-assortative network on the network robustness will be most obvious9the effect will be less for neutral network9and there will be no obvious effect for assorta­tive network.Key words:complex network；robustness；clustering coefficient大数据在人们生活中扮演的角色越来越重要,复杂网络和复杂系统也得到人们进一步的重视•生活中的各种复杂系统都可以抽象作为复杂网络，复杂网络中节点数目众多，节点与节点之间的关系也千差万别•复杂网络的一项研究领域是网络部分结构失效对网络整体结构和功能的影响⑴，称为鲁棒性分析.Albert等⑵分析了小世界网（WS模型）和无标度网（EA模型）在遭到蓄意攻击或随机故障时的网收稿日期：2017-09-27基金项目：国家自然科学基金（11361033）作者简介：卢鹏丽（1973-）,女，甘肃酒泉人，博士，教授.络鲁棒性，并对万维网的鲁棒性进行了分析•结果显示小世界网在蓄意攻击和随机故障两种情况下的鲁棒性差异不是很大，无标度网和万维网对于随机故障的鲁棒性明显优于对蓄意攻击的鲁棒性，主要原因是两种网络的结构分布差异较大.Paolo等⑶建立了一种基于动态方法的模型，在动态模型下对WS小世界网和EA无标度网进行鲁棒性分析，提出了无标度网的不均匀性・Liu等⑷对中国九江炼油系统进行了鲁棒性分析，得出真实系统中具有均匀分布的网络鲁棒性更高.Bansanl等⑸提出了同配网络、异配网络和中性网络的概念,并分析了三种网络的鲁棒性.Schultz等⑹提出变量梯度法对复杂・102・兰州理工大学学报第45卷网络进行稳定性的判断・Iyer等⑺除了采用介数中心性,还加入了紧密度和特征向量等全局指标,分析合成网络和真实网络遭到随机故障和蓄意攻击时的鲁棒性.已有的文献多是对于复杂网络最大连通度的分析，本文主要采取最大连通子图的直径和最大连通子图的平均路径长度作为衡量标准⑻，全面分析了具有相同度分布且聚类系数不同的中性网络、异配网络和同配网络的鲁棒性•复杂网络的随机故障和蓄意攻击在文献［9-11］中已经有详细的描述，本文重点分析聚类系数在不同网络攻击中的表现.1基本概念G（V,E）表示一个无向无权的简单网络，其中V =｛“，巳2，•・•，"｝表示G中节点的集合；E｛（v i9Vj）I3伯GV｝是G中边的集合，且|V|=〃|E|=m；A是其对应的邻接矩阵，如果节点s和口之间有边存在,则其元素Aij=\,否则Aij=0.定义1（网络的直径D）—般定义两节点G汀）间的最短距离心［⑵为连接两者的最短路径的边的数目；网络的直径为所有两点间的最大距离,记为D：13］,即：D=max（1）（心）定义2（平均路径长度L）网络的平均路径长度L是所有节点对之间距离的平均值，即:L=——--------工右（2）y N（N—1）3其中:N为网络节点的总数目；平均路径长度L：13］描述网络中节点间的离散程度.定义3（聚类系数C）聚类系数C用来描述网络中节点的聚集情况，即网络有多紧密•一般地，假设网络中的一个节点i通过局条边与其他节点相连接虫是节点z的邻居节点数目•如果局个节点之间互相连接，它们之间存在局（化一1）/2条边，而这局个节点之间实际存在的边数£与总的可能存在边数之比就是节点，的聚类系数G,即：G=怂（铝1）（3）一个网络的聚类系数a⑶就是网络中所有节点的聚类系数的平均值，即：C气％⑷显然有O<C<1,只有在全连通网络中,聚类系数才能等于1,通常情况下一般均小于1.在完全随机网络中,C〜NT,其中N为网络节点的总数目.定义4（最大连通度&喚）最大连通度Gnax M 是指当网络受到攻击或者干扰时，在所剩仍具有连接能力网络中，其中所含节点数目最多的子网络中的节点数占所剩下节点数目的比例，即:其中:是最大连通子图的节点个数;N'是所有连通子图的节点数总和.2复杂网络的鲁棒性分析2.1复杂网络的结构对于一个复杂网络，如果网络中连接度大的节点总是倾向于与连接度大的节点连接，那么这种网络称为同配网络;如果网络中连接度大的节点总是倾向于与连接度小的节点连接，那么这种网络称为异配网络;如果网络中两个节点之间是否有边相连与这两个节点的连接度无关，那么这种网络称为中性网络•图1〜3形象地描述了中性网络、同配网络和异配网络在受到蓄意攻击和随机故障后网络的连通状况的仿真结果•其中蓄意攻击是指网络中的特定节点（即关键节点）发生故障以后网络的连通情况，而随机故障是指网络中任意节点发生故障以后网络的连通情况.（C）中性网随机故障图1中性网在受到蓄意攻击或随机故障前后的连通状态Fig.1Connective state of neutral network before andafter intentional attack or random fault通过仿真结果的对照可知,蓄意攻击对网络连通度的影响明显大于随机故障对网络连通度的影响.中性网络节点之间的连接并无明确的规律，故对抗蓄意攻击和随机故障时表现出很大的不明确性.第3期卢鹏丽等:聚类系数指标对复杂网络鲁棒性的影响分析・103・同配网中关键节点总是相互连接在一起，故同配网络在蓄意攻击时显得异常脆弱•而异配网在蓄意攻击时显示出很强的健壮性.(b)同配网蓄意攻击(c)同配网随机故障图2同配网在受到蓄意攻击或随机故障前后的连通状态Fig.2Connective state of assortative network before and after intentional attack or random fault(b)异配网蓄意攻击图3异配网在受到蓄意攻击或随机故障前后的连通状态Fig.3Connective state of hetero-assortative network be­fore and after intentional attack or random fault对多数实际网络进行研究显示,互联网以及蛋白质交换网络等生物网络是异配网络，而人际关系网以及电影演员合作网络等许多现实网络是同配网络，包括复杂网络中著名的无标度网络也属于同配网络•而不同的在线社会网络可能是同配、异配或者中性网络•例如包含7亿多节点的Facebook网络呈现出同配性特征，大型在线社交网络Cyworld却是异配网络⑸.2.2复杂网络的鲁棒性对于现实中的复杂系统,总是希望复杂系统拥有一定的鲁棒性，也就是复杂系统对外界的各种干扰具备一定的抗干扰能力•在实际生活中，系统面临各种各样的主观或者客观的干扰是不可避免的，鲁棒性和脆弱性分别是从稳定指标与失效指标的角度来表征网络的特性,两者相辅相成•鲁棒性越大，其脆弱性就越小,即抗毁能力越强;鲁棒性越小,其脆弱性越大，即抗毁能力越弱.先前的各种鲁棒性分析中都围绕着网络的最大连通度进行•网络的鲁棒性通常与网络的最大连通子图有关，所以网络中最大连通子图的直径和平均路径长度是网络鲁棒性分析的指标.2.3算法介绍复杂网络由于节点众多且结构复杂，网络在构造时很难出现构造的两个网络结构一样的情况，往往构造出来的网络结构之间有较大的差异•为了更准确地分析聚类系数指标对复杂网络鲁棒性的影响,本文选取待分析网络时让待分析网络具有相同的节点度分布，使得构造出的网络之间结构差异较小•网络都由I000个节点度已知情况下的节点，根据同配网、异配网和中性网的连接规律，将节点连接成所对应的同配网、异配网和中性网•为确保生成网络聚类系数的一般性,根据以上网络生成规则，生成100组同配网、异配网和中性网，并对它们的初始聚类系数进行了统计•统计发现节点数为I000的中性网络，初始聚类系数主要分布在0.001到0.003之间，同配网的初始聚类系数主要分布在0.008到0.012之间，异配网的初始聚类系数主要分布在0.0015到0.0025之间，故实验中采用了具有特殊初始聚类系数的网络作为待分析网络.本文主要采取网络的最大连通子图的直径和最大连通子图的平均路径长度作为衡量标准，全面地分析了具有相同度分布且聚类系数不同的中性网络、异配网络和同配网络的鲁棒性，攻击方式分为随机故障和蓄意攻击•算法如下：1)随机生成具有I000个节点的网络，计算网络的度分布.2)根据随机网络的度分布，对网络进行重连,生成多组相应的同配网络、中性网络和异配网络.3)计算多组同配网络、异配网络和中性网络的聚类系数，并在同一种网络中取出三组聚类系数不同的网络，作为待分析网络.・104・兰州理工大学学报第45卷4)分别按照随机故障和蓄意攻击两种方式确定需要在待分析网络中删除的节点，随机故障时随机选取节点进行删除，蓄意攻击时选取节点度较大的节点优先进行删除.同时将待删除节点以及待删除节点所连接的边删除.5)判断当前网络的最大连通子图，计算f、D 和L.其中J为受到攻击时节点数与原网络节点数的比值Q为网络受到攻击后最大子图的直径丄为网络受到攻击后最大子图的平均最短路径.6)计算待分析网络中的节点数•若节点数为0,则进行下一步，否则返回步骤4.7)算法结束.3实验与分析实验结果分别如图4〜6所示，其中L为当前最大连通子图的平均路径长度,D为当前最大连通子图的直径.图4分别为初始聚类系数为0.001、0.002和0.003的中性网络在随机故障和蓄意攻击时网络中L和D的变化情况•图5分别为初始聚类系数为0.008,0.010和0.012的同配网络在受到随机故障和蓄意攻击时网络中L和D的变化情况•图6分别为初始聚类系数为0・0015、0・0020和0.0025的异配网络在受到随机故障和蓄意攻击时网络中L 和D的变化情况.3.1中性网络图4a比较了三种中性网络在受到随机故障时网络中D的变化情况•在丢失少量节点时，网络中D变化不明显.初始聚类系数越大的中性网络,D的起伏越大•图4b比较了三种中性网络在受到随机故障时网络中L的变化情况•图4c比较了三种中性网络在受到蓄意攻击时网络中D的变化情况，且初始聚类系数越大的网络,D越先出现起伏现象.图4d比较了三种中性网络在受到蓄意攻击时网络中L的变化情况，相比于网络随机故障,初始聚类系数越小的网络在被攻击时所产生的L的最大值要高于初始聚类系数大的网络.根据以上分析得出，当移除节点数目较少时,无论是随机故障还是蓄意攻击，网络的D和L都呈现出一个缓慢增值的趋势,但在移除节点数目到达一定数量时,D和L的变化明显，蓄意攻击在移除节点40%左右出现浮动，随机故障在移除节点60%左右出现浮动，蓄意攻击对网络的破坏明显大于随机故障•中性网络在少量节点丢失时，网络最大子图的直径与平均最短路径都缓慢地增长•但在大量节点丢失时，网络的D和L都发生剧烈的变化•这是由于中性网络的不确定性造成的，中性网络中节点之间的连边没有什么明确的关系,在受到大量节点丢失时,节点之间的离散程度明显变化很大,且聚类系数越大的中性网络，在丢失节点后网络中的D和L 越大，节点间的离散程度越高,网络的鲁棒性越差.目M画中V*鴉屋移除节点百分比(a)随机故障时的D(b)随机故障时的厶目M画中V*鴉屋移除节点百分比(c)蓄意攻击时的D(d)蓄意攻击时的厶图4三种初始聚类系数不同的中性网络在受到攻击时网络中最大连通子图中的最大直径和平均最短路径变化情况Fig.4The maximum diameter and average shortest path change in the most Dalian subgraph of the neutralnetwork with three initial clustering coefficients inthe network underattack第3期卢鹏丽等:聚类系数指标对复杂网络鲁棒性的影响分析• 105 •1412108 6 4 2目M 画中V *鴉屋移除节点百分比(a)随机故障时的DO0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0移除节点百分比(a)随机故障时的D移除节点百分比(b)随机故障时的厶■■■■■■■ O6 5 4 3 2 1移除节点百分比(b)随机故障时的厶粗M 画中V *鴉屋粗M 画中V *鴉屋— 0.001 5——0.002 0一 0.002 5移除节点百分比(c)蓄意攻击时的D0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0移除节点百分比(c)蓄意攻击时的D移除节点百分比移除节点百分比(d)蓄意攻击时的厶(d)蓄意攻击时的厶图5三种初始聚类系数不同的同配网络在受到攻击时网络中最大连通子图中的最大直径和平均最短路径变化情况Fig. 5 the maximum diameter and average shortest pathchange of the most Dalian pass subgraph in the as ­sortative network under three different initial clus ­tering coefficients when attacked3.2同配网络同配网络的D 和L 在随机故障中体现出很高的稳定性•在丢失大量节点之前,D —直保持着一个图6三种初始聚类系数不同的异配网络在受到攻击时网络中最大连通子图中的最大直径和平均最短路径变化情况Fig. 6 The maximum diameter and average shortest pathchange of the most Dalian pass subgraph in thedisassortative network under three different initialclustering coefficients when attacked稳定的状态•随着丢失节点数目的增加(在丢失节点数50%时)网络中最大连通子图的直径才有微弱的 变化，且初始聚类系数越大的同配网络，直径的起伏・106・兰州理工大学学报第45卷越大•在丢失节点数75%左右时，网络被分割为大量小碎片,直径剧烈下降•这是由于同配网络中连接度大的节点优先与连接度大的节点连接，这种网络结构类似于无标度网，所以同配性网络在随机故障时,网络的D和L都呈现出一种稳定的态势.且初始聚类系数越大，网络在随机故障时出现的直径最大值越高,网络的离散程度越高•同配网络在遭到蓄意攻击时,网络的D和L变化剧烈,且在丢失节点占比35%左右时就出现明显的变化，所产生的D和L的最大值都远高于同配网络在受到随机故障时所产生的D和L.在同配网中初始聚类系数越大，网络中D和L的浮动越大，节点间的离散程度越高,网络的鲁棒性越差.3.3异配网络图6a比较了三种异配网络在受到随机故障时网络中D的变化情况.图6b比较了三种异配网络在受到随机故障时网络中L的变化情况•图6c比较了三种异配网络在受到蓄意攻击时D的变化情况.图6d比较了三种异配网络在受到蓄意攻击时，网络中L的变化情况•初始聚类系数越大，异配网络在遭到蓄意攻击时网络的直径起伏越大.异配网络的D和L在随机故障中体现的相对较稳定•在丢失大量节点前，网络的D和L—直保持着一个相对稳定的状态,但有小的起伏•随着丢失节点数目的增加(在丢失节点的数60%左右时)网络中的最大连通子图的直径开始剧烈下降，且下降的程度与初始的聚类系数无关•异配网络在少量的节点遭到蓄意攻击时，网络的最大连通子图的直径缓慢增长，在移除节点占比50%左右时，直径明显地增长，且初始聚类系数越大的网络直径的变化越大•网络的最大连通子图的平均最短路径呈现出一种缓慢增加的趋势，在移除节点占比60%左右时出现极值，平均路径长度开始缓慢下降•在异配网络的蓄意攻击中可以明显地看出聚类系数对网络中D 和L的影响较大，聚类系数越大的异配网D和L的变化越明显，节点间的离散程度越高,网络的鲁棒性越差.3.4综合分析中性网络和异配网络的初始网络直径都在9左右，初始平均路径长度都在5左右，且不会随着初始聚类系数的变化而发生变换.同配网的初始网络直径和初始平均路径长度相对较大，初始直径在22左右,初始平均路径长度在8.10左右,也不会随着初始聚类系数的变化而改变.由表1分析可知，在度分布相同的情况下，同配网的初始聚类系数明显大于中性网和异配网，中性网和异配网的初始聚类系数相似•为了进一步分析聚类系数指标对不同系统的鲁棒性的影响，取初始聚类系数相同的中性网和异配网进行比较.图7为移除节点百分比(b)随机故障时的厶目M画中V*鴉屋161412108642—•中性网络-一异配网络00.10.20.30.40.50.60.70.80.9 1.0移除节点百分比(d)蓄意攻击时的厶图7相同聚类系数的中性网和异配网在受到攻击时网络中最大连通子图中的最大直径和平均最短路径变化情况Fig.7The maximum diameter and the average shortest path change in the most Dalian pass graph in the networkwhen the neutral network and the disassortative net­work with the same clustering coefficients are attacked第3期卢鹏丽等:聚类系数指标对复杂网络鲁棒性的影响分析・107・表1三种不同网络在不同聚类系数下网络的初始直径和初始聚类系数Tab.1Initial diameter and initial clustering coefficient of three different networks under different clusteringcoefficients网络类型初始聚类系数初始直径初始路径长度0.00110 5.20中性网0.0029 5.450.00310 5.460.00820&10同配网0.01022&160.01224&140.00159 4.93异配网0.002010 4.890.00259 4.87初始聚类系数都为0.002的中性网和异配网在受到随机故障和蓄意攻击时网络中L和D的变化情况.图7a比较了中性网络和异配网络在受到随机故障时网络中D的变化情况.图7b比较了中性网络和异配网络在受到随机故障时网络中L的变化情况•图7c比较了中性网络和异配网络在受到蓄意攻击时D的变化情况.图7d比较了中性网络和异配网络在受到蓄意攻击时，网络中L的变化情况.图7中网络的节点数目、节点度分布和初始聚类系数都相同，只因为节点之间连线的方式存在差异就使得网络的鲁棒性有着巨大的差异.由以上分析可知，聚类系数越大，网络中D和L的起伏越大，网络的鲁棒性越差•对于异配网，初始聚类系数对网络受到随机故障和蓄意攻击时的影响最大，中性网络次之,同配网络虽然有很大初始聚类系数，却在随机故障和蓄意攻击时对其最大连通子图的直径和平均路径长度都没有很大的影响.4结语通过分析聚类系数对复杂网络中最大连通子图的直径和平均路径长度的影响可知•在度分布相同的情况下,聚类系数越大，网络的鲁棒性越差•且聚类系数在不同的网络中所体现出作用的大小也不同，在异配网中聚类系数对网络的鲁棒性的作用明显，中性网次之，同配网中受到聚类系数的影响最小•除聚类系数对网络中最大连通子图的直径和最大连通子图的平均路径长度的影响外，聚类系数对网络中其他相关方面的影响将来需要做进一步的研究和验证.参考文献：[1]PATEL S J,PATTEWAR T M.Software birthmark basedtheft detection of JavaScript programs using agglomerative clustering and frequent subgraph minming[C]//Embedded System(ICES),2014International Conference on.[S.1.]:IEEE,2014.[2]ALBERT R,JEONG H,SI A L.Error and attack tolerance ofcomplex networks[J].Nature,2000,406(4)：378-382.[3]CRUCITTI P,LATORA V,MARCHIORI Error andattack tolerance of complex networks[J].Physica 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robustnessRobustnessIntroduction:Robustness is a fundamental concept in various fields, including engineering, computer science, and biology. It refers to the ability of a system or organism to withstand and adapt to changing conditions, disturbances, or uncertainties. In this document, we will explore the concept of robustness, its significance in different domains, and strategies to enhance it.1. Importance of Robustness:Robustness plays a crucial role in ensuring the stability, reliability, and efficiency of systems. In engineering, it is essential to design robust structures and machines that can endure extreme conditions without significant damage. For example, in civil engineering, structures like buildings and bridges must be robust enough to withstand earthquakes, heavy winds, and other external forces.Similarly, in computer science, robustness is critical for ensuring the resilience and availability of software systems.Robust software is capable of handling unexpected user inputs, recovering from errors, and preventing crashes or system failures. This is especially important in mission-critical systems such as aviation, healthcare, and financial sectors.2. Factors Affecting Robustness:Various factors contribute to the robustness or vulnerability of a system. One crucial factor is redundancy, which involves duplicating critical components or functionalities. Redundancy can provide backup mechanisms that help the system to continue functioning even if some parts fail. For example, in electric power distribution systems, redundancy is employed to minimize the impact of a single point of failure.Another factor is adaptability, which refers to the system's ability to adjust its behavior or configuration in response to changing conditions. Adaptive systems can detect deviations, learn from them, and modify their strategies accordingly. This adaptability is particularly important in dynamic environments, such as autonomous vehicles navigating through traffic or robotics systems faced with unpredictable scenarios.Furthermore, fault tolerance is a measure of a system's ability to continue operating even when certain components orprocesses fail. Fault-tolerant systems utilize mechanisms such as error detection, error recovery, and error correction to prevent complete system failures. These techniques are commonly employed in communication networks, where the failure of individual nodes should not disrupt the entire network.3. Enhancing Robustness:There are several strategies that can be employed to enhance the robustness of a system or organism:a. Redundancy: Introducing redundancies in critical components or processes can provide backup mechanisms to ensure continuous operation. Redundancy can be achieved through component duplication, introducing backup systems, or implementing failover mechanisms.b. Testing and Validation: Thorough testing and validation processes can help identify vulnerabilities and weaknesses in a system. By subjecting the system to various testing scenarios and analyzing its responses, developers can strengthen the system's resilience and prepare it for unexpected challenges.c. Error Handling and Recovery: Implementing robust error handling and recovery mechanisms is crucial for preventing system failures. Techniques such as exception handling, error logging, and graceful degradation can help the system recover from errors in a controlled manner while minimizing the impact on its overall operation.d. Adaptive Control: Adaptive control strategies enable the system to continuously monitor its performance and make real-time adjustments to optimize its behavior. This can involve algorithms that learn from past experiences, machine learning techniques, or feedback control systems that adjust parameters based on environmental changes.e. Security Measures: Enhancing the security of a system can significantly contribute to its robustness. Implementing secure authentication protocols, encryption algorithms, and access control mechanisms can safeguard the system against various threats and minimize vulnerabilities.Conclusion:Robustness is a vital characteristic that ensures the stability, reliability, and adaptability of systems in various domains. By incorporating redundancy, adaptability, fault tolerance, testing/validation, error handling, and security measures,system developers can enhance robustness and ensure optimal performance even under challenging conditions. As technology continues to advance, robustness will remain a critical consideration in designing and managing complex systems.。

仿真花不同类型的英文术语在仿真领域中，有许多不同类型的英文术语。

下面是一些常见的术语及其解释：1. Simulation (仿真): The imitation or representation of the operation or features of one system through the use of another system, typically a computer program. It is used to study, analyze, and predict the behavior of complexreal-world systems.2. Virtual Reality (虚拟现实): A computer-generated simulation of a three-dimensional environment that can be interacted with and experienced by a person. It typically involves the use of a head-mounted display and other sensory devices to create a sense of presence in thevirtual world.3. Augmented Reality (增强现实): An interactive experience that combines real-world elements with computer-generated sensory inputs, such as graphics, sound, or GPSdata. It enhances the user's perception of the real world by overlaying digital information onto the physical environment.4. Agent-based Modeling (基于代理的建模): A simulation technique that models the behavior of individual agents or entities and their interactions within a system. Agents can represent individuals, organizations, or other entities, and their behavior is governed by predefined rules or algorithms.5. Monte Carlo Simulation (蒙特卡洛仿真): A statistical technique that uses random sampling to model and analyze the behavior of complex systems. It is particularly useful for assessing the risk and uncertainty associated with decision-making processes.6. Discrete Event Simulation (离散事件仿真): A simulation technique that models the behavior of a system as a sequence of discrete events in time. It is commonly used to study systems with dynamic, time-dependent processes, such as manufacturing systems or transportationnetworks.7. Continuous Simulation (连续仿真): A simulation technique that models the behavior of a system as a continuous function of time. It is often used to study systems with continuous, time-dependent processes, such as fluid dynamics or electrical circuits.8. Sensitivity Analysis (敏感性分析): A technique used to assess the impact of changes in input parameters or assumptions on the output of a simulation model. It helps identify the most influential factors and understand the robustness of the model.9. Validation (验证): The process of comparing the behavior of a simulation model to the real-world system it represents. It involves verifying that the model accurately reproduces the observed behavior and meets the intended objectives.10. Optimization (优化): The process of finding the best possible solution to a problem within a given set ofconstraints. In simulation, optimization techniques are often used to identify the optimal configuration or parameter values that maximize or minimize a certain objective.这些术语涵盖了仿真领域的一些关键概念和技术。

基于社团划分的复杂网络级联抗毁攻击策略作者：丁超姚宏杜军彭兴钊李浩敏来源：《计算机应用》2014年第06期摘要：为研究在社团划分基础上复杂网络的级联抗毁攻击策略，采用节点及其邻居节点介数定义初始负荷，这种定义方式综合考虑了节点的信息，采用局部择优分配策略处理故障节点负荷，研究了网络耦合强度，WS（WattsStrogatz）小世界网络、BA（BarabásiAlbert）无标度网络、ER（ErdsRényi）随机网络、局域世界（WL）网络在社团划分攻击策略下抗毁性，以及不同攻击策略下具有重叠和非重叠社团结构网络的抗毁性。

仿真结果表明，网络的耦合强度与抗毁性成负相关；不同类型网络在快速分裂算法识别社团前提下，攻击介数最大节点时网络抗毁性最弱；具有重叠社团结构的网络在集团渗流算法（CPM）识别后，采用攻击重叠部分介数最大节点的策略时网络抗毁性最弱。

结论表明采用社团划分的攻击策略可以最大规模破坏网络。

关键词：攻击策略；社团划分；复杂网络；级联抗毁性；网络模型中图分类号： TP393；N945.1文献标志码：A6 结语级联故障普遍存在现实网络中，研究网络的攻击策略对网络抗毁性的影响对于有效打击敌方网络，指导我方网络建设提高网络抗毁性具有重要意义。

本文提出了一种基于社团划分的网络级联抗毁攻击策略，节点初始负荷根据节点及其邻居节点介数定义的“负荷容量”模型，故障负荷分配方式采用局部择优策略。

仿真分析了网络的负荷分配指数对抗毁性的影响，结果表明当α=1时网络的抗毁性最强。

研究分析了WS、BA、ER、WL四种网络模型在社团划分下的攻击策略，重点研究了具有重叠和非重叠社团结构网络的抗毁攻击策略，仿真结果表明基于社团划分的蓄意攻击策略在四种网络模型中均具有较好攻击效果。

研究了社团结构参数对网络抗毁性的影响，结果表明网络的耦合强度与抗毁性成正相关。

在非重叠社团网络中首先快速实现社团划分，然后分社团攻击介数最大的节点取得了最好的攻击策略；在重叠社团结构网络中，实现社团划分后蓄意攻击重叠部分介数最大的节点，然后分社团攻击介数最大节点为最有效网络攻击策略。

英语作文-重大工程项目成功背后的技术支持In the realm of monumental engineering projects, success is not merely a product of grand vision or meticulous planning, but also hinges crucially on the technical underpinnings that support these endeavors. Whether constructing a skyscraper that defies gravity, a bridge that spans oceans, or a complex network of tunnels threading through rugged terrain, the ultimate achievement of these feats is fundamentally rooted in the robustness and innovation of their technical foundations.The backbone of any major engineering project lies in its technical infrastructure. This encompasses a vast array of disciplines, from structural engineering and materials science to advanced computing and environmental analysis. Each facet contributes uniquely to ensuring the project's viability and success. For instance, in the construction of high-rise buildings, engineers must meticulously calculate load distributions, wind resistance, and seismic stability to guarantee both safety and longevity.Moreover, the advent of cutting-edge technologies has revolutionized the landscape of engineering projects. Computer-aided design (CAD) software allows for precise modeling and simulation, enabling engineers to foresee potential challenges and optimize designs before construction commences. This preemptive approach not only minimizes risks but also enhances efficiency, reducing both time and costs associated with the project.In the realm of infrastructure development, particularly in transportation networks, technological advancements have facilitated the creation of bridges and tunnels that seamlessly integrate with their natural surroundings while accommodating heavy traffic flows. Advanced materials such as carbon fiber composites and high-strength alloys have expanded the possibilities of construction, enabling engineers to build structures that are both lightweight and resilient.Furthermore, the role of information technology cannot be overstated in modern engineering projects. Real-time monitoring systems equipped with sensors provide continuous feedback on structural health and performance, allowing for proactivemaintenance and timely interventions. This proactive approach not only extends the lifespan of infrastructure but also enhances safety standards, ensuring public trust and confidence in the completed project.Beyond physical structures, the success of major engineering projects also hinges on sustainable practices and environmental stewardship. The integration of renewable energy sources, such as solar and wind power, into the design of buildings and infrastructure reduces carbon footprints and promotes energy efficiency. Similarly, initiatives like green roofs and rainwater harvesting systems contribute to mitigating urban heat islands and alleviating strain on municipal water resources.In conclusion, the realization of major engineering projects is a testament to human ingenuity and technical prowess. It is the harmonious convergence of innovative design, rigorous analysis, and meticulous execution that defines their success. By embracing the ultimate in technological advancements and sustainable practices, engineers not only shape the physical landscape but also pave the way for a more resilient and interconnected world. Thus, behind every towering skyscraper, sprawling bridge, and intricate tunnel lies a symphony of technical support that transforms visionary concepts into tangible realities for generations to come.。

2018年第3期 (总第183期）f e息通信INFORMATION&COMMUNICATIONS2018(Sum. N o 183)X波段线性调频连续波雷达低相噪P L L源设计肖辉春S周文欣2,肖涵3(1.威海技师学院智能制造系；2.荣成市实验中学，山东威海264300;3中国海洋大学电子系，山东青岛266100)摘要:X波段线性调频雷达（Linear FrequencyModulation Continue Wave，LFMCW)被广泛应用于海上目标的导航和追踪。

为了改善雷达环路相位噪声，文章提出了一种混频式P L L电路设计方案。

所设计的P L L电路通过调整锁相环的分 频比达到改善环路相位嗓声的目的。

验证结果表明，通过调整锁相环的环路带宽，可以显著改善相位嗓声。

关键词:X波段;线性调频;连续波雷达;P L L电路;锁相环中图分类号:TN957.51 文献标识码:A文章编号：1673-1131(2018)03-0016-03〇引言连续波雷达通过连续发射频率不变的信号，利用移动目 标对信号造成的多普勒频移，实现相关目标的侦测[1]。

线性调 频连续波（Linear Frequency Modulation Continue Wave, LFMCW)雷达，是目前应用最为广泛的一种连续波雷达。

LFM C W雷达具有发射功率较低、灵敏度较高、结构简单、机动 性好、适应恶劣环境等突出优势已被广泛应用于海上目标导航 与跟踪、船舶成像等领域[2_5]。

对于X波段LFM C W雷达，其系 统噪声，特别是信号载波的相位噪声是影响系统性能的关进因 素'系统相位噪声的存在使输出信号的相位随机变化，导致 雷达无法对具有弱反射特性的目标进行有效测量[7]。

因此，为 了实现LFM CW雷达对目标的有效识别，必须保证雷达信号源 的频率稳定性，即保证多数情况下雷达系统具有较低的相位噪 声，达到提高载噪比的目的[8_9]。