1、Internet全局相互作用晶格扩散平均场?复杂系统不能够用分析的方法去研究，必须考虑个体之间的关联和作用；理解复杂系统的行为应该从理解系统相互作用网络的拓扑结构开始；网络拓扑结构的信息是构建系统模型、研究系统性质和功能的基础。为什么研究复杂网络？复杂网络是研究复杂系统的一种角度和方法，它关注系统中个体相互关联的作用的拓扑结构，是理解复杂系统性质和功能的基础。为什么研究复杂网络？WWW电力网因特网朋友关系网性关系网科学引文网演员网科学家合著网航空网道路交通网城市公共交通网神经网络基因网络蛋白质相互作用网络生态网络新陈代谢网络A food webA Unified Approachtowards
2、 the Connection Topology of various Complex SystemsK=5C=0K=5C=1一般情况下，聚集系数较大，平均最短路径较长。当p不太小时，聚集系数较小，平均最短路径较短。kNllnlnrandNkpCrandC(p):平均聚集系数平均聚集系数 L(p):平均最短路径平均最短路径!)()(kkekpNekPkkkpN10000个顶点p=0.0015 kkP)(kkPln)(ln=-3World Wide WebWorld Wide Web800 million documents(S.Lawrence,2019)Nodes:WWW documents
3、 Links:URL linksNWWW 109 N(k=500)103INTERNET BACKBONE(Faloutsos,Faloutsos and Faloutsos,2019)Nodes:computers,routers Links:physical linesNodes:actors Links:cast jointlyN=212,250 actors k =28.78P(k)k-=2.3SCIENCE CITATION INDEXNodes:papers Links:citations1736 PRL papers(1988)Nodes:scientist(authors)Li
4、nks:write paper together(Newman,2000,H.Jeong et al 2019)Sex-webNodes:people(Females;Males)Links:sexual relationshipsLiljeros et al.Nature 20194781 Swedes;18-74;59%response rate.Metabolic networkOrganisms from all three domains of life are scale-free networks!H.Jeong,B.Tombor,R.Albert,Z.N.Oltvai,and
5、A.L.Barabasi,Nature,407 651(2000)ArchaeaBacteriaEukaryotes kkP)(kkPln)(ln)()()(kPkkkP同向匹配无标度网络反向匹配无标度网络Physics collaboration networkPalla et al.Nature 435,9(2019)复杂网络中的社团结构Community Structures male femaleRhesus monkey network(WEO)066065004DLECACEKR006022KECY076KDCNEZERABCD(b)模体在网络中密度明显较高的子图（基本结构单元）M
6、ost real world networks have the same internal structure:Scale-free networks(1)The number of nodes(N)is NOT fixed.Networks continuously expand by the addition of new nodes(2)The attachment is NOT uniform.A node is linked with higher probability to a node that already has a large number of links.Exam
7、ples:WWW:new documents link to well known sites (CNN,YAHOO,NewYork Times,etc)Citation:well cited papers are more likely to be cited againScale-free model(1)GROWTH:At every timestep we add a new node with m edges(connected to the nodes already present in the system).(2)PREFERENTIAL ATTACHMENT:The pro
8、bability that a new node will be connected to node i depends on the connectivity ki of that nodeA.-L.Barabsi,R.Albert,Science 286,509(2019)jjiikkk)(P(k)k-3给定度分布P(k)的复杂网络上的点缺陷：顶点以概率q被随机的占据（工作状态良好）或以概率1-q被空置(被破坏)然后考察系统存在无限大连通集团的临界概率qc对无标度网络当=3,临界概率 qc 是0或负值.无标度网络对于顶点的随机移除非常稳健!无标度网络对有目的的最大度攻击非常脆弱！SW:SW
9、:网络上的动力学疾病传播nEpidemic Dynamics in Complex NetworksReachability in Colorado Springs(Sexual contact only)High-risk actors over 4 years695 people representedLongest path is 17 stepsAverage distance is about 5 stepsAverage person is within 3 steps of 75 other people137 people connected through 2 indepen
10、dent paths,core of 30 people connected through 4 independent paths(Node size=log of degree)Drug sharing network传染病动力学：SI 模型接触数目 I S固定人口总数的疾病传播模型:I(t)易受感染者 传染者:易受感染者人数:总人数:传染者人数:传染比例t010020030040050060000.511.522.533.5x 105Epidemic Dynamics：SI Model()()(1()()()1di ti ti tdti ts t01()1(1/1)ti tie02468
11、100.00.20.40.60.81.0di(t)/dtTime传染病动力学：SIS 模型()()()()()()()()()1di ti t s ti tdtds ti t s tdti ts t 有效扩散速率：1()1 1/i 1()0i 02468100.00.20.40.60.81.0Y Axis TitleX Axis Title02468100.00.20.40.60.81.0Y Axis TitleX Axis Title传染病动力学：SIR模型()()()()()()()()()()()()1di ti t s ti tdtds ti t s tdtdr ti tdti ts
12、tr t 0.00.20.40.60.81.00.00.20.40.60.81.0is1/疾病扩散条件：01/sOR01/s01s 1网络上的疾病传播n完全混合 网络拓扑结构的影响规则网格复杂网络SIS model on Networks网络上的疾病传播n小世界网上的SIS 模型利用平均场理论计算被感染顶点密度随时间的变化：稳态解:传播阈值:有效扩散速率And Let =1小世界网上的SIS 模型n计算机数值模拟N=103_-N=3*106,=6,10 个不同的网络,100 个不同的初始分布BA无标度网络上的SIS 模型利用平均场理论计算被感染顶点密度随时间的变化：SWBA稳态解:23()2/P kmkBA无标度网度分布EvolutionBA无标度网络上的SIS 模型Disease spreading PercolationDisease spreading Percolation!无标度网络上的SIR模型少数者博弈网络结构的影响？数值模拟结果但还没有做到与其他学科领域的有机结合 例如，对Intenet的容错性和鲁棒性的研究100101102103100101102identical power-law degrees高度值的中枢节点核心低度值的网状核心度分布相同的网络的细致结构未必相同
