Complex network and its applications高忠科Apr 13, 2011Outline社团结构及其探寻算法4复杂系统与复杂网络1描述复杂网络基本统计量2小世界和无标度网络模型35复杂网络应用举例7关于复杂性关于复杂性我们所关心的问题：大量个体（更典型的是具有适应性的主体）所组成的复杂系统，在没有中心控制、非完全信息、仅仅存在局域相互作用的条件下，通过个体之间的非线性相互作用，可以在宏观层次上涌现出一定的结构和功能。
相互作用与复杂性Internet全局相互作用晶格扩散平均场什么是复杂网络？1复杂网络是对复杂系统的抽象和描述方式，任何包含大量组成单元（或子系统）的复杂系统，当把构成单元抽象成节点、单元之间的相互关系抽象为边时，都可以当作复杂网络来研究。
1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作用的拓扑结构，是理解复杂系统性质和功能的基础。
什么是复杂网络？1Watts DJ and Strogatz SH, Nature393, 440 (1998)Citation: 4911 (Small-world network)Barabási AL and Albert R, Science286, 509 (1999)Citation: 5474（Scale-free network）1复杂网络为研究复杂系统提供了一个全新的视角，对理解真实系统的复杂行为起着重要的作用。
1复杂网络研究的兴起，广泛应用于社会学，物理统计学，经济学，控制学，工程学，生物医学等多个跨学科研究领域。
Emergence of a networked lifeAtomMoleculeCellTissueOrgans OrganismsCommunities为什么研究复杂网络？1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；1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；1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？1复杂网络是构成复杂系统的基本框架( backbone )，每一个复杂系统都可以看作是单元或个体之间的相互作用网络；1复杂网络在刻画复杂性方面的重要性是由于结构和功能之间是相互影响的。
Examples of Complex NetworksThe worldwide air transportation network: a real socio-economic networkGuimera, Mossa, Turtschi, Amaral, PNAS(2005)The protein interactome of yeast: a real biochemical networkJeong, Mason, Barabasi, Oltvai, Nature(2001)生命金字塔不同领域的复杂网络1社会网：演员合作网，友谊网，科研合作网，Email网1生物网：食物链网，神经网，新陈代谢网，蛋白质网，基因调控网络1信息网络：WWW，专利使用，论文引用，计算机共享1技术网络：电力网，Internet，电话线路网1交通运输网：航线网，铁路网，自然河流网1时间序列信号复杂网络：流型复杂网络，脑功能网络，金融网络等网络研究的历史11736，欧拉：哥尼斯堡七桥11950，Erdos, Renyi: 随机图论11998，Strogatz, Barabasi: 小世界和无标度网络为什么现在才开始研究复杂网络？1计算机技术的发展：h 使我们拥有各种网络的数据库，并有可能对大规模的网络进行实证研究1普适性的发现：h 许多实际网络具有相同的定性性质h 且已有的理论不能描述和解释1理论研究的发展h 小世界网络(Small World Network), 无标度网络(Scale-free Network)h 统计物理学的研究手段复杂网络研究所关心的问题How to investigate Complex Networks ?1如何定量刻画复杂网络？h 网络结构的描述及其性质1网络是如何发展成现在这种结构的？h 网络演化模型1网络特定结构的后果是什么？h 网络结构的鲁棒性h 网络上的动力学行为和过程Outline社团结构及其探寻算法4复杂系统与复杂网络1描述复杂网络基本统计量2小世界和无标度网络模型35复杂网络应用举例7Describing a network formally1N nodes and E edges,1where E ≤N (N -1)/21N = 7, E = 9Note: In graph theory language this graph is of order 7 and size 9.Directed networksMore edges: E≤N(N-1)Much more complex topology.Adjacency matrixThe most convenient way of describing a network is the adjacency matrix a ij.A link between node i to node j is recorded by a ‘1’in the i th row and the j th column.Adjacency matrixUndirected networks havea symmetric adjacency matrix a ij.Directed networks in generalhave asymmetric a ij.Weighted networksIn a weighted network a real number is attached to each edge, so that we obtain a real adjacency matrix, usually denoted as w ij.DegreeIn an undirected network the degree k i of a node i is the number of nodes i is connected to:k i= Σj a ij= Σj a jiHere k1= 2, k2= 4, k3= 1, k4= 3 and k5= 2.In-degree and out-degreeIn a directed network the in-degree k i(in)of a node i is the number of directed edges pointing to node i:k i(in)= Σj a jiwhile the out-degree k i(out)of a node i is the number of directed edges pointing from node i:k i(out)= Σj a ijIn-degree and out-degreeThus, in a directed network, nodes can be highly connected, yet also isolated (e.g. in terms of sending or receiving information.)In-degree and out-degreeCitationsThe network of scientific citations provide examples illustrating two extremes:High in-degree and low out-degree:much-cited research articleLow in-degree and high out-degree:Book or review articleStrengthIn a weighted, undirected network the strength is the sum of the weights for the edges connecting to a node:s i= Σj w ij= Σj w jiHence s1= 4,s2= 18,s3= 2,s4= 13 and s5= 15.Shortest path lengthThe distance between two nodes i and j is the shortest path connecting the two nodes.i jd ij= 4i jAverage shortest path length：DiameterThe diameter of a network is the largest distance in the network -in other words it is the maximum shortest path connecting any two nodes.D= 2 D= 1Note: Fully connected networks (like the one on the right) have diameter D = 1.Clustering coefficientThe clustering coefficient measures how densely connected the neighborhood of a node is.It does this by counting the number of triangles of which a given node i is a part of, and dividing this value by the number of edge pairs.Often the clustering coefficient is averaged over the entire network: Where N is the number of nodes.BetweennessThe communication of two non-adjacent nodes, say j and k, depends on the nodes belonging to the paths connecting j and k. Consequently, a measure of the relevance of a given node can be obtained by counting the number of geodesics going through it, and defining the so-called node betweenness, defined as:where nis the number of shortest paths connecting j and k, while jk(i) is the number of shortest paths connectingj and k and passing njkthrough i.Standard measures of node centralityAssortativityAssortativity describes the correlation between the degree of a node and the degree of its neighbors.Networks in which highly connected nodes are linked to other nodes with a high degree are termed assortative. Such networks include social networks.Networks in which highly connected nodes are only linked to nodes with a low degree are termed disassortative. Such networks include the World Wide Web and biological networks.Assortativity CoefficientOne way of measuring assortativity is to determine the Pearson correlation coefficient between the degrees of pairs of connected nodes. This is termed the associativity coefficient r :r = (1/σq ) Σjk jk (e jk -q j q k )and lies between -1 (disassortative) and 1 (assortative).Some values for real networks:Physics coauthorship: 0.363Company directors: 0.276Internet: -0.189Marine food web: -0.247Degree distributionOutline社团结构及其探寻算法4复杂系统与复杂网络1描述复杂网络基本统计量2小世界和无标度网络模型35复杂网络应用举例7规则网络1规则网络是指平移对称性晶格,任何一个格点的近邻数目都相同。