Graph Theory and Network Measurment Social and Economic Networks MohammadAmin Fazli Social and Economic Networks 1
ToC • Network Representation • Basic Graph Theory Definitions • (SE) Network Statistics and Characteristics • Some Graph Theory • Readings: • Chapter 2 from the Jackson book • Chapter 2 from the Kleinberg book Social and Economic Networks 2
Network Representation • N = {1,2, … ,n} is the set of nodes (vertices) • A graph (N,g) is a matrix [g ij ] n×n where g ij represents a link (relation, edge) between node i and node j • Weighted network: 𝑗𝑘 ∈ 𝑆 • Unweighted network: 𝑗𝑘 ∈ {0,1} • Undirected network: 𝑗𝑘 = 𝑘𝑗 Social and Economic Networks 3
Network Representation • Edge list representation: = 12, 23 • Edge addition and deletion: g+ij, g-ij • Network isomorphism between (N, g) and (N’, g’) : ∃ 𝑔:𝑂→𝑂 ′ 𝑗𝑘 = ′ 𝑔 𝑗 𝑔(𝑘) • ( N’,g’) is a subnetwork of g’ if 𝑂 ′ ⊆ 𝑂, ′ ⊆ • Induced (restricted graphs): 𝑇 𝑗𝑘 = 𝑗𝑘 𝑗𝑔 𝑗 ∈ 𝑇, 𝑘 ∈ 𝑇 0 Social and Economic Networks 4
Path and Cycles • A Walk is a sequence of edges connecting a sequence of nodes 𝑋 = 𝑗 1 𝑗 2 , 𝑗 2 𝑗 3 , 𝑗 3 𝑗 4 , … , 𝑗 𝑜−1 𝑗 𝑙 ∀ 𝑞 : 𝑗 𝑞 𝑗 𝑞+1 ∈ • A Path is a walk in which no node repeats • A Cycle is a path which starts and ends at the same node 𝑗 𝑙 = 𝑗 1 • The number of walks between two nodes: Social and Economic Networks 5
Components & Connectedness • (N,g) is connected if every two nodes in g are connected by some path. • A component of a network (N,g) is a non-empty subnetwork ( N’,g’) which is • ( N’,g’) is connected • If 𝑗 ∈ 𝑂′ and 𝑗𝑘 ∈ then 𝑘 ∈ 𝑂 ′ and 𝑗𝑘 ∈ ′ • Strongly connectivity and strongly connected components for directed graphs. • C(N,g) = C(g) = set of g ’s connected components • The link ij is a bridge iff g-ij has more components than g • Giant component is a component which contains a significant fraction of nodes. • There is usually at most one giant component Social and Economic Networks 6
Special Kinds of Graphs • Star: • Complete Graph: Social and Economic Networks 7
Special Kinds of Graphs • Tree: a connected network with no cycle • A connected network is a tree iff it has n-1 links • A tree has at least two leaves • In a tree, there is a unique path between any pair of nodes • Forest: a union of trees • Cycle: a connected graph with n edges in which the degree of every node is 2. Social and Economic Networks 8
Neighborhood • 𝑂 𝑗 = 𝑘: 𝑗𝑘 = 1 2 = 𝑂 𝑗 ∪ • 𝑂 𝑗 𝑘∈𝑂 𝑗 𝑂 𝑘 𝑙 = 𝑂 𝑗 () ∪ 𝑙−1 • 𝑂 𝑗 𝑘∈𝑂 𝑗 𝑂 𝑘 𝑙 = 𝑗∈𝑇 𝑂 𝑗 𝑙 • 𝑂 𝑇 • Degree: 𝑒 𝑗 = #𝑂 𝑗 () • For directed graphs out-degree and in-degree is defined Social and Economic Networks 9
Degree Distribution • Degree distribution of a network is a description of relative frequencies of nodes that have different degrees. • P(d) is the fraction of nodes that have degree d under the degree distribution P . • Most of social and economical networks have scale-free degree distribution • A scale-free (power-law) distribution P(d) satisfies: 𝑄 𝑒 = cd −𝛿 • Free of Scale: P(2) / P(1) = P(20)/P(10) Social and Economic Networks 10
Degree Distribution Social and Economic Networks 11
Degree Distribution • Scale-free distributions have fat-tails • For large degrees the number of nodes that degree is much more than the random graphs. log 𝑄 𝑒 = log 𝑑 − 𝛿log(𝑒) Social and Economic Networks 12
Diameter & Average Path Length • The distance between two nodes is the length of the shortest path between them. • The diameter of a network is the largest distance between any two nodes. • Diameter is not a good measure to path lengths, but it can work as an upper-bound • Average path length is a better measure. Social and Economic Networks 13
Diameter & Average Path Length • The tale of Six-degrees of Separation • The diameter of SENs is 6!!! • Based on Milgram’s Experiment • The true story: • The diameter of SENs may be high • The average path length is low [ 𝑃(log 𝑜 ) ] Social and Economic Networks 14
Diameter & Average Path Length • The distance distribution in graph of all active Microsoft Instant Messenger user accounts Social and Economic Networks 15
Cliquishness & Clustering • A clique is a maximal complete subgraph of a given network ( 𝑇 ⊆ 𝑂 , 𝑇 is a complete network and for any 𝑗 ∈ 𝑂 ∖ 𝑇: 𝑇∪ 𝑗 is not complete. • Removing an edge from a network may destroy the whole clique structure (e.g. consider removing an edge from a complete graph). • An approximation: Clustering coefficient, • This is the overall clustering coefficient Social and Economic Networks 16
Cliquishness & Clustering • Individual Clustering Coefficient for node i: • Average Clustering Coefficient: • These values may differ Social and Economic Networks 17
Cliquishness & Clustering Social and Economic Networks 18
Cliquishness & Clustering • Average clustering goes to 1 • Overall clustering goes to 0 Social and Economic Networks 19
Transitivity • Consider a directed graph g, one can keep track of percentage of transitive triples: Social and Economic Networks 20
Centrality • Centrality measures show how much central a node is. • Different measures for centrality have been developed. • Four general categories: • Degree: how connected a node is • Closeness: how easily a node can reach other nodes • Betweenness: how important a node is in terms of connecting other nodes • Neighbors ’ characteristics: how important, central or influential a node ’ s neighbors are Social and Economic Networks 21
Degree Centrality • A simple measure: 𝑒 𝑗 𝑜 − 1 Social and Economic Networks 22
Closeness Centrality • A simple measure: −1 𝑘≠𝑗 𝑚 𝑗, 𝑘 𝑜 − 1 • Another measure (decay centrality) 𝜀 𝑚(𝑗,𝑘) 𝑘≠𝑗 • What does it measure for 𝜀 = 1 ? Social and Economic Networks 23
Betweenness Centrality • A simple measure: Social and Economic Networks 24
Neighbor-Related Measures • Katz prestige: 𝐿 () 𝑄 𝐿 = 𝑘 𝑄 𝑗𝑘 𝑗 𝑒 𝑘 𝑘≠𝑗 𝑗𝑘 • If we define 𝑗𝑘 = 𝑒 𝑘 , we have 𝑄 𝐿 = 𝑄 𝐿 or 𝑄 𝐿 = 0 𝐽 − • Calculating Katz prestige reduces to finding the unit eigenvector. Social and Economic Networks 25
Eigenvectors & Eigenvalues • For an 𝑜 × 𝑜 matrix T an eigenvector v is a 𝑜 × 1 vector for which ∃ 𝜇 𝑈𝑤 = 𝜇𝑤 • Left-hand eigenvector: 𝑤𝑈 = 𝜇𝑤 • Perron-Ferobenius Theorem: if T is a non-negative column stochastic matrix (the sum of entries in each column is one), then there exists a right-hand eigenvector v and has a corresponding eigenvalue 𝜇 = 1. • The same is true for right-hand eigenvectors and row stochastic matrixes. Social and Economic Networks 26
Eigenvectors & Eigenvalues • How to calculate: 𝑈 − 𝜇𝐽 𝑤 = 0 • For this equation to have a non-zero solution v, T − 𝜇𝐽 must be singular (non-invertible): det 𝑈 − 𝜇𝐽 = 0 Social and Economic Networks 27
Neighbor-Related Measures • Computing Katz prestige for the following • Katz prestige ≈ degree! • Not interesting on undirected networks, but interesting on directed networks. Social and Economic Networks 28
Neighbor-Related Measures 𝑓 = 𝑘 𝑗𝑘 𝐷 𝑓 • To solve the problem: Eigenvector Centrality: 𝜇𝐷 𝑗 𝑘 𝜇𝐷 𝑓 = 𝐷 𝑓 () • Katz2: 𝑄 𝐿2 , 𝑏 = 𝑏𝐽 + 𝑏 2 2 𝐽 + 𝑏 3 3 𝐽 + ⋯ 𝑄 𝐿2 , 𝑏 = 1 + 𝑏 + 𝑏 2 2 + ⋯ 𝑏𝐽 = 𝐽 − 𝑏 −1 𝑏𝐽 • Bonacich: 𝐶 , 𝑏, 𝑐 = 1 − 𝑐 −1 𝑏𝐽 𝐷 𝑓 Social and Economic Networks 29
Final Discussion about Centrality Measures Social and Economic Networks 30
Matching • A matching is a subset of edges with no common end-point. • Finding the maximum matching is an interesting problem specially in bipartite graphs (recall Matching Markets) • A bipartite network (N,g) is one for which N can be partitioned into two sets A and B such that each edge in g resides between A and B. • A perfect matching infects all vertices. • Philip-Hall Theorem: For a bipartite graph (N,g), there exists a matching of a set 𝐷 ⊆ 𝐵 , if and only if ∀ 𝑇⊆𝐷 𝑂 𝑇 ≥ 𝑇 Proof: see the whiteboard. Social and Economic Networks 31
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