Graph Theory and Network Measurment Social and Economic Networks - PowerPoint PPT Presentation

Graph Theory and Network Measurment Social and Economic Networks Jafar Habibi 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 = 𝑏𝑕𝐽 + 𝑏 2 𝑕 2 𝐽 + 𝑏 3 𝑕 3 𝐽 + ⋯ 𝑄 𝐿2 𝑕, 𝑏 = 1 + 𝑏𝑕 + 𝑏 2 𝑕 2 + ⋯ 𝑏𝑕𝐽 = 𝐽 − 𝑏𝑕 −1 𝑏𝑕𝐽 • Bonacich: 𝐶 𝑕, 𝑏, 𝑐 = 1 − 𝑐𝑕 −1 𝑏𝑕𝐽 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 Social and Economic Networks 31

Matching • Proof sketch: Social and Economic Networks 32