Social Network Analysis

Transcription

1 Social Network Analysis Mathematics of Networks Manar Mohaisen Department of EEC Engineering

2 Adjacency matrix Network types Edge list Adjacency list Graph representation 2

3 Adjacency matrix Adjacency matrix Undirected graph: A ij & A ji = 1 if nodes i and j are connected, A ij = 0 otherwise. Directed graph: A ij and/or A ji = 1 if nodes i and j are connected, A ij = 0 otherwise. A complete network has an all-ones adjacency matrix However, in real networks a tiny fraction of the elements is non-zero. The adjacency matrix is sparse. 3

4 Adjacency matrix A B A B C C D undirected graph D directed graph m number of edges n number of nodes 4

5 Adjacency matrix Double-edge and self-edge Multiple edges between nodes i and j Aij = Aji = number of edges between i and j Self-edge Aii = 2 (undirected graphs)

6 Adjacency matrix Weighted networks In real networks Certain relationships are more valued than others. Weight might represent the number of minutes two mobile phone users talk to each other. A C 1 D B 6

7 Adjacency matrix directed graphs Directed graph A ij = 1 if there is an edge from i to j, 0 otherwise This is counter-intuitive, but mathematically useful

8 Edge list Edge list (1, 2) (2, 3) (3, 1) (3, 4) (4, 3)

9 Adjacency list Adjacency list 1: 2 2: 3 3: 1, 4 4:

10 Cocitation and bibliographic coupling Turn a directed graph into an undirected one Cocitation of vertices i and j in a directed graph Number of nodes that have outgoing edges pointing to both i and j. Cocitation coefficient k Cocitation matrix i j Cocitation of i and j is 3. 10

11 Cocitation and bibliographic coupling Cocitation network The adjacency matrix is the cocitation matrix with difference that diagonal elements are given by Since C is symmetric, the cocitation network is undirected. There is an edge between nodes i and j if C ij > 0. The cocitation network is weighted network with edge weights equal to the cocitation coefficient. 11

12 Cocitation and bibliographic coupling Bibliographic coupling of vertices i and j in a directed graph Number of nodes to which both nodes point. The bibliographic coupling of i and j is The bibliographic coupling matrix k Diagonal elements i Bibliographic coupling between i and j is 3. j 12

13 Cocitation and bibliographic coupling Cocitation vs. bibliographic coupling Cocitation is limited to influential papers, review articles, books, etc. Bibliographic coupling of two papers is high if they cite many others. Bibliographies vary less than the number of citations a paper receives. Bibliographic coupling is a more uniform indicator of similarity between papers Science Citation Index (SCI) uses bibliographic coupling. 13

14 Acyclic directed networks A cycle It is a closed loop Nodes with no outgoing edges are places at the bottom. The adjacency matrix is strictly upper triangular

15 Hypergraphs Hyperedge An edge that joins more than two nodes. Family links are an example of hyperedges Hypergraph It is a network that includes hyperedges 15

16 Examples Bipartite networks Affiliation and cocitation grags. Adjacency matrix Called incidence matrix, where

17 Bipartite networks A B C D A B C D 7 Two-mode bipartite to one-mode projection Information is lost; groups actors belonged to, number of those groups. Weighted projection captures part of the information

18 Properties Trees Connected: every node is reachable from any node in the tree via some path. It does not have loops A tree of n nodes has n-1 edges Any connected network with n nodes and n-1 edges is therefore a tree. 18

19 Planar networks Properties Planar network can be drawn on a plane without having any edges cross. Road network is to a good approximation a planar network. Planar network Four-color theorem It is possible to color any set of regions on a two-dimensional map with at most four colors with adjacent regions having different colors. K5 UG 19

20 Planar networks Kuratowski s reduction theorem Every non-planar network contains at least one subgraph that is an expansion of K5 or UG. Planar network Measure of planarity 99% planar network? Open research topic K5 UG 20

21 Network measures Network measures are used to better understand the networks, often, hidden features. 21

22 Degree of node i Degree: Undirected graph Number of edges connecting it to other nodes Degree sequence [k A, k B, k C, k D ] [2, 2, 3, 1] Total number of edges A n = 4 m = 4 C D B Average degree 22

23 Degree: Undirected graph Maximum possible number of edges In a simple graph (with no multi-edges or self-edges) The connectance or density of the graph A B n = 4 m = 4 C D 23

24 In-degree of node i Degree: Directed graph Number of edges incoming to node i Out-degree of node i Number of edges leaving node i A B Total number of edges directed graph C D Average degree 24

25 Degree A B A B C C D D 25

26 Degree distribution Degree distribution p k The probability that a randomly selected node has degree k. A B A B C C D D

27 In- and out-degree - again In- and out-degree In-degree A B C D Out-degree 27

28 In- and out-degree - again In-degree list [1, 2, 1, 1, 1] Ordered list: [2, 1, 1, 1, 1] Out-degree list [0, 1, 2, 1, 2] Ordered list: [2, 2, 1, 1, 0] 28

29 Density How many edges are there compared to the maximum possible number of edges. Here, V = n. Undirected graph Directed graph 29

30 Undirected graph Clustering coefficient N i is the neighborhood for node v i. k i = N i is the number of neighbors for node v i. L i is the number of links between the neighbors of v i / /

31 directed graph Clustering coefficient v i and v j are neighbors if they share the same edge regardless of the direction; if either e ij or e ji existed. 31

32 Path i A path A series of connected nodes. j Total number of paths of length 2 between nodes i and j 32

33 Path i j Total number of paths of length 3 between nodes i and j Total number of paths of length r between nodes i and j 33

34 Path i Total number of paths of length r that start and end at node i (loops) j The total number of loops of length r 34

35 Undirected graph A is symmetric matrix Path Via Eigen-value decomposition Directed graph Via Schur decomposition i j K: diagonal real matrix Q: unitary matrix i j T: upper-diagonal matrix Q: unitary matrix 35

36 Shortest path and average path length Shortest path The path with the shortest distance between two nodes; multiple shortest paths might exist Average path length (undirected graph) 5 36

37 Average path length directed graph Shortest path Average path length 5 37

38 Diameter The longest shortest path in the graph i.e., the distance between the two furthest away nodes

39 A component Components There is a path between each pair of nodes A disconnected graph of two disconnected components A connected graph (via a bridge edge)

40 Connectedness Strongly connected graph/component There is a path between every pair of nodes in the graph following the direction of the edges. Weakly connected graph/component If a path can t be found by following the direction of the edges but can be found if directed edges are treated as undirected. A B A B strongly connected G F E H D C weakly connected G F E H D C 40

41 Diffusion The graph Laplacian Spread of gas, an idea, a disease etc. in a network. The substance moves from node j to a neighbor node i at a rate The rate at which the substance changes at node i: 41

42 Diffusion contd. The graph Laplacian In matrix form 42

43 The graph Laplacian The graph Laplacian is defined as Therefore, 43

44 The graph Laplacian Let v i as the i-th eigenvector of the Laplacian L, a i (t) is a time-varying coefficient. Since v i is not a zero vector, Solution is given by 44

45 Eigenvalues of L The graph Laplacian L is symmetric: Eigenvalues are non-negative and real. For each edge, designate one edge to be edge 1 and the other to be edge 2. Consider the mxn matrix B Therefore 45

46 The graph Laplacian Now let v i be an eigenvector of L with eigenvalue l i Eigenvalues are non-negative. Therefore, diffusion contains decaying exponents (system converges as t à ) Eigenvalues can be zero, however. At least one eigenvalue = 0. 46

47 Example The graph Laplacian A B C D The Laplacian 47

48 Components and algebraic connectivity Let a network consist of c components The number of components per n 1, n 2,, n c Then, There are c eigenvectors with eigenvalue zero. For instance, If the network is connected, i.e., consists of a single component The second eigenvalue is non-zero and is called the algebraic connectivity of the network. 48

49 A random walk Random walks A path a cross a network created by taking repeated random steps. A node might be visited more than once (self-avoiding walks does not do that) Or 49

50 Random walks As t à Or Therefore, 50

51 Random walks Conclusion In the limit of long time, nodes with high degree are more likely to be visited by the random walk. 51

52 Summary Graph representation A taste of graph measures The graph Laplancian Random walks 52