Introduction Centrality Measures Implementation Applications Limitations Homework. Centrality Metrics. Ron Hagan, Yunhe Feng, and Jordan Bush

Transcription

1 Centrality Metrics Ron Hagan, Yunhe Feng, and Jordan Bush University of Tennessee Knoxville April 22, 2015

2 Outline 1 Introduction 2 Centrality Metrics 3 Implementation 4 Applications 5 Limitations

3 Introduction

4 Introduction to Centrality Motivating Question: Given a graph, which nodes are the most important?

5 What defines importance? 1 Network Flow 2 Network Cohesiveness

6 What defines importance? 1 Network Flow 2 Network Cohesiveness

7 History of Centrality Measures Concept originally used in fields studying human communication and social networks

8 History of Centrality Measures Concept originally used in fields studying human communication and social networks Examples: Identifying most important person in a group Creating best group arrangements from group projects Identifying central places for urban planning

9 History of Centrality Measures First centrality measure was formally proposed by Bavelas in 1948 Eigenvector centrality was developed by Bonacich in 1972 Freeman (1976) condensed the existing literature to develop a formal mathematical framework for centrality (Degree, Closeness, and Betweenness)

10 Centrality Measures

11 Degree Centrality Definition Intuitive Definition: The node with the most connections is the most important Mathematical Definition: for graph G = (V, E), the degree centrality of a given node v is C D (v) = degree(v)

12 Degree Centrality Example v C D (v) Table: Degree Centrality

13 Degree Centrality When the network is directed, we can distinguish between indegree and outdegree

14 Degree Centrality When the network is directed, we can distinguish between indegree and outdegree Can normalize C D to make it comparable between graphs C D(v) = degree(v) n 1

15 Degree Centrality When the network is directed, we can distinguish between indegree and outdegree Can normalize C D to make it comparable between graphs C D(v) = degree(v) n 1 Can calculate a metric of degree centrality for the whole network, called Centralization

16 Betweenness Centrality Definition Intuitive Definition: The node with the most strategic location is the most important. The frequency with which a node is on the shortest paths between all other nodes

17 Betweenness Centrality Definition Mathematical Definition: Define g ij as the number of shortest paths between node v i and node v j. Let g ij (v k ) be the number of shortest paths between v i and v j that contain node v k. Then the betweenness centrality of node v k is given as C B (v k ) = n i n j g ij (v k ) g ij

18 Betweenness Centrality Definition Mathematical Definition: Define g ij as the number of shortest paths between node v i and node v j. Let g ij (v k ) be the number of shortest paths between v i and v j that contain node v k. Then the betweenness centrality of node v k is given as C B (v k ) = n i n j g ij (v k ) g ij Computed used All Pairs, Shortest Paths Algorithm

19 Betweenness Centrality Example v C B (v) = = = Table: Betweenness Centrality

20 Closeness Centrality Definition Intuitive Definition: The most important node is the one that is the most independent Nodes that require the fewest number of edges to transfer information to all other nodes are considered the most independent

21 Closeness Centrality Definition Mathematical Definition: Let d(v i, v j ) be the number of edges in the shortest path between nodes v i and v j. Note that as v i and v j get farther apart ( less central ), d(v i, v j ) gets larger Then we define the closeness centrality as C C (v k ) = 1 N d(v i, v k ) i

22 Closeness Centrality Definition Mathematical Definition: Let d(v i, v j ) be the number of edges in the shortest path between nodes v i and v j. Note that as v i and v j get farther apart ( less central ), d(v i, v j ) gets larger Then we define the closeness centrality as C C (v k ) = 1 N d(v i, v k ) i Computed used All Pairs, Shortest Paths Algorithm

23 Closeness Centrality Example v C C (v) = = = = = = 1 11 Table: Closeness Centrality

24 Closeness Centrality Can normalize C C to make it comparable between graphs C C(v k ) = C C(v k ) n 1

25 Eigenvector Centrality Definition Intuitive Definition: The node that has the most information flowing through it is the most important You are only important if you have important neighbors

26 Eigenvector Centrality Definition Intuitive Definition: The node that has the most information flowing through it is the most important You are only important if you have important neighbors

27 Eigenvector Centrality Definition Mathematical Definition: Let A be the adjacency matrix of a graph, such that a ij = 1 if node v i is connected to v j and a ij = 0 if not. Then we define the eigenvector centrality of node v k as x k = 1 a vt x t λ t G where λ is a constant This can be rewritten in matrix form as Ax = λx

28 Eigenvector Centrality Definition Mathematical Definition: Let A be the adjacency matrix of a graph, such that a ij = 1 if node v i is connected to v j and a ij = 0 if not. Then we define the eigenvector centrality of node v k as x k = 1 a vt x t λ t G where λ is a constant This can be rewritten in matrix form as Ax = λx This can be solved using basic linear algebra

29 Eigenvector Centrality Example v C EV (v) Table: Eigenvector Centrality

30 Implementation

31 Degree Centrality 1 for i from 1 to V 2 for j from 1 to V 3 if matrix[i][j]!= 0 then 4 degree[i]++; 5 end if

32 All Pairs Shortest Paths Algorithm Description Goal: Determine the set of shortest paths between each pair of vertices in a graph. Floyd-Warshall algorithm Run Dijktra s from each vertex

33 Floyd-Warshall algorithm pseudocode 1 let dist be a V V array of minimum distances 2 initialized to infinity 3 for each edge (u,v) 4 dist[u][v] <- w(u,v) 5 for k from 1 to V 6 for i from 1 to V 7 for j from 1 to V 8 if dist[i][k] + dist[k][j] < dist[i][j] 9 then dist[i][j] <- dist[i][k] + dist[k][j] 10 end if

34 Floyd-Warshall algorithm example

35 All Pairs Shortest Paths Floyd-Warshall: O(V 3 ) Dijkstra s: O(V 3 ) using an adjacency matrix, O(EVlogV ) using adjacency lists. For our implementation we used Dijktra s ran from each vertex.

36 Betweenness Centrality 1 for i from 1 to V 2 excute dijkstra(i) -> vector path[i] 3 for j from 1 to V 4 vector all_path <- path[i][j] 5 add path[i] into vector all_path 6 for i from 1 to V 7 for j from 1 to all_path.size() 8 if i in all_path[j] then path_num[i]++ 9 end if 10 betweenness[i]=path_num[i]/2/(( V -1)( V -2)/2)

37 Closeness Centrality 1 for i from 1 to V 2 excute dijkstra(i) -> vector path[i] 3 for j from 1 to V 4 path_length[i] += path[i][j] 5 closeness_cent[i] = 1.0/path_length[i];

38 Eigenvector Centrality We implemented Power Iteration to get the eigenvector centrality. The method is described by the iteration: b k+1 = Ab k Ab k Starting with a vector b 0, which may be an approximation to the dominant eigenvector or a random vector.at every iteration, the vector b k is multiplied by the matrix A and normalized.

39 Eigenvector Centrality Power Iteration under the assumptions: A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues The starting vector b 0 has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue. Then a subsequence of b k converges to an eigenvector associated with the dominant eigenvalue

40 Eigenvector Centrality 1 for i from 1 to V 2 eigenvector[i] <- 1; 3 for n from 1 to MAX_TIMES 4 for i from 1 to V 5 tmp_eigen[i] <- 0 6 for j from 1 to V 7 tmp_eigen[i] += matrix[i][j]*eigenvector[j] 8 norm_sq <- 0; 9 for i from 1 to V 10 norm_sq <- tmp_eigen[i]*tmp_eigen[i] 11 norm <- sqrt(norm_sq) 12 for i from 1 to V 13 eigenvector[i] <- tmp_eigen[i]/norm

41 Comparison of the 4 Centrality Approaches Graph Designing

42 Comparison of the 4 Centrality Approaches Graph Designing

43 Comparison of the 4 Centrality Approaches Degree Centrality Hicks Lu(Zheng) Wang Langston Lu(Allan) Rodriguez Hagan Bush Feng Phillips Almami Maksov Eberius Butler Duggan Grady Tong Li Cox Sherman

44 Comparison of the 4 Centrality Approaches Betweenness Centrality Hicks Lu(Zheng) Wang Langston Lu(Allan) Rodriguez Hagan Bush Eberius Feng Phillips Almami Maksov Butler Duggan Grady Tong Li Cox Sherman

45 Comparison of the 4 Centrality Approaches Closeness Centrality Hagan Bush Hicks Lu(Zheng) Wang Langston Feng Phillips Lu(Allan) Rodriguez Almami Maksov Eberius Butler Duggan GradyTong Cox Sherman Li

46 Comparison of the 4 Centrality Approaches Eigenvector Centrality Hagan Bush Eberius Hicks Lu(Zheng) Wang Langston Feng Lu(Allan) Rodriguez Phillips Almami Maksov Butler Duggan Grady Tong Cox Sherman Li

47 Applications

48 Applications Gene Networks Ecological Networks Social Networks Epidemiology PageRank algorithm in Google Search Engine

49 Emperical Examples We performed empirical testing using the R igraph package on a variety of graphs including: random graphs correlation graphs built from gene expression food webs

50 Random Graph We created an Erdős-Rényi random graph with 1000 vertices and an edge probability of 0.3 using the igraph package built in generator.

51 Random Graph

52 Random Graph

53 Random Graph

54 Random Graph

55 Gene Expression Data Prostate Cancer We built graphs from a set of gene expression data from two sets of tissue samples from prostate cancer tumors and normal tissue by: Calculating the pairwise Pearson Corellation coefficients for the gene probes. Creating a weighted correlation graph. Thresholding the graph to create an unweighted graph.

56 Prostate Cancer Correlation Graphs Betweenness

57 Prostate Cancer Correlation Graphs Eigenvector Centrality

58 DAVID Enrichment 100 probes with largest negative change in eigenvector centrality (normal vs cancer)

59 DAVID Enrichment

60 Food Web 125 vertex directed graph representing carbon exchanges in the Florida Bay ecosystem. Networks for both the wet and dry seasons.

61 Food Web Betweenness

62 Food Web Eigenvector Centrality 18 Florida Bay Eigenvector Centrality Nodes Wet Season Dry Season Eigenvector Centrality

63 Limitations

64 Limitations Degree Centrality Degree Centrality s fatal flaw : it can only measure node and a node directly connected, but will not be calculated indirectly connected. For example, in Influence network, if A can affect the 10 individuals, 10 individuals can no longer influence each other, while another person B can also affect the 10 individuals, 10 individuals may continue to affect 100 per person, in this case, if measured by Degree Centrality words, A and B will influence are equivalent, but that is clearly not the case.

65 Limitations Betweenness Centrality, Closeness Centrality, Eigenvector Centrality Limitations of these Centrality approaches are that they are harder to compute than, for example, degree centrality. For Closeness Centrality, another disadvantage is that this score is mostly suitable for connected graphs.

66 Homework

67 Homework Find the degree, betweenness, closeness, and eigenvector centralities for each vertex in the graph. answers to Ron Hagan

68 References Bavelas, A A mathematical model for group structures. Human Organization 7(3): Bonacich, P. (1972). Factoring and weighting approaches to status scores and clique identification. Journal of Mathematical Sociology 2(1): Freeman, L. C Centrality in social networks conceptual clarification. Social Networks 1(3): Floyd, Robert W. (June 1962). Algorithm 97: Shortest Path. Communications of the ACM 5 (6): 345 Csardi G, Nepusz T: The igraph software package for complex network research, InterJournal, Complex Systems Ulanowicz, R.E., C. Bondavalli and M.S. Egnotovich. (1998) Network Analysis of Trophic Dynamics in South Florida Ecosystem, FY 97: The Florida Bay Ecosystem. Ref. No. [UMCES]CBL Chesapeake Biological Laboratory, Solomons, MD USA

69 Questions?