Betweenness Centrality : Algorithms and Lower Bounds

arXiv:0809.1906v2 [cs.DS] 19 Oct 2008 Betweenness Centrality : Algorithms and Lower Bounds Shiva Kintali∗ Abstract One of the most fundamental problems in large-scale network analysis is to determine the importance of a particular node in a network. Betweenness centrality is the most widely used metric to measure the importance of a node in a network. In this paper, we present a randomized parallel algorithm and an algebraic method for computing betweenness centrality of all nodes in a network. We prove that any path-comparison based algorithm cannot compute betweenness in less than O(nm) time. Keywords: all-pairs shortest paths, between- ness centrality, lower bounds, parallel graph al- gorithms, social networks. 1 Introduction One of the most fundamental problems in large- scale network analysis is to determine the im- portance of a particular node (or an edge) in a network. For example, in social networks we wish to know agents that have very short con- nections to large portions of the population. In communication networks we wish to know the links that carry a lot of traffic, ISPs that at- tract a lot of business, links that, if disconnected, decrease network performance dramatically, and so on. A particular way to measure the impor- tance of network elements (nodes or edges) is us- ing centrality metrics such as closeness centrality [29], graph centrality [19], stress centrality [31] and betweenness centrality ([16], [2]). An impor- ∗College of Computing, Georgia Institute of Technol- ogy, Atlanta, GA-30332. Email : kintali@cc.gatech.edu tant application of centrality arises in the study epidemic phenomena in networks when an infec- tious disease or a computer virus is disseminated. The power of a node to spread the epidemic is related to its centrality [28]. Centrality metrics also find applications in natural language pro- cessing [14], to compute relative importance of textual units. Betweenness centrality (introduced by Free- man [16] and Anthonisse [2]) is the most pop- ular (and computationally expensive) centrality metric. Some recent applications of between- ness include the study of biological networks [20, 26, 12], study of sexual networks and AIDS [24], identifying key actors in terrorist networks [22, 10], organizational behavior [6], supply chain management [9], and transportation networks [18]. Betweenness can also be used as a heuristic to solve NP-hard problems like graph clustering. For example, Newman and Girvan [25] developed a heuristic to find community structure in large networks, based on betweenness of the edges of the network. Since the networks of interest are huge, it is important to develop algorithms that compute these metrics efficiently. Brandes [4] showed that betweenness centrality can be computed in the same asymptotic time bounds as n Single Source Shortest Path (SSSP) computations. Brandes and Pich [5] presented experimental results of estimating different centrality measures under various node-selection strategies. Eppstein and Wang [13] presented a randomized approxima- tion algorithm for closeness centrality. 1 1.1 Betweenness Centrality We denote a network by an undirected graph G(V, E), with vertex set {v1, v2, . . . , vn} (or {1, 2, . . . , n}), with |V | = n vertices and |E| = m edges, representing the relationships between the vertices. In this paper, we refer to connected undirected graphs, unless otherwise stated. Each edge e ∈E has a positive integer weight w(e). Unweighted graphs have w(e) = 1 for all edges. A path from s to t is defined as a sequence of edges (vi, vi+1), 0 ≤i ≤l, where v0 = s and vl = t. The length of a path is the sum of weights of edges in this sequence. We use d(s, t) to de- note the distance (the minimum length of any path connecting s and t in G) between vertices s and t. We set d(i, i) = 0 by convention. We de- note the total number of shortest paths between vertices s and t by λst = λts. We set λss = 1 by convention. The number of shortest paths between s and t, passing through a vertex v, is denoted by λst(v). Let Diam(G) be the diam- eter (the longest shortest path) of the graph G. Let A = (aij) be the adjacency matrix of the graph, i.e., A is a 0-1 matrix with aij = 1 iff (i, j) ∈E. Let δst(v) denote the fraction of shortest paths between s and t that pass through a partic- ular vertex v i.e., δst(v) = λst(v) λst . We call δst(v) the pair-dependency of s, t on v. Betweenness centrality of a vertex v is defined as BC(v) = X s,t:s̸=v̸=t δst(v) The dependency of a source vertex s ∈V on a vertex v ∈V is defined as δs∗(v) = X t:t̸=s,t̸=v δst(v). The betweenness centrality of a vertex v can be then expressed as BC(v) = X s:s̸=v δs∗(v). Define the set of predecessors of a vertex v on shortest paths from s as Ps(v) = {u ∈V : (u, v) ∈E, d(s, v) = d(s, u) + w(u, v)}. The fol- lowing theorem, states that the dependencies of the closer vertices can be computed from the de- pendencies of the farther vertices. Theorem 1.1. [4] The dependency of s ∈V

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