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A Measure of Betweenness Centrality Based on Random Walks M.E.J Social Networks 27 (2005) 39–54 A measure of betweenness centrality based on random walks M.E.J. Newman Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109-1120, USA Abstract Betweenness is a measure of the centrality of a node in a network, and is normally calculated as the fraction of shortest paths between node pairs that pass through the node of interest. Betweenness is, in some sense, a measure of the influence a node has over the spread of information through the network. By counting only shortest paths, however, the conventional definition implicitly assumes that information spreads only along those shortest paths. Here, we propose a betweenness measure that relaxes this assumption, including contributions from essentially all paths between nodes, not just the shortest, although it still gives more weight to short paths. The measure is based on random walks, counting how often a node is traversed by a random walk between two other nodes. We show how our measure can be calculated using matrix methods, and give some examples of its application to particular networks. © 2004 Elsevier B.V. All rights reserved. Keywords: Centrality; Betweenness; Random walks; Current flow 1. Introduction Over the years, network researchers have introduced a large number of centrality indices, measures of the varying importance of the vertices in a network according to one criterion or another (Wasserman and Faust, 1994; Scott, 2000). These indices have proved of great value in the analysis and understanding of the roles played by actors in social networks, as well 0378-8733/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.socnet.2004.11.009 40 M.E.J. Newman / Social Networks 27 (2005) 39–54 as by vertices in networks of other types, including citation networks, computer networks, and biological networks. Perhaps the simplest centrality measure is degree, which is the number of edges incident on a vertex in a network—the number of ties an actor has in social network parlance. Degree is a measure in some sense of the popularity of an actor. A more sophisticated centrality measure is closeness, which is the mean geodesic (i.e., shortest- path) distance between a vertex and all other vertices reachable from it.1 Closeness can be regarded as a measure of how long it will take information to spread from a given vertex to others in the network. Another important class of centrality measures is the class of betweenness measures. Betweenness, as one might guess, is a measure of the extent to which a vertex lies on the paths between others. The simplest and most widely used betweenness measure is that of Freeman (1977, 1979), usually called simply betweenness. (Where necessary, to distinguish this measure from other betweenness measures considered in this paper, we will refer to it as shortest-path betweenness.) The betweenness of a vertex i is defined to be the fraction of shortest paths between pairs of vertices in a network that pass through i. If, as is frequently the case, there is more than one shortest path between a given pair of vertices, then each such path is given equal weight such that the weights sum to unity. To be precise, suppose (st) that gi is the number of geodesic paths from vertex s to vertex t that pass through i, and suppose that nst is the total number of geodesic paths from s to t. Then, the betweenness of vertex i is (st) s<t gi /nst bi = , (1) (1/2)n(n − 1) where n is the total number of vertices in the network.2 We may, or may not, according to taste, consider the end-points of a path to fall on that path; the choice makes only the difference of an additive constant in the values for bi. In this paper, we will generally include the end-points. Betweenness centrality can be regarded as a measure of the extent to which an actor has control over information flowing between others. In a network in which flow is entirely or at least mostly along geodesic paths, the betweenness of a vertex measures how much flow will pass through that particular vertex. Betweenness can be calculated for all vertices in time O(mn) for a network with m edges and n vertices (Newman, 2001; Brandes, 2001). In most networks, however, information (or anything else) does not flow only along geodesic paths (Stephenson and Zelen, 1989; Freeman et al., 1991). News or a rumor or a message or a fad does not know the ideal route to take to get from one place to another; more likely it wanders around more randomly, encountering who it will. And even in a case such as the famous small-world experiment of Milgram (1967) and Travers and Milgram (1969), or its modern-day equivalent (Dodds et al., 2003), in which participants are explicitly instructed to get a message to a target by the most direct route possible, there is no evidence that people are especially successful in this task. Thus, we would imagine that in most cases 1 Some define closeness to be the reciprocal of this quantity, but either way the information communicated by the measure is the same. 2 Alternatively, bi may be normalized by dividing by its maximum possible value, which it achieves for a “star graph” in which one central vertex is connected to every other by a single edge (Freeman, 1979). M.E.J. Newman / Social Networks 27 (2005) 39–54 41 Fig. 1. (a) Vertices A and B will have high (shortest-path) betweenness in this configuration, while vertex C will not. (b) In calculations of flow betweenness, vertices A and B in this configuration will get high scores while vertex C will not. a realistic betweenness measure should include non-geodesic paths in addition to geodesic ones. Furthermore, by giving all the weight to the geodesic paths, and none to any other paths, no matter how closely competitive they are, the shortest-path betweenness measure can produce some odd effects. Consider the network sketched in Fig. 1a, for instance, in which two large groups are bridged by connections among just a few of their members. Vertices A and B will certainly get high betweenness scores in this case, since all shortest paths between the two communities must pass through them. Vertex C, on the other hand, will get a low score, since none of those shortest paths pass through it, taking instead the direct route from A to B. It is plausible, however, that in many real-world situations C would have quite a significant role to play in information flows. Certainly, it is possible for information to flow between two individuals via a third mutual acquaintance, even when the two individuals in question are themselves well acquainted. To address these problems, Freeman et al. (1991) suggested a more sophisticated be- tweenness measure, usually known as flow betweenness, that includes contributions from some non-geodesic paths. Flow betweenness is based on the idea of maximum flow. Imag- ine each edge in a network as a pipe that can carry a unit flow of some fluid. We can ask what the maximum possible flow then is between a given source vertex s and target vertex t through these pipes. In general, the answer is that more than a single unit of flow can be carried between source and target by making simultaneous use of several different paths through the network. The flow betweenness of a vertex i is defined as the amount of flow through vertex i when the maximum flow is transmitted from s to t, averaged over all s and t.3 Maximum flow from a given s to all reachable targets t can be calculated in worst-case time O(m2) using, for instance, the augmenting path algorithm (Ahuja et al., 1993), and hence the flow betweenness for all vertices can be calculated in time O(m2n).4 In practical terms, one can think of flow betweenness as measuring the betweenness of vertices in a network in which a maximal amount of information is continuously pumped between all sources and targets. Necessarily, that information still needs to “know” the ideal route (or one of the ideal routes) from each source to each target, in order to realize the 3 Technically, this definition is not unique, because there need not be a unique solution to the flow problem. To get around this difficulty, Freeman et al. define their betweenness measure as the maximum possible flow through i over all possible solutions to the st maximum flow problem, averaged over all s and t. 4 One can do somewhat better, particularly on networks like those discussed here in which all edges have the same capacity, by using more advanced algorithms (see Ahuja et al., 1993). 42 M.E.J. Newman / Social Networks 27 (2005) 39–54 maximum flow. Although the flow betweenness does take account of paths other than the shortest path (and indeed need not take account of the shortest path at all), this still seems an unrealistic definition for many practical situations: flow betweenness suffers from some of the same drawbacks as shortest-path betweenness, in that it is often the case that flow does not take any sort of ideal path from source to target, be it the shortest path, the maximum flow path, or another kind of ideal path. Moreover, like the shortest-path measure, flow betweenness can give counterintuitive results in some cases. Consider, for example, the network sketched in Fig. 1b, which again has two large groups joined by a few contacts.
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