A measure of betweenness centrality based on ... - Semantic Scholar
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
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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., shortestpath) 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) g /nst , (1) bi = s
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., shortestpath) 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) g /nst , (1) bi = s
