  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.
