  Betweenness measures provide quantitative tools to pick out fine details from
the massive amount of interaction data that is available from large complex
networks. They allow us to study the extent to which a node takes part when
information is passed around the network. Nodes with high betweenness may be
regarded as key players that have a highly active role. At one extreme,
betweenness has been defined by considering information passing only through
the shortest paths between pairs of nodes. At the other extreme, an alternative
type of betweenness has been defined by considering all possible walks of any
length. In this work, we propose a betweenness measure that lies between these
two opposing viewpoints. We allow information to pass through all possible
routes, but introduce a scaling so that longer walks carry less importance.
This new definition shares a similar philosophy to that of communicability for
pairs of nodes in a network, which was introduced by Estrada and Hatano (Phys.
Rev. E 77 (2008) 036111). Having defined this new communicability betweenness
measure, we show that it can be characterized neatly in terms of the
exponential of the adjacency matrix. We also show that this measure is closely
related to a Frechet derivative of the matrix exponential. This allows us to
conclude that it also describes network sensitivity when the edges of a given
node are subject to infinitesimally small perturbations. Using illustrative
synthetic and real life networks, we show that the new betweenness measure
behaves differently to existing versions, and in particular we show that it
recovers meaningful biological information from a protein-protein interaction
network.
