Total communicability as a centrality measure

We examine node centrality measures based on the notion of total communicability, defined in terms of the row sums of matrix functions of the adjacency matrix of the network. Our main focus is on the matrix exponential and the resolvent, which have natural interpretations in terms of walks on the underlying graph. While such measures have been used before for ranking nodes in a network, we show that they can be computed very rapidly even in the case of large networks. Furthermore, we propose the (normalized) total sum of node communicabilities as a useful measure of network connectivity. Extensive numerical studies are conducted in order to compare this centrality measure with the closely related ones of subgraph centrality [E. Estrada and J. A. Rodriguez-Velazquez, Phys. Rev. E, 71 (2005), 056103] and Katz centrality [L. Katz, Psychometrica, 18 (1953), pp. 39-43]. Both synthetic and real-world networks are used in the computations.