Accounting for the role of long walks on networks via a new matrix function

We introduce a new matrix function for studying graphs and real-world networks based on a double-factorial penalization of walks between nodes in a graph. This new matrix function is based on the matrix error function. We find a very good approximation of this function using a matrix hyperbolic tangent function. We derive a communicability function, a subgraph centrality and a double-factorial Estrada index based on this new matrix function. We obtain upper and lower bounds for the double-factorial Estrada index of graphs, showing that they are similar to those of the single-factorial Estrada index. We then compare these indices with the single-factorial one for simple graphs and real-world networks. We conclude that for networks containing chordless cycles-holes-the two penalization schemes produce significantly different results. In particular, we study two series of real-world networks representing urban street networks, and protein residue networks. We observe that the subgraph centrality based on both indices produces a significantly different ranking of the nodes. The use of the double-factorial penalization of walks opens new possibilities for studying important structural properties of real-world networks where long-walks play a fundamental role, such as the cases of networks containing chordless cycles.