Kirchhoffian indices for weighted digraphs

The resistance indices, namely the Kirchhoff index and its generalisations, have undergone intense critical scrutiny in recent years. Based on random walks, we derive three Kirchhoffian indices for strongly connected and weighted digraphs. These indices are expressed in terms of (i) hitting times and (ii) the trace and eigenvalues of suitable matrices associated to the graph, namely the asymmetric Laplacian, the diagonally scaled Laplacian and their MoorePenrose inverses. The appropriateness of the generalised Kirchhoff index as a measure of network robustness is discussed, providing an alternative interpretation which is supported by an empirical application to the World Trade Network.