Simplicial Kirchhoff index of random complexes

Kirchhoff index is an electrical network-theoretic invariant which is defined as the sum of effective resistances between all pairs of vertices. As a robustness measure of simplicial networks, a simplicial analogue of the Kirchhoff index is defined to be the sum of simplicial effective resistances for all subsets of vertices of size dimension plus one. In this paper, we investigate the Kirchhoff index of random simplicial complexes as a generalization of random graphs. We present a formula for the expectation of the random variable and show how it concentrates around the expectation. We also perform numerical experiments revealing that the expectation and the fluctuation are still valid for realizations of the random simplicial Kirchhoff index.