Wave operators and Green´s functions on random graphs

Differential operators on random graphs have utility in solving graph diffusion and related problems. Such methods are applied for instance in influence propagation and search and rank methods. In this paper we examine approaches to building the inverse of differential operators via Green´s functions for such problems. In particular, we show the utility of the graph-Helmholtz equation on random graphs, and build an equivalent of the wavenumber on a graph, enabling rough and rapid O(1) approximation to the solution for certain regimes. Continuing work focuses on enhanced approximations at sub-quadratic costs in the number of nodes.