Commute Time for a Gaussian Wave Packet on a Graph

This paper presents a novel approach to quantifying the information flow on a graph. The proposed approach is based on the solution of a wave equation, which is defined using the edge-based Laplacian of the graph. The initial condition of the wave equation is a Gaussian wave packet on a single edge of the graph. To measure the information flow on the graph, we use the average return time of the Gaussian wave packet, referred to as the wave packet commute time. The advantage of using the edge-based Laplacian of a graph over its vertex-based counterpart is that it translates results from traditional analysis to graph theoretic domain in a more natural way. Therefore it can be useful in applications where distance and speed of propagation are important.