Graph Characterization Using Gaussian Wave Packet Signature

Abstract

In this paper we present a new approach for characterizing graphs using the solution of the wave equation. The wave equation provides a richer and potentially more expressive means of characterizing graphs than the more widely studied heat equation. Unfortunately the wave equation whose solution gives the kernel is less easily solved than the corresponding heat equation. There are two reasons for this. First, the wave equation can not be expressed in terms of the familiar node-based Laplacian, and must instead be expressed in terms of the edge-based Laplacian. Second, the eigenfunctions of the edge-based Laplacian are more complex than that of the node-based Laplacian. In this paper we present a solution to the wave equation, where the initial condition is Gaussian wave packets on the edges of the graph. We propose a global signature of the graph which is based on the amplitudes of the waves at different edges of the graph over time. We apply the proposed method to both synthetic and real world datasets and show that it can be used to characterize graphs with higher accuracy.