Spectral Properties of Laplacian and Stochastic Diffusion Process for Edge Expansion in Hypergraphs

There has been recent work [Louis STOC 2015] to analyze the spectral properties of hypergraphs with respect to edge expansion. In particular, a diffusion process is defined on a hypergraph such that within each hyperedge, measure flows from nodes having maximum weighted measure to those having minimum. The diffusion process determines a Laplacian, whose spectral properties are related to the edge expansion properties of the hypergraph. It is suggested that in the above diffusion process, within each hyperedge, measure should flow uniformly in the complete bipartite graph from nodes with maximum weighted measure to those with minimum. However, we discover that this method has some technical issues. First, the diffusion process would not be well-defined. Second, the resulting Laplacian would not have the claimed spectral properties. In this paper, we show that the measure flow between the above two sets of nodes must be coordinated carefully over different hyperedges in order for the diffusion process to be well-defined, from which a Laplacian can be uniquely determined. Since the Laplacian is non-linear, we have to exploit other properties of the diffusion process to recover a spectral property concerning the “second eigenvalue” of the resulting Laplacian. Moreover, we show that higher order spectral properties cannot hold in general using the current framework. Inspired from applications in finance, we consider a stochastic diffusion process, in which each node also experiences Brownian noise from outside the system. We show a relationship between the second eigenvalue and the convergence behavior of the process. In particular, for the special case when the Brownian noise at each node has zero variance, the process reduces to the (deterministic) diffusion process within a closed system, and we can recover an upper bound on the mixing time in terms of the second eigenvalue. Department of Computer Science, the University of Hong Kong. {hubert,zhtang,czzhang}@cs.hku.hk