High Dimensional Random Walks and Colorful Expansion

TL;DR

This paper introduces high order random walks on high dimensional simplicial complexes, establishes a local-to-global criterion for their rapid convergence, and demonstrates that Ramanujan complexes satisfy this criterion, ensuring fast mixing of these walks.

Contribution

It defines high order random walks on simplicial complexes, introduces colorful expansion as a new combinatorial expansion notion, and proves Ramanujan complexes meet the criteria for rapid convergence.

Findings

High order random walks on simplicial complexes converge rapidly under certain spectral conditions.

A new notion of colorful expansion generalizes graph expansion to high dimensions.

Ramanujan complexes are explicit examples satisfying the rapid convergence criterion.

Abstract

Random walks on bounded degree expander graphs have numerous applications, both in theoretical and practical computational problems. A key property of these walks is that they converge rapidly to their stationary distribution. In this work we {\em define high order random walks}: These are generalizations of random walks on graphs to high dimensional simplicial complexes, which are the high dimensional analogues of graphs. A simplicial complex of dimension $d$ has vertices, edges, triangles, pyramids, up to $d$-dimensional cells. For any $0 \leq i < d$, a high order random walk on dimension $i$ moves between neighboring $i$-faces (e.g., edges) of the complex, where two $i$-faces are considered neighbors if they share a common $(i+1)$-face (e.g., a triangle). The case of $i=0$ recovers the well studied random walk on graphs. We provide a {\em local-to-global criterion} on a complex which implies {\em rapid convergence of all high order random walks} on it. Specifically, we prove that if the $1$-dimensional skeletons of all the links of a complex are spectral expanders, then for {\em all} $0 \le i < d$ the high order random walk on dimension $i$ converges rapidly to its stationary distribution. We derive our result through a new notion of high dimensional combinatorial expansion of complexes which we term {\em colorful expansion}. This notion is a natural generalization of combinatorial expansion of graphs and is strongly related to the convergence rate of the high order random walks. We further show an explicit family of {\em bounded degree} complexes which satisfy this criterion. Specifically, we show that Ramanujan complexes meet this criterion, and thus form an explicit family of bounded degree high dimensional simplicial complexes in which all of the high order random walks converge rapidly to their stationary distribution.