[Paper Review] High Dimensional Random Walks and Colorful Expansion

Tali Kaufman, David Mass|arXiv (Cornell University)|Apr 11, 2016

Limits and Structures in Graph Theory23 references22 citations

TL;DR

This paper introduces high-dimensional random walks on simplicial complexes, generalizing classical random walks on graphs to higher-dimensional faces. It establishes a local-to-global condition—spectral expansion in all 1D links—ensuring rapid mixing of all high-order random walks, and proves that Ramanujan complexes satisfy this criterion, providing an explicit family of bounded-degree complexes with fast-converging high-order walks.

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.

Motivation & Objective

• To define and study high-order random walks on high-dimensional simplicial complexes, generalizing classical random walks on graphs.
• To identify a local condition on a complex that guarantees rapid mixing of all high-order random walks across all dimensions.
• To construct an explicit family of bounded-degree simplicial complexes where all high-order random walks converge quickly to their stationary distribution.
• To establish a connection between a new notion of 'colorful expansion' and the convergence rate of high-order random walks.

Proposed method

• Define high-order random walks on $d$-dimensional simplicial complexes, where transitions occur between $i$-faces that share a common $(i+1)$-face, for $0 \leq i < d$.
• Introduce the adjacency matrix $A_i$ for $i$-faces, indexed by their incidence, to model the transition dynamics of the walk.
• Establish a local-to-global criterion: if the 1D skeletons of all links of the complex are spectral expanders, then all high-order random walks mix rapidly.
• Define 'colorful expansion' as a combinatorial generalization of graph expansion, directly tied to the mixing rate of high-order walks.
• Prove that spectral expansion in the 1D skeletons of links implies rapid mixing by relating the spectral gap to the walk's convergence rate.
• Demonstrate that Ramanujan complexes satisfy the spectral link condition, making them an explicit family of bounded-degree complexes with fast-mixing high-order walks.

Experimental results

Research questions

• RQ1 Can high-dimensional analogues of random walks on graphs be defined such that they mix rapidly on bounded-degree simplicial complexes?
• RQ2 What local condition on a simplicial complex ensures rapid mixing of high-order random walks across all dimensions?
• RQ3 How does the spectral expansion of link complexes relate to the convergence rate of high-order random walks?
• RQ4 Is there an explicit, bounded-degree family of high-dimensional complexes where all high-order walks mix rapidly?
• RQ5 Can a new combinatorial expansion notion—colorful expansion—characterize the mixing behavior of high-order random walks?

Key findings

• If the 1D skeletons of all links in a simplicial complex are spectral expanders, then all high-order random walks on the complex mix rapidly to their stationary distribution.
• The mixing rate of high-order random walks is bounded by $ \sqrt{\frac{d_{\text{max}}}{d_{\text{min}}}} \lambda^t $, where $ \lambda = \max\{ |\lambda_2|, |\lambda_n| \} $ is the second-largest spectral magnitude of the normalized adjacency matrix.
• Colorful expansion is introduced as a natural generalization of combinatorial expansion in graphs and is shown to be equivalent to rapid mixing of high-order walks.
• Ramanujan complexes satisfy the spectral link condition, making them an explicit, bounded-degree family of high-dimensional simplicial complexes with fast-mixing high-order random walks.
• The result provides a local criterion (spectral expansion in links) that implies global rapid mixing, establishing a local-to-global principle for high-dimensional expansion.
• The framework extends classical random walk theory on graphs to higher dimensions, offering a new tool for applications in coding theory, property testing, and PCP constructions.