Random Walks and Evolving Sets: Faster Convergences and Limitations

TL;DR

This paper introduces new combinatorial analyses of random walks and evolving sets, establishing faster convergence bounds and highlighting their limitations in disproving the small-set expansion hypothesis.

Contribution

It defines a combinatorial spectral gap, proves convergence bounds for non-lazy random walks, and demonstrates the limitations of evolving sets in small-set expansion conjectures.

Findings

Tight lower bounds on small-set expansion for graph powers.

Faster convergence of random walks with higher vertex expansion.

Evolving set process cannot disprove small-set expansion hypothesis.

Abstract

Analyzing the mixing time of random walks is a well-studied problem with applications in random sampling and more recently in graph partitioning. In this work, we present new analysis of random walks and evolving sets using more combinatorial graph structures, and show some implications in approximating small-set expansion. On the other hand, we provide examples showing the limitations of using random walks and evolving sets in disproving the small-set expansion hypothesis. - We define a combinatorial analog of the spectral gap, and use it to prove the convergence of non-lazy random walks. A corollary is a tight lower bound on the small-set expansion of graph powers for any graph. - We prove that random walks converge faster when the robust vertex expansion of the graph is larger. This provides an improved analysis of the local graph partitioning algorithm using the evolving set process. - We give an example showing that the evolving set process fails to disprove the small-set expansion hypothesis. This refutes a conjecture of Oveis Gharan and shows the limitations of local graph partitioning algorithms in approximating small-set expansion.