Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes

TL;DR

This paper explores the spectral properties of random walks on simplicial complexes and their relation to hypergraph expansion, revealing surprising limitations and establishing new Cheeger-like inequalities.

Contribution

It bridges spectral theory of simplicial complexes with hypergraph expansion, providing new insights and bounds, and highlighting limitations of existing spectral methods.

Findings

Spectral gaps of certain walks do not bound hypergraph conductance.

A Cheeger-like inequality relates spectral properties to hypergraph expansion.

Link expansion cannot be used to bound hypergraph expansion in a Cheeger-like manner.

Abstract

There are a lot of recent works on generalizing the spectral theory of graphs and graph partitioning to hypergraphs. There have been two broad directions toward this goal. One generalizes the notion of graph conductance to hypergraph conductance [LM16, CLTZ18]. In the second approach one can view a hypergraph as a simplicial complex and study its various topological properties [LM06, MW09, DKW16, PR17] and spectral properties [KM17, DK17, KO18a, KO18b, Opp20]. In this work, we attempt to bridge these two directions of study by relating the spectrum of {\em up-down walks} and {\em swap-walks} on the simplicial complex to hypergraph expansion. In surprising contrast to random-walks on graphs, we show that the spectral gap of swap-walks and up-down walks between level $m$ and $l$ with $1 < m \leq l$ can not be used to infer any bounds on hypergraph conductance. Moreover, we show that the spectral gap of swap-walks between $X(1)$ and $X(k-1)$ can not be used to infer any bounds on hypergraph conductance, whereas we give a Cheeger-like inequality relating the spectral of walks between level $1$ and $l$ for any $l \leq k$ to hypergraph expansion. This is a surprising difference between swaps-walks and up-down walks! Finally, we also give a construction to show that the well-studied notion of {\em link expansion} in simplicial complexes can not be used to bound hypergraph expansion in a Cheeger-like manner.