Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes

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 up-down walks and 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 can not be used to infer any bounds on hypergraph conductance. For the up-down walks, we show that spectral gap of walks between levels $m, l$ satisfying $1 < m < l$ can not be used to bound hypergraph expansion. We give a Cheeger-like inequality relating the spectral of walks between level 1 and $l$ to hypergraph expansion. Finally, we also give a construction to show that the well-studied notion of link expansion in simplicial complexes can not be used to bound hypergraph expansion in a Cheeger like manner.