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 link expansion in simplicial complexes can not be used to bound hypergraph expansion in a Cheeger like manner.

翻译：近期大量工作致力于将图的谱理论与图划分推广至超图。针对这一目标，学界主要形成了两大研究方向：其一是将图传导率概念推广至超图传导率[LM16, CLTZ18]；其二是将超图视为单纯复形，研究其拓扑性质[LM06, MW09, DKW16, PR17]与谱性质[KM17, DK17, KO18a, KO18b, Opp20]。本文尝试通过建立单纯复形上“上下行走”与“交换行走”的谱与超图扩张之间的联系，来沟通这两个研究方向。与图上的随机游走形成惊人反差的是，我们证明：当层数m与l满足1 < m ≤ l时，交换行走与上下行走的谱间隙无法用于推断超图传导率的任何界。此外，我们指出X(1)与X(k-1)之间交换行走的谱间隙同样不能用于推断超图传导率的界，但给出了一个类Cheeger不等式，将任意l ≤ k时第1层与第l层之间的行走谱与超图扩张联系起来——这揭示了交换行走与上下行走之间意料之外的差异。最后，我们通过构造实例证明：单纯复形中已被广泛研究的“链路扩张”概念无法以Cheeger方式界定超图扩张。