Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes

Louis, A and Paul, R and Ray, A (2024) Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes. In: Scandinavian Symposium on Algorithm Theory, SWAT 2024, 12-14 June 2024, Helsinki.

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Abstract

There are a lot of recent works on generalizing the spectral theory of graphs and graph partitioning to k-uniform hypergraphs. There have been two broad directions toward this goal. One generalizes the notion of graph conductance to hypergraph conductance Louis, Makarychev � TOC�16; Chan, Louis, Tang, Zhang � JACM�18. In the second approach, one can view a hypergraph as a simplicial complex and study its various topological properties Linial, Meshulam � Combinatorica�06; Meshulam, Wallach � RSA�09; Dotterrer, Kaufman, Wagner � SoCG�16; Parzanchevski, Rosenthal � RSA�17 and spectral properties Kaufman, Mass � ITCS�17; Dinur, Kaufman � FOCS�17; Kaufman, Openheim � STOC�18; Oppenheim � DCG�18; Kaufman, Openheim � Combinatorica�20. 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, a downward closed set system, to hypergraph expansion. More precisely, we study the simplicial complex obtained by downward closing the given hypergraph and random walks between its levels X(l), i.e., the sets of cardinality l. 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 < l cannot 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) cannot be used to infer any bounds on hypergraph conductance. In contrast, we give a Cheeger-like inequality relating the spectra of walks between level 1 and l for any l < 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 cannot be used to bound hypergraph expansion in a Cheeger-like manner. © Anand Louis, Rameesh Paul, and Arka Ray; licensed under Creative Commons License CC-BY 4.0.

Item Type:	Conference Paper
Publication:	Leibniz International Proceedings in Informatics, LIPIcs
Publisher:	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Additional Information:	The copyright for this article belongs to Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Keywords:	Graph theory; Random processes, High dimensional expander; High-dimensional; Higher-dimensional; Hyper graph; Hypergraph expansion; Link expansion; Random Walk; Simplicial complex; Sparsest cut; Threshold rank, Expansion
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	17 Aug 2024 11:47
Last Modified:	17 Aug 2024 11:47
URI:	http://eprints.iisc.ac.in/id/eprint/85455

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