Sparse High Dimensional Expanders via Local Lifts

TL;DR

This paper introduces a new elementary, deterministic method to construct high dimensional expanders with bounded degree, expanding the understanding and availability of such structures without relying on algebraic techniques.

Contribution

The authors present a novel algebra-free, local lift-based construction of bounded degree high dimensional expanders, extending previous algebraic methods and enabling new complex structures.

Findings

Constructed HDXs with bounded face degree using local lifts.

Demonstrated that local lifts of HDXs often preserve expansion properties.

Produced families of HDXs with links of arbitrarily large diameter.

Abstract

High dimensional expanders (HDXs) are a hypergraph generalization of expander graphs. They are extensively studied in the math and TCS communities due to their many applications. Like expander graphs, HDXs are especially interesting for applications when they are bounded degree, namely, if the number of edges adjacent to every vertex is bounded. However, only a handful of constructions are known to have this property, all of which rely on algebraic techniques. In particular, no random or combinatorial construction of bounded degree HDXs is known. As a result, our understanding of these objects is limited. The degree of an $i$-face in an HDX is the number of $(i+1)$-faces containing it. In this work we construct HDXs whose higher dimensional faces have bounded degree. This is done by giving an elementary and deterministic algorithm that takes as input a regular $k$-dimensional HDX $X$ and outputs another $k$-dimensional HDX $\widehat{X}$ with twice as many vertices. While the degree of vertices in $\widehat{X}$ grows, the degree of the $(k-1)$-faces in $\widehat{X}$ stays the same. As a result, we obtain a new `algebra-free' construction of HDXs whose $(k-1)$-face degree is bounded. Our algorithm is based on a simple and natural generalization of the construction by Bilu and Linial (Combinatorica, 2006), which build expanders using lifts coming from edge signings. Our construction is based on local lifts of HDXs, where a local lift is a complex whose top-level links are lifts of links in the original complex. We demonstrate that a local lift of an HDX is an HDX in many cases. In addition, combining local lifts with existing bounded degree constructions creates new families of bounded degree HDXs with significantly different links than before. For every large enough $D$, we use this technique to construct families of bounded degree HDXs with links that have diameter $\geq D$.