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Sparse High Dimensional Expanders via Local Lifts

Inbar Ben Yaacov, Yotam Dikstein, Gal Maor

TL;DR

A new `algebra-free' construction of HDXs whose $(k-1)$-face degree is bounded, 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.

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 $\geq D$.

Sparse High Dimensional Expanders via Local Lifts

TL;DR

A new `algebra-free' construction of HDXs whose -face degree is bounded, 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.

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 -face in an HDX is the number of -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 -dimensional HDX and outputs another -dimensional HDX with twice as many vertices. While the degree of vertices in grows, the degree of the -faces in stays the same. As a result, we obtain a new `algebra-free' construction of HDXs whose -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 , we use this technique to construct families of bounded degree HDXs with links that have diameter .
Paper Structure (38 sections, 19 theorems, 40 equations)

This paper contains 38 sections, 19 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on High Dimensional Expanders
  3. Our Results
  4. Comparing to Random Constructions of HDXs
  5. The Construction
  6. Understanding Vertex vs. Edge Degree in Bounded-Degree Constructions
  7. Related Work
  8. Bounded degree HDX
  9. Graph and HDX lifts
  10. Open Questions
  11. Combinatorial constructions
  12. Links with other properties
  13. Other notions of expansion
  14. Better local spectral expansion
  15. Organization of This Paper
  16. ...and 23 more sections

Key Result

Theorem 1.2

There exists a randomized algorithm $\mathcal{A}$ that takes as input a $k$-dimensional complex $X_0$ and an integer $i \geqslant 1$, runs in expected time $\mathop{\mathrm{poly}}\nolimits((2^i|X_0(0)|)^k)$ at most, and outputs a $k$-dimensional complex $X_i$ with $2^i |X_0(0)|$ vertices. The algori For every $j \leqslant k-2$, $|X_i(j)|=2^{(j+1)i}|X_0(j)|$.

Theorems & Definitions (49)

  • Definition 1.1: High dimensional expander
  • Theorem 1.2: See thm:rand-alg-lll-full for a more precise statement
  • Theorem 1.3: See \ref{['cor:explicit_hdx']}
  • Theorem 1.4
  • Definition 2.1: spectral expander
  • Definition 2.3: lift
  • Definition 2.4: Function induced lift
  • Lemma 2.5
  • Lemma 2.6: BiluL2006
  • Definition 2.7: simplicial complex
  • ...and 39 more