From Erdos-Renyi graphs to Linial-Meshulam complexes via the multineighbor construction

TL;DR

This paper studies the properties of m-neighbor complexes derived from Erdős-Rényi graphs, revealing probabilistic structures and correlations, and compares them to Linial-Meshulam complexes, proposing a convergence conjecture.

Contribution

It introduces the analysis of m-neighbor complexes for Erdős-Rényi graphs and compares their probabilistic properties to Linial-Meshulam complexes, highlighting differences and conjecturing convergence.

Findings

Complexes are (t-1)-dimensional with high probability for certain parameters.

All (t-2)-faces are present in these complexes.

Pairs of (t-1)-faces exhibit correlations, unlike in Linial-Meshulam complexes.

Abstract

The $m$-neighbor complex of a graph is the simplicial complex in which faces are sets of vertices with at least $m$ common neighbors. We consider these complexes for Erdos-Renyi random graphs and find that for certain explicit families of parameters the resulting complexes are with high probability $(t-1)$-dimensional with all $(t-2)$-faces and each $(t-1)$-face present with a fixed probability. Unlike the Linial-Meshulam measure on the same complexes there can be correlations between pairs of $(t-1)$-faces but we conjecture that the two measures converge in total variation for certain parameter sequences.