Shotgun Assembly of Linial-Meshulam Model

TL;DR

This paper extends graph reconstruction results to simplicial complexes, showing that Linial-Meshulam models are reconstructable from their 1-neighbourhoods under certain probability regimes, similar to Erdős-Rényi graphs.

Contribution

It generalizes the concept of graph reconstruction to simplicial complexes and establishes reconstructability thresholds for the Linial-Meshulam model.

Findings

Reconstructable for 0<α<1/3

Not reconstructable for 1/2<α<1

Extends Erdős-Rényi results to higher-dimensional complexes

Abstract

In a recent paper [6], J. Gaudio and E. Mossel studied the shotgun assembly of the Erd\H{o}s-R\'enyi graph $\mathcal G(n,p_n)$ with $p_n=n^{-\alpha}$, and showed that the graph is reconstructable form its $1$-neighbourhoods if $0<\alpha < 1/3$ and not reconstructable from its $1$-neighbourhoods if $1/2 <\alpha<1$. In this article, we generalise the notion of reconstruction of graphs to the reconstruction of simplicial complexes. We show that the Linial-Meshulam model $Y_{d}(n,p_n)$ on $n$ vertices with $p_n=n^{-\alpha}$ is reconstructable from its $1$-neighbourhoods when $0< \alpha < 1/3$ and is not reconstructable form its $1$-neighbourhoods when $1/2 < \alpha < 1$.