Shotgun Assembly of Erdos-Renyi Random Graphs

TL;DR

This paper investigates the conditions under which Erdős-Rényi random graphs can be reconstructed from local neighborhoods, identifying specific regimes of edge probability decay where reconstruction is possible or impossible.

Contribution

It provides the first rigorous analysis of graph shotgun assembly for ER random graphs, establishing thresholds for exact reconstructability based on neighborhood size and edge probability.

Findings

Reconstruction from distance-1 neighborhoods is possible for  < /3.

Reconstruction from distance-2 neighborhoods is possible for  < /2 and /2 < /5.

Reconstruction is impossible for higher /5 regimes.

Abstract

Graph shotgun assembly refers to the problem of reconstructing a graph from a collection of local neighborhoods. In this paper, we consider shotgun assembly of \ER random graphs $G(n, p_n)$, where $p_n = n^{-\alpha}$ for $0 < \alpha < 1$. We consider both reconstruction up to isomorphism as well as exact reconstruction (recovering the vertex labels as well as the structure). We show that given the collection of distance-$1$ neighborhoods, $G$ is exactly reconstructable for $0 < \alpha < \frac{1}{3}$, but not reconstructable for $\frac{1}{2} < \alpha < 1$. Given the collection of distance-$2$ neighborhoods, $G$ is exactly reconstructable for $\alpha \in \left(0, \frac{1}{2}\right) \cup \left(\frac{1}{2}, \frac{3}{5}\right)$, but not reconstructable for $\frac{3}{4} < \alpha < 1$.