Shotgun threshold for sparse Erd\H{o}s-R\'enyi graphs

TL;DR

This paper determines the precise threshold depth for reconstructing Erdős-Rényi graphs from local neighborhood profiles, improving previous bounds and resolving an open question.

Contribution

It establishes the exact shotgun assembly threshold for Erdős-Rényi graphs, refining earlier results and providing a sharper constant factor.

Findings

Derived the shotgun assembly threshold r* for Erdős-Rényi graphs.

Improved the constant factor in the threshold compared to previous work.

Solved an open problem from Mossel and Ross (2019).

Abstract

In the shotgun assembly problem for a graph, we are given the empirical profile for rooted neighborhoods of depth $r$ (up to isomorphism) for some $r\geq 1$ and we wish to recover the underlying graph up to isomorphism. When the underlying graph is an Erd\H{o}s-R\'enyi $\mathcal G(n, \frac{\lambda}{n})$, we show that the shotgun assembly threshold $r_* \approx \frac{ \log n}{\log (\lambda^2 \gamma_\lambda)^{-1}}$ where $\gamma_\lambda$ is the probability for two independent Poisson-Galton-Watson trees with parameter $\lambda$ to be rooted isomorphic with each other. Our result sharpens a constant factor in a previous work by Mossel and Ross (2019) and thus solves a question therein.