Distance Reconstruction of Sparse Random Graphs

TL;DR

This paper presents an algorithm that reconstructs Erdős-Rényi random graphs from distance queries efficiently, especially for sparse graphs with average degree growing with the number of vertices.

Contribution

It introduces a query-efficient method for reconstructing sparse random graphs in the distance query model, applicable above the connectivity threshold.

Findings

Reconstruction algorithm works with $O( ext{average degree}^2 imes n imes ext{log} n)$ queries.

For certain sparse regimes, the algorithm uses $O(n ext{log}^5 n)$ queries.

Effective for $p$ values slightly above the connectivity threshold.

Abstract

In the distance query model, we are given access to the vertex set of a $n$-vertex graph $G$, and an oracle that takes as input two vertices and returns the distance between these two vertices in $G$. We study how many queries are needed to reconstruct the edge set of $G$ when $G$ is sampled according to the $G(n,p)$ Erd\H{o}s-Renyi-Gilbert distribution. Our approach applies to a large spectrum of values for $p$ starting slightly above the connectivity threshold: $p \geq \frac{2000 \log n}{n}$. We show that there exists an algorithm that reconstructs $G \sim G(n,p)$ using $O( \Delta^2 n \log n )$ queries in expectation, where $\Delta$ is the expected average degree of $G$. In particular, for $p \in [\frac{2000 \log n}{n}, \frac{\log^2 n}{n}]$ the algorithm uses $O(n \log^5 n)$ queries.