Graph Reconstruction from Noisy Random Subgraphs

TL;DR

This paper studies reconstructing an unknown graph from multiple noisy, randomly sampled subgraphs, providing bounds on the number of traces needed for successful reconstruction under various conditions.

Contribution

It introduces bounds on the number of noisy subgraph traces required to reconstruct a random graph, highlighting the difference between random and arbitrary graphs.

Findings

O(p_e^{-1} p_v^{-2} log n) traces suffice for random graphs

Reconstruction requires exponentially many traces for arbitrary graphs

Reconstruction success depends on noise levels and sampling probabilities

Abstract

We consider the problem of reconstructing an undirected graph $G$ on $n$ vertices given multiple random noisy subgraphs or "traces". Specifically, a trace is generated by sampling each vertex with probability $p_v$, then taking the resulting induced subgraph on the sampled vertices, and then adding noise in the form of either (a) deleting each edge in the subgraph with probability $1-p_e$, or (b) deleting each edge with probability $f_e$ and transforming a non-edge into an edge with probability $f_e$. We show that, under mild assumptions on $p_v$, $p_e$ and $f_e$, if $G$ is selected uniformly at random, then $O(p_e^{-1} p_v^{-2} \log n)$ or $O((f_e-1/2)^{-2} p_v^{-2} \log n)$ traces suffice to reconstruct $G$ with high probability. In contrast, if $G$ is arbitrary, then $\exp(\Omega(n))$ traces are necessary even when $p_v=1, p_e=1/2$.