Recovery thresholds for hidden weighted sparse graphs

2026-06-12

AI Summary

This work investigates the information-theoretic limits of recovering a hidden sparse subgraph from a noisy, high-dimensional weighted complete graph. Focusing on edge weights drawn from Bernoulli, exponential, or Gaussian distributions, the study introduces a unified analytical framework based on the local Lipschitz continuity of Rényi divergence, integrating Kullback–Leibler divergence with random graph theory to rigorously characterize thresholds for both almost exact and partial recovery. The key contribution lies in establishing a precise connection between these recovery thresholds and the first-moment threshold of Erdős–Rényi graphs, revealing an exponential-scale “all-or-nothing” phase transition under typical distributions. Furthermore, the paper provides necessary and sufficient conditions for almost exact recovery and extends them to the setting of partial recovery.

Abstract

Recovering structural information from noisy high-dimensional data is a fundamental task in statistical inference. We investigate the recovery thresholds for a graph hidden in a randomly weighted complete graph. Specifically, an unknown graph $H^* \in H_n$ is chosen uniformly at random, and hidden in a complete graph of $n$ vertices as follows: the weight of an edge $e \in H$ is distributed independently according to $P_n$; otherwise the weight is distributed independently according to $Q_n$. The goal is to recover almost all of $H$ from these edge weights. Assuming a local Lipschitzness of the Rényi divergence between distributions $P_n$ and $Q_n$, and a mild density condition for the graphs $H_n$, we give a unified characterization of the information-theoretic limit for recovering almost all of $H$ (also known as almost exact recovery). Our characterization connects the KL divergence between $P_n$ and $Q_n$ to the logarithm of the first moment threshold of $H$ in the Erdős-Rényi random graph model $G(n,p)$. Our lower bound also extends to the task of partial recovery, in which only a constant $λ$-fraction of $H$ needs to be recovered. Last but not least, for certain Bernoulli and Exponential regimes, and for Gaussian distributions, we are able to show an All-or-Nothing (AoN) threshold phenomenon at the exponential scale.

Problem

• hidden graph
• recovery threshold
• weighted sparse graph
• statistical inference
• almost exact recovery

Innovation

• recovery threshold
• hidden sparse graph
• information-theoretic limit
• All-or-Nothing phenomenon
• Rényi divergence