Information-Theoretic and Computational Limits of Correlation Detection under Graph Sampling

TL;DR

This work investigates the information-theoretic limits and algorithmic feasibility of detecting latent correlation between two Erdős-Rényi graphs when only induced subgraphs are observed. It derives sharp sample-size thresholds for detection, showing the optimal rate $s \asymp \sqrt{\frac{n\log n}{p^2 h(\gamma)}}$ with a subpolynomial gap in certain dense-graph regimes. On the algorithmic side, it introduces polynomial-time tests based on counts of trees and bounded-degree motifs, and provides lower-bound evidence via the low-degree framework that, in many regimes, these tests are rate-optimal and a computational gap persists. Numerical experiments on synthetic data and a real coauthor network corroborate the theoretical predictions, demonstrating strong empirical performance of the proposed tests and illustrating the impact of subgraph size and correlation strength. The results advance understanding of when correlation detection is statistically possible and when computationally efficient procedures can attain the information-theoretic limits, with implications for network data analysis under sampling constraints.

Abstract

Correlation analysis is a fundamental problem in statistics. In this paper, we consider the correlation detection problem between a pair of Erdos-Renyi graphs. Specifically, the problem is formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are independent; under the alternative hypothesis, the two graphs are edge-correlated through a latent permutation. We focus on the scenario where only two induced subgraphs are sampled, and characterize the sample size threshold for detection. At the information-theoretic level, we establish the sample complexity rates that are optimal up to constant factors over most parameter regimes, and the remaining gap is bounded by a subpolynomial factor. On the algorithmic side, we propose polynomial-time tests based on counting trees and bounded degree motifs, and identify the regimes where they succeed. Moreover, leveraging the low-degree conjecture, we provide evidence of computational hardness that matches our achievable guarantees, showing that the proposed polynomial-time tests are rate-optimal. Together, these results reveal a statistical--computational gap in the sample size required for correlation detection. Finally, we validate the proposed algorithms on synthetic data and a real coauthor network, demonstrating strong empirical performance.

Figures (7)

• Figure 1: An example of induced subgraph sampling with $n=10$ and $s=7$.
• Figure 2: Comparison of the proposed test statistic $\mathcal{T}_{\mathcal{M}_{BD}(N_{\mathsf{e}},d)}$ with $N_{\mathsf{e}}=d=4$ for fixed $p=0.05$, $\rho\in \left\{ 0.85,0.9,0.95,0.99 \right\}$ and varying subgraph size $s\in\{80,85,90,95,100\}$.
• Figure 3: Comparison of the proposed test statistic $\mathcal{T}_{\mathcal{M}_{T}(N_{\mathsf{e}})}$ with $N_{\mathsf{e}}=4$ for fixed $p=0.05$, $\rho\in \left\{ 0.85,0.9,0.95,0.99 \right\}$ and varying subgraph size $s\in\{80,85,90,95,100\}$.
• Figure 4: Comparison of the proposed test statistic by varying subgraph size $s\in\{80,85,90,95,100\}$ in coauthor network.
• Figure 5: Comparison of the proposed test statistic $\mathcal{T}_{\mathcal{M}_{BD}(N_{\mathsf{e}},d)}$ with $N_{\mathsf{e}}=d=3$ for fixed $p=0.05$, $\rho\in \left\{ 0.85,0.9,0.95,0.99 \right\}$ and varying subgraph size $s\in\{80,85,90,95,100\}$.

Theorems & Definitions (27)

• Definition 1: Correlated Erdős-Rényi graph
• Theorem 1: Information-theoretic thresholds
• Definition 2
• Theorem 2: Polynomial-time algorithm
• Lemma 1
• Proposition 1
• Proposition 2
• Definition 3: Correlated functional digraph
• Lemma 2
• Definition 4: Core set