Sample Complexity of Correlation Detection in the Gaussian Wigner Model

TL;DR

This paper investigates the sample complexity of detecting correlations between two Gaussian Wigner graphs with unlabeled vertices, establishing optimal detection thresholds and proposing an efficient algorithm.

Contribution

It characterizes the minimal sample size needed for correlation detection and introduces a computationally efficient approximate algorithm.

Findings

Derived the optimal sample complexity rate for correlation detection.

Proposed an efficient approximate detection algorithm.

Analyzed the conditional second moment to establish detection thresholds.

Abstract

Correlation analysis is a fundamental step in uncovering meaningful insights from complex datasets. In this paper, we study the problem of detecting correlations between two random graphs following the Gaussian Wigner model with unlabeled vertices. Specifically, the task is formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are independent, while under the alternative hypothesis, they are edge-correlated through a latent vertex permutation, yet maintain the same marginal distributions as under the null. We focus on the scenario where two induced subgraphs, each with a fixed number of vertices, are sampled. We determine the optimal rate for the sample size required for correlation detection, derived through an analysis of the conditional second moment. Additionally, we propose an efficient approximate algorithm that significantly reduces running time.