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Testing Dependency of Weighted Random Graphs

Mor Oren, Vered Paslev, Wasim Huleihel

TL;DR

This work studies the problem of detecting edge dependency between two weighted random graphs, framing it as a hypothesis test between independence and a permutation-induced dependence. It extends prior results for Gaussian and dense ER graphs to general weight distributions, deriving information-theoretic thresholds and proposing a polynomial-time detector using GLRT-type strategies. The paper proves lower and upper bounds that pinpoint when detection is possible or impossible and provides a low-degree polynomial analysis suggesting a fundamental statistical–computational gap. It also introduces a tractable correlation-based test and analyzes it alongside the GLRT, offering practical insights for efficient detection. Overall, the results illuminate the precise regime where detection transitions occur and argue, under a plausible conjecture, that computationally efficient methods cannot close the gap to the information-theoretic limit in certain regimes.

Abstract

In this paper, we study the task of detecting the edge dependency between two weighted random graphs. We formulate this task as a simple hypothesis testing problem, where under the null hypothesis, the two observed graphs are statistically independent, whereas under the alternative, the edges of one graph are dependent on the edges of a uniformly and randomly vertex-permuted version of the other graph. For general edge-weight distributions, we establish thresholds at which optimal testing becomes information-theoretically possible or impossible, as a function of the total number of nodes in the observed graphs and the generative distributions of the weights. Finally, we identify a statistical-computational gap, and present evidence suggesting that this gap is inherent using the framework of low-degree polynomials.

Testing Dependency of Weighted Random Graphs

TL;DR

This work studies the problem of detecting edge dependency between two weighted random graphs, framing it as a hypothesis test between independence and a permutation-induced dependence. It extends prior results for Gaussian and dense ER graphs to general weight distributions, deriving information-theoretic thresholds and proposing a polynomial-time detector using GLRT-type strategies. The paper proves lower and upper bounds that pinpoint when detection is possible or impossible and provides a low-degree polynomial analysis suggesting a fundamental statistical–computational gap. It also introduces a tractable correlation-based test and analyzes it alongside the GLRT, offering practical insights for efficient detection. Overall, the results illuminate the precise regime where detection transitions occur and argue, under a plausible conjecture, that computationally efficient methods cannot close the gap to the information-theoretic limit in certain regimes.

Abstract

In this paper, we study the task of detecting the edge dependency between two weighted random graphs. We formulate this task as a simple hypothesis testing problem, where under the null hypothesis, the two observed graphs are statistically independent, whereas under the alternative, the edges of one graph are dependent on the edges of a uniformly and randomly vertex-permuted version of the other graph. For general edge-weight distributions, we establish thresholds at which optimal testing becomes information-theoretically possible or impossible, as a function of the total number of nodes in the observed graphs and the generative distributions of the weights. Finally, we identify a statistical-computational gap, and present evidence suggesting that this gap is inherent using the framework of low-degree polynomials.
Paper Structure (26 sections, 9 theorems, 122 equations)

This paper contains 26 sections, 9 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Related work.
  3. Notation.
  4. Problem Formulation
  5. Probabilistic model.
  6. Learning problem.
  7. Main Results
  8. Lower bounds.
  9. Upper bounds.
  10. Computational lower bound.
  11. Proofs
  12. Proof of Theorem \ref{['th:2LB']}
  13. Preliminaries on permutations
  14. A formula for the second moment
  15. Truncation technique
  16. ...and 11 more sections

Key Result

Theorem 1

Consider the detection problem in eqn:testing, and fix $\alpha\in(0,1)$. Weak detection is statistically impossible, i.e., $\mathsf{R}_n^\star=1-o(1)$, if for any $\omega(1)=\delta_n=o(\log n)$, and any constant $\epsilon>0$.

Theorems & Definitions (20)

  • Theorem 1: Weak detection lower bound
  • Remark 1
  • Example 1: Gaussian case
  • Example 2: Bernoulli case
  • Theorem 2: GLRT strong detection
  • Remark 2
  • Theorem 3: Comparison test weak detection
  • Conjecture 1: Low-degree conj., informal
  • Theorem 4
  • Lemma 1
  • ...and 10 more