Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Low-Degree Hardness of Detection for Correlated Erdős-Rényi Graphs

Jian Ding, Hang Du, Zhangsong Li

TL;DR

This work analyzes detection in the correlated Erdős-Rényi graph model through the lens of low-degree polynomial algorithms. By formulating a rigorous low-degree framework and exploiting subgraph-count based polynomial bases, it derives computational lower bounds in both dense and sparse regimes: hardness for degree- $O(\rho^{-1})$ tests in general, and, in the sparse setting with $nq=n^{o(1)}$ and $\rho^2<\alpha-\varepsilon$ (Otter's constant $\alpha\approx 0.338$), hardness for degree up to $d$ with $\log d=o\big(\frac{\log n}{\log nq}\wedge \sqrt{\log n}\big)$. These results align with the performance of state-of-the-art correlation-detection and exact-matching algorithms, suggesting that current methods are near-optimal within the low-degree paradigm. The analysis connects detection hardness to implications for exact graph matching, highlighting a likely information-computation gap and providing a structural hardness landscape across regimes via careful truncation to admissible graphs and subgraph-based polynomial counts. Overall, the paper advances understanding of the computational limits for detection and matching in correlated random graphs and clarifies when existing algorithms may be essentially optimal.

Abstract

Given two Erdős-Rényi graphs with $n$ vertices whose edges are correlated through a latent vertex correspondence, we study complexity lower bounds for the associated correlation detection problem for the class of low-degree polynomial algorithms. We provide evidence that any degree-$O(ρ^{-1})$ polynomial algorithm fails for detection, where $ρ$ is the edge correlation. Furthermore, in the sparse regime where the edge density $q=n^{-1+o(1)}$, we provide evidence that any degree-$d$ polynomial algorithm fails for detection, as long as $\log d=o\big( \frac{\log n}{\log nq} \wedge \sqrt{\log n} \big)$ and the correlation $ρ<\sqrtα$ where $α\approx 0.338$ is the Otter's constant. Our result suggests that several state-of-the-art algorithms on correlation detection and exact matching recovery may be essentially the best possible.

Low-Degree Hardness of Detection for Correlated Erdős-Rényi Graphs

TL;DR

This work analyzes detection in the correlated Erdős-Rényi graph model through the lens of low-degree polynomial algorithms. By formulating a rigorous low-degree framework and exploiting subgraph-count based polynomial bases, it derives computational lower bounds in both dense and sparse regimes: hardness for degree- tests in general, and, in the sparse setting with and (Otter's constant ), hardness for degree up to with . These results align with the performance of state-of-the-art correlation-detection and exact-matching algorithms, suggesting that current methods are near-optimal within the low-degree paradigm. The analysis connects detection hardness to implications for exact graph matching, highlighting a likely information-computation gap and providing a structural hardness landscape across regimes via careful truncation to admissible graphs and subgraph-based polynomial counts. Overall, the paper advances understanding of the computational limits for detection and matching in correlated random graphs and clarifies when existing algorithms may be essentially optimal.

Abstract

Given two Erdős-Rényi graphs with vertices whose edges are correlated through a latent vertex correspondence, we study complexity lower bounds for the associated correlation detection problem for the class of low-degree polynomial algorithms. We provide evidence that any degree- polynomial algorithm fails for detection, where is the edge correlation. Furthermore, in the sparse regime where the edge density , we provide evidence that any degree- polynomial algorithm fails for detection, as long as and the correlation where is the Otter's constant. Our result suggests that several state-of-the-art algorithms on correlation detection and exact matching recovery may be essentially the best possible.
Paper Structure (13 sections, 19 theorems, 148 equations, 1 table)

This paper contains 13 sections, 19 theorems, 148 equations, 1 table.

Table of Contents

  1. Introduction and main result
  2. Backgrounds on the correlated random graph model
  3. The low-degree polynomial framework
  4. Proof of Theorem \ref{['thm-dense-regime']}
  5. Proof of Theorem \ref{['thm-sparse-regime']}
  6. Truncating on admissible graphs
  7. Reduction to admissible polynomials
  8. Completing the proof
  9. Preliminary facts about graphs
  10. Complimentary proofs
  11. Proof of Proposition \ref{['prop-another-bound-conditional-expectation']}.
  12. Proof of Lemma \ref{['lemma-containment-maximum-graph']} and Lemma \ref{['lemma-expand-graph']}
  13. Verification of \ref{['eq-quanlitative-estimation']}.

Key Result

Theorem 1.2

For sufficiently large integer $n$ and parameters $q,\rho\in (0,1)$, there is evidence suggesting that algorithms based on polynomials of degree $O(\rho^{-1})$ fail for detection in the correlated Erdős-Rényi graph model. Furthermore, if $q,\rho$ satisfies $nq=n^{o(1)}$ and $\rho^2<{\alpha}-\vareps

Theorems & Definitions (50)

  • Definition 1.1: Correlated Erdős-Rényi graph model
  • Theorem 1.2: Informal
  • Definition 1.3: Polynomial algorithms for detection
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6
  • Definition 1.7
  • Remark 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 40 more