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Optimal recovery of correlated Erdős-Rényi graphs

Hang Du

TL;DR

The paper analyzes optimal partial recovery of a planted vertex-matching π^* between two correlated Erdős–Rényi graphs G1 and G2 in the sparse regime p=n^{-α+o(1)} with nps^2=λ. It links recoverability to the balanced-load structure of the true intersection graph and employs a posterior-based viewpoint, an iterative heavy-vertex recovery scheme, and an orbit-decomposition toolkit to derive tight upper and lower bounds on the recoverable fraction via the limiting distribution μ_λ and the function F_λ; the results are sharp except at atoms of μ_λ, and they extend prior all-or-nothing thresholds to a nuanced partial-recovery regime. The approach combines combinatorial conditioning, FKG-type monotonicity, and nuanced large-deviation controls, yielding new insights into when a non-vanishing fraction of π^* can be recovered and when recovery remains fundamentally impossible. This advances the understanding of information-theoretic thresholds in sparse correlated graphs and informs algorithmic and statistical perspectives on graph alignment problems in non-dense regimes.

Abstract

For two unlabeled graphs $G_1,G_2$ independently sub-sampled from an Erdős-Rényi graph $\mathbf G(n,p)$ by keeping each edge with probability $s$, we aim to recover \emph{as many as possible} of the corresponding vertex pairs. We establish a connection between the recoverability of vertex pairs and the balanced load allocation in the true intersection graph of $ G_1 $ and $ G_2 $. Using this connection, we analyze the partial recovery regime where $ p = n^{-α+ o(1)} $ for some $ α\in (0, 1] $ and $ nps^2 = λ= O(1) $. We derive upper and lower bounds for the recoverable fraction in terms of $ α$ and the limiting load distribution $ μ_λ$ (as introduced in \cite{AS16}). These bounds coincide asymptotically whenever $ α^{-1} $ is not an atom of $ μ_λ$. Therefore, for each fixed $ λ$, our result characterizes the asymptotic optimal recovery fraction for all but countably many $ α\in (0, 1] $.

Optimal recovery of correlated Erdős-Rényi graphs

TL;DR

The paper analyzes optimal partial recovery of a planted vertex-matching π^* between two correlated Erdős–Rényi graphs G1 and G2 in the sparse regime p=n^{-α+o(1)} with nps^2=λ. It links recoverability to the balanced-load structure of the true intersection graph and employs a posterior-based viewpoint, an iterative heavy-vertex recovery scheme, and an orbit-decomposition toolkit to derive tight upper and lower bounds on the recoverable fraction via the limiting distribution μ_λ and the function F_λ; the results are sharp except at atoms of μ_λ, and they extend prior all-or-nothing thresholds to a nuanced partial-recovery regime. The approach combines combinatorial conditioning, FKG-type monotonicity, and nuanced large-deviation controls, yielding new insights into when a non-vanishing fraction of π^* can be recovered and when recovery remains fundamentally impossible. This advances the understanding of information-theoretic thresholds in sparse correlated graphs and informs algorithmic and statistical perspectives on graph alignment problems in non-dense regimes.

Abstract

For two unlabeled graphs independently sub-sampled from an Erdős-Rényi graph by keeping each edge with probability , we aim to recover \emph{as many as possible} of the corresponding vertex pairs. We establish a connection between the recoverability of vertex pairs and the balanced load allocation in the true intersection graph of and . Using this connection, we analyze the partial recovery regime where for some and . We derive upper and lower bounds for the recoverable fraction in terms of and the limiting load distribution (as introduced in \cite{AS16}). These bounds coincide asymptotically whenever is not an atom of . Therefore, for each fixed , our result characterizes the asymptotic optimal recovery fraction for all but countably many .

Paper Structure

This paper contains 22 sections, 27 theorems, 142 equations, 1 algorithm.

Table of Contents

  1. Introduction and main results
  2. Related works
  3. Intuitions and proof overview
  4. Notations
  5. Preliminaries
  6. Balanced load
  7. The limiting load distribution
  8. The orbit decomposition
  9. Proof of the positive result
  10. Reduction to few mismatches
  11. The conditioning scheme
  12. The orbit decomposition under conditioning
  13. Proof of Proposition \ref{['prop-reduced-conditional-probability']}
  14. Proof of Negative result
  15. Deferred proofs in Section \ref{['sec-positive']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

Fix two constants $\alpha,\lambda$ such that $0<\alpha\le 1$ and $\lambda>1$. Let $\mu_\lambda$ be the limiting load distribution introduced in AS16 (as formally defined in Proposition prop-weak-convergence-of-empirical-measure) and $F_\lambda(x)\equiv\mu_\lambda((x,+\infty))$. Let $p=p(n),s=s(n)$ s

Theorems & Definitions (51)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 41 more