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Information-Theoretic Thresholds for the Alignments of Partially Correlated Graphs

Dong Huang, Xianwen Song, Pengkun Yang

TL;DR

This work introduces partially correlated graph models, namely the partially correlated Erdős-Rényi and Gaussian Wigner models, to study recovering a latent correlated subgraph and the vertex mapping between two graphs. It develops a correlated functional digraph representation to capture edge-wise correlations, and proposes a total-similarity edge-overlap estimator that exploits the decomposition into independent path and cycle components. The authors derive sharp information-theoretic thresholds for both partial and exact recovery in the ER and GW settings, revealing how the required number of correlated nodes $m$ scales with the ambient size $n$ and parameters $p$ and $\\rho$ (via $\\gamma$ and $\\phi$). They provide matching upper and lower bounds and discuss implications for graph sampling, correlation detection, and computational questions, offering a phase diagram perspective and directions for future work.

Abstract

This paper studies the problem of recovering the hidden vertex correspondence between two correlated random graphs. We propose the partially correlated Erdős-Rényi graphs model, wherein a pair of induced subgraphs with a certain number are correlated. We investigate the information-theoretic thresholds for recovering the latent correlated subgraphs and the hidden vertex correspondence. We prove that there exists an optimal rate for partial recovery for the number of correlated nodes, above which one can correctly match a fraction of vertices and below which correctly matching any positive fraction is impossible, and we also derive an optimal rate for exact recovery. In the proof of possibility results, we propose correlated functional digraphs, which partition the edges of the intersection graph into two types of components, and bound the error probability by lower-order cumulant generating functions. The proof of impossibility results build upon the generalized Fano's inequality and the recovery thresholds settled in correlated Erdős-Rényi graphs model.

Information-Theoretic Thresholds for the Alignments of Partially Correlated Graphs

TL;DR

This work introduces partially correlated graph models, namely the partially correlated Erdős-Rényi and Gaussian Wigner models, to study recovering a latent correlated subgraph and the vertex mapping between two graphs. It develops a correlated functional digraph representation to capture edge-wise correlations, and proposes a total-similarity edge-overlap estimator that exploits the decomposition into independent path and cycle components. The authors derive sharp information-theoretic thresholds for both partial and exact recovery in the ER and GW settings, revealing how the required number of correlated nodes scales with the ambient size and parameters and (via and ). They provide matching upper and lower bounds and discuss implications for graph sampling, correlation detection, and computational questions, offering a phase diagram perspective and directions for future work.

Abstract

This paper studies the problem of recovering the hidden vertex correspondence between two correlated random graphs. We propose the partially correlated Erdős-Rényi graphs model, wherein a pair of induced subgraphs with a certain number are correlated. We investigate the information-theoretic thresholds for recovering the latent correlated subgraphs and the hidden vertex correspondence. We prove that there exists an optimal rate for partial recovery for the number of correlated nodes, above which one can correctly match a fraction of vertices and below which correctly matching any positive fraction is impossible, and we also derive an optimal rate for exact recovery. In the proof of possibility results, we propose correlated functional digraphs, which partition the edges of the intersection graph into two types of components, and bound the error probability by lower-order cumulant generating functions. The proof of impossibility results build upon the generalized Fano's inequality and the recovery thresholds settled in correlated Erdős-Rényi graphs model.
Paper Structure (39 sections, 20 theorems, 162 equations, 5 figures)

This paper contains 39 sections, 20 theorems, 162 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Interpretation of Recovery Thresholds
  4. Related Work
  5. Graph sampling.
  6. Subgraphs isomorphism problem.
  7. Efficient algorithms and computational hardness.
  8. Correlation detection
  9. Other graph models.
  10. Notations
  11. Correlated functional digraph
  12. Recovery by maximizing total similarity score
  13. Bad event of signal.
  14. Bad event of noise.
  15. Erdős-Rényi model
  16. ...and 24 more sections

Key Result

Theorem 1

There exists an estimator $\hat{\pi}$ such that, for any constant $\delta\in(0,1)$ and $\pi^*\in \mathcal{S}_{n,m}$, when $m\ge \frac{c_1(\delta)\log n}{p^2 \phi(\gamma)}$, where $c_1(\delta)$ is a constant depending on $\delta$. Conversely, for any constant $c,\delta \in(0,1)$, there exists a constant $c_2(c,\delta)$ such that, when $m\le \frac{c_2(c,\delta)\log n}{p^2 \phi(\gamma)}$, for any es

Figures (5)

  • Figure 1: Phase diagram for recovery thresholds with $p=n^{-a_1}$, $\rho = n^{-a_2}$, and $m = n^{a_3}$.
  • Figure 2: Examples of the mapping $\pi$ and the underlying correlation $\pi^*$, where the domain and range of $\pi$ and $\pi^*$ could be different.
  • Figure 3: The connected components in the correlated functional digraph.
  • Figure 4: Illustration of a path of size $\ell$.
  • Figure 5: Illustration of a cycle of size $\ell$.

Theorems & Definitions (42)

  • Definition 1: Erdős-Rényi graph
  • Definition 2: Correlated Erdős-Rényi graphs
  • Definition 3: Correlated Gaussian Wigner model
  • Definition 4: Partially correlated Erdős-Rényi graphs
  • Definition 5: Partially correlated Gaussian Wigner model
  • Theorem 1: Erdős-Rényi model, partial recovery
  • Theorem 2: Erdős-Rényi model, exact recovery
  • Theorem 3: Gaussian Wigner model
  • Remark 1
  • Definition 6: Correlated functional digraph
  • ...and 32 more