Contextual Graph Matching with Correlated Gaussian Features

Abstract

We investigate contextual graph matching in the Gaussian setting, where both edge weights and node features are correlated across two networks. We derive precise information-theoretic thresholds for exact recovery, and identify conditions under which almost exact recovery is possible or impossible, in terms of graph and feature correlation strengths, the number of nodes, and feature dimension. Interestingly, whereas an all-or-nothing phase transition is observed in the standard graph-matching scenario, the additional contextual information introduces a richer structure: thresholds for exact and almost exact recovery no longer coincide. Our results provide the first rigorous characterization of how structural and contextual information interact in graph matching, and establish a benchmark for designing efficient algorithms.

Figures (2)

• Figure 1: Recovery phase diagram in the regime where $\rho, \eta \to 0$. The blue line corresponds to the exact recovery threshold (\ref{['exc']}), the green line to the almost exact recovery threshold (\ref{['almexc']}), and the red region to the information-theoretic impossibility of achieving overlap greater than $50\%$ (\ref{['partial']}).
• Figure 2: Complete recovery phase diagram under \ref{['conjecture']}, in the regime where $\rho, \eta \to 0$. The blue and green regions are unchanged from \ref{['fig:phase_proved']}. The red region corresponds to infeasibility of partial recovery. The point at $(x,y)=(4,0)$ is now a triple point.

Theorems & Definitions (36)

• Theorem 2.1: Exact Recovery
• Theorem 2.2: Achievability of almost exact recovery
• Theorem 2.3: Impossibility of recovering more than $50\%$ of the nodes
• Conjecture 2.1
• Proposition 3.1
• Lemma 3.2
• Remark 3.3
• Lemma 3.4
• Remark 3.5
• Lemma 3.6