[Paper Review] Partial Recovery of Erd\H{o}s-R\'enyi Graph Alignment via $k$-Core Alignment

Daniel Cullina, Negar Kiyavash+2 more|arXiv (Cornell University)|Sep 10, 2018

Advanced Graph Neural Networks26 references4 citations

TL;DR

This paper introduces the k-core alignment estimator to achieve partial recovery of vertex alignment in sparse, correlated Erdłs-Rényi graphs. It proves that when the average degree of the intersection graph tends to infinity, alignment of a vanishing fraction of vertices can be recovered with high probability, establishing matching achievability and converse bounds for partial recovery in the information-theoretic regime.

ABSTRACT

We determine information theoretic conditions under which it is possible to partially recover the alignment used to generate a pair of sparse, correlated Erd\H{o}s-R\'enyi graphs. To prove our achievability result, we introduce the $k$-core alignment estimator. This estimator searches for an alignment in which the intersection of the correlated graphs using this alignment has a minimum degree of $k$. We prove a matching converse bound. As the number of vertices grows, recovery of the alignment for a fraction of the vertices tending to one is possible when the average degree of the intersection of the graph pair tends to infinity. It was previously known that exact alignment is possible when this average degree grows faster than the logarithm of the number of vertices.

Motivation & Objective

• To determine information-theoretic conditions for partial recovery of vertex alignment in correlated sparse ErdŁs-Rényi graphs.
• To introduce and analyze the k-core alignment estimator as a novel method for achieving partial recovery.
• To establish matching achievability and converse bounds for partial alignment recovery, bridging the gap between exact recovery and detection.
• To extend the understanding of graph alignment limits beyond exact recovery, focusing on recovery of almost all vertices with one-sided error.
• To provide theoretical foundations for efficient graph matching in correlated random graph models with practical applications.

Proposed method

• Proposes the k-core alignment estimator, which seeks an alignment where the intersection graph has minimum degree k.
• Uses a combinatorial optimization framework to maximize edge overlap under the k-core constraint.
• Employs generating functions and matrix trace methods to analyze the expected number of valid alignments.
• Applies spectral and probabilistic bounds using eigenvalue analysis and Jordan decomposition of transition matrices.
• Derives upper bounds on the probability of incorrect alignment using moment generating functions and exponential tilting.
• Establishes matching converse bounds via a refined analysis of the likelihood ratio and tail behavior of alignment counts.

Experimental results

Research questions

• RQ1 Under what conditions is partial recovery of vertex alignment possible in correlated ErdŁs-Rényi graphs?
• RQ2 Can the k-core alignment estimator achieve high-probability partial recovery when the average degree of the intersection graph grows?
• RQ3 What is the fundamental information-theoretic limit for partial recovery, and how does it compare to exact recovery?
• RQ4 How does the k-core alignment method compare to existing algorithms in terms of theoretical guarantees and computational efficiency?
• RQ5 Can the proposed method achieve partial recovery with sub-exponential time complexity?

Key findings

• Partial recovery of the vertex alignment is possible with high probability when the average degree of the intersection graph tends to infinity as the number of vertices grows.
• The k-core alignment estimator achieves this partial recovery by enforcing a minimum degree k in the intersection graph under the alignment.
• A matching converse bound shows that partial recovery is impossible when the average degree of the intersection graph remains bounded.
• The paper establishes that partial recovery is possible even when exact recovery is not, demonstrating a phase transition between partial and exact recovery.
• The analysis reveals that the threshold for partial recovery is significantly lower than that for exact recovery, which requires the average degree to grow faster than log n.
• The method achieves information-theoretic optimality, with both achievability and converse results matching up to constant factors.