[Paper Review] Partial Recovery in the Graph Alignment Problem

Georgina Hall, Laurent Massoulié|arXiv (Cornell University)|Jul 1, 2020

Nanocluster Synthesis and Applications4 citations

TL;DR

This paper establishes necessary and sufficient conditions for partial recovery in the graph alignment problem on correlated Erdős-Rényi graphs. It shows that partial recovery is possible when $ nqs = \Theta(1) $, under specific sparsity and correlation constraints, resolving a key gap in understanding information-theoretic limits for this problem in applications like network deanonymization and systems biology.

ABSTRACT

In this paper, we consider the graph alignment problem, which is the problem of recovering, given two graphs, a one-to-one mapping between nodes that maximizes edge overlap. This problem can be viewed as a noisy version of the well-known graph isomorphism problem and appears in many applications, including social network deanonymization and cellular biology. Our focus here is on partial recovery, i.e., we look for a one-to-one mapping which is correct on a fraction of the nodes of the graph rather than on all of them, and we assume that the two input graphs to the problem are correlated Erdős-Rényi graphs of parameters $(n,q,s)$. Our main contribution is then to give necessary and sufficient conditions on $(n,q,s)$ under which partial recovery is possible with high probability as the number of nodes $n$ goes to infinity. In particular, we show that it is possible to achieve partial recovery in the $nqs=Θ(1)$ regime under certain additional assumptions.

Motivation & Objective

• To address the lack of theoretical understanding of partial recovery in graph alignment, a regime that is practically relevant but underexplored in prior work.
• To model real-world applications such as social network deanonymization and protein interaction network alignment using correlated Erdős-Rényi graphs with parameters $ (n, q, s) $.
• To derive information-theoretic conditions under which a constant fraction $ \alpha $ of nodes can be correctly aligned with high probability as $ n \to \infty $.
• To bridge the gap between exact and almost-exact recovery regimes by characterizing the $ nqs = \Theta(1) $ regime as feasible for partial recovery.
• To provide both necessary and sufficient conditions using Kullback-Leibler divergence and spectral analysis, offering a complete characterization of the phase transition.

Proposed method

• Formalize the graph alignment problem as recovering a hidden permutation between nodes of two correlated Erdős-Rényi graphs with parameters $ (n, q, s) $, where $ n $ is the number of nodes, $ q $ controls edge density, and $ s $ controls correlation.
• Define partial recovery as correctly aligning a constant fraction $ \alpha $ of nodes, rather than all or all but a vanishing fraction.
• Use the Kullback-Leibler divergence between the joint distribution of independent edges and correlated edges as a necessary condition: if this divergence is too small (sub-constant), partial recovery is impossible.
• Establish a sufficient condition via spectral analysis and moment generating function techniques, showing that $ nqs = \Theta(1) $ enables partial recovery under mild sparsity assumptions.
• Apply matrix recurrence relations and eigenvalue decomposition to compute the moment generating function $ \mathbb{E}[e^{tW}] $, leading to closed-form expressions for the expected value of the alignment score.
• Leverage the solution of a quadratic equation to derive the optimal $ z^* $ in a variational bound, which is used to control tail probabilities and establish concentration.

Experimental results

Research questions

• RQ1 Under what conditions on $ (n, q, s) $ is partial recovery of the node alignment possible with high probability as $ n \to \infty $?
• RQ2 Is the regime $ nqs = \Theta(1) $ sufficient for partial recovery, and how does it compare to the regimes for exact and almost-exact recovery?
• RQ3 What is the role of the Kullback-Leibler divergence between edge correlation and independence in determining the feasibility of partial recovery?
• RQ4 Can the information-theoretic limits of partial recovery be characterized using spectral and moment-based methods in the correlated Erdős-Rényi model?
• RQ5 How do sparsity and correlation parameters interact to determine the threshold for partial recovery in graph alignment?

Key findings

• A necessary condition for partial recovery is that the Kullback-Leibler divergence between the joint distribution of independent edges and correlated edges must be bounded away from zero; if it is $ o(1) $, partial recovery is impossible.
• Partial recovery is possible when $ nqs = \Theta(1) $, provided that the graphs are sufficiently sparse and the correlation structure satisfies mild regularity conditions.
• The result extends the known information-theoretic phase transition: while $ nqs = \log n + \omega(1) $ enables exact recovery and $ nqs = \omega(1) $ enables almost-exact recovery, $ nqs = \Theta(1) $ is shown to be the threshold for partial recovery.
• The analysis uses a moment generating function approach and eigenvalue decomposition of a recurrence matrix to derive exact expressions for the expected alignment score.
• The optimal variational parameter $ z^* $ in the bound is derived by solving a quadratic equation, yielding tight bounds on the tail probability of the alignment score.
• The final expression for the moment generating function simplifies to $ \mathbb{E}[e^{tW}] = \alpha^{n+1} + \beta^{n+1} $, confirming the exponential decay of misalignment probability under the derived conditions.