Information-Theoretic Limits on Exact Subgraph Alignment Problem

TL;DR

This work addresses subgraph alignment, the problem of locating a random small pattern inside a larger Erdős-Rényi graph and recovering its vertex set and labeling. It introduces the ER subgraph pair model $(G,H_\pi)$ and develops information-theoretic limits for two recovery tasks: exact set recovery and exact permutation recovery, deriving achievability and converse conditions. Achievability results rely on the first-moment method and MAP equivalence, while the converse uses a structural-entropy framework to argue fundamental limits. The findings establish asymptotically tight thresholds in regimes where $\frac{m}{2}h(p) - \log n$ and $mp - \log m$ meet specific growth criteria, guiding when reliable subgraph localization and labeling is information-theoretically feasible and informing future algorithmic approaches.

Abstract

The graph alignment problem aims to identify the vertex correspondence between two correlated graphs. Most existing studies focus on the scenario in which the two graphs share the same vertex set. However, in many real-world applications, such as computer vision, social network analysis, and bioinformatics, the task often involves locating a small graph pattern within a larger graph. Existing graph alignment algorithms and analysis cannot directly address these scenarios because they are not designed to identify the specific subset of vertices where the small graph pattern resides within the larger graph. Motivated by this limitation, we introduce the subgraph alignment problem, which seeks to recover both the vertex set and/or the vertex correspondence of a small graph pattern embedded in a larger graph. In the special case where the small graph pattern is an induced subgraph of the larger graph and both the vertex set and correspondence are to be recovered, the problem reduces to the subgraph isomorphism problem, which is NP-complete in the worst case. In this paper, we formally formulate the subgraph alignment problem by proposing the Erdos-Renyi subgraph pair model together with some appropriate recovery criterion. We then establish almost-tight information-theoretic results for the subgraph alignment problem and present some novel approaches for the analysis.

Figures (3)

• Figure 1: Example of Erdös-Rényi subgraph pair model with $(n,m,p) = (9,4,0.3)$: $G \sim \mathrm{ER}(n,p)$ is generated according to the Erdös-Rényi measure. Induced subgraph $H$ is obtained through the vertex subset $\mathcal{S} = \{3,4,6,8\}$. Anonymized subgraph $H_\pi$ is obtained through applying the bijection $\pi: \mathcal{S} \to [m]$ defined as $\pi(3) = 1$, $\pi(4) = 6$, $\pi(6) = 3$, $\pi(8) = 4$. Note that the receiver will only be given the base graph $G$ and the pattern $H_\pi$, without knowing the bijection $\pi$.
• Figure 2: The green region represents the information‑theoretically achievable regime, while the red region corresponds to the non‑achievable regime. The grey region indicates cases whose achievability remains unknown. It is worth noting that the achievability and converse conditions coincide to form a tight threshold under certain conditions.
• Figure 3: The encoder encodes the source $\mathcal{S}$ by compressing the unlabaled graph $H = G[\mathcal{S}]$, and the decoder uses both $G$ and $H$ to reconstruct the estimation $\hat{\mathcal{S}}$. Unlike the general lossless compression setup, the encoder has no freedom to design the encoding function.

Theorems & Definitions (29)

• Definition 1: Brute-force Estimator
• Definition 2: MAP Estimator
• Proposition 3
• proof
• Theorem 4: Exact Set Recovery Achievability
• Theorem 5: Exact Permutation Recovery Achievability
• Theorem 6: Converse
• Remark 1
• Theorem 7: Achievability-Converse pair for Exact Set Recovery
• Theorem 8: Achievability-Converse pair for Exact Permutation Recovery