Exact Recovery in the Data Block Model

TL;DR

This work studies exact recovery in the Data Block Model (DBM), a data-augmented stochastic block model, and introduces the Chernoff--TV divergence to sharply characterize the recovery threshold in the logarithmic-degree regime. It proves achievability via a polynomial-time two-stage algorithm that jointly leverages graph structure and vertex attributes, and provides a matching information-theoretic converse, establishing when exact recovery is impossible. The framework recovers the classical SBM threshold in the uninformative-data limit and reduces to data-only limits when the graph carries little information, thereby unifying several prior models of side information. Simulations in the two-community setting validate the phase transition and demonstrate substantial gains from vertex data, highlighting practical implications for incorporating side information in network clustering.

Abstract

Community detection in networks is a fundamental problem in machine learning and statistical inference, with applications in social networks, biological systems, and communication networks. The stochastic block model (SBM) serves as a canonical framework for studying community structure, and exact recovery, identifying the true communities with high probability, is a central theoretical question. While classical results characterize the phase transition for exact recovery based solely on graph connectivity, many real-world networks contain additional data, such as node attributes or labels. In this work, we study exact recovery in the Data Block Model (DBM), an SBM augmented with node-associated data, as formalized by Asadi, Abbe, and Verdú (2017). We introduce the Chernoff--TV divergence and use it to characterize a sharp exact recovery threshold for the DBM. We further provide an efficient algorithm that achieves this threshold, along with a matching converse result showing impossibility below the threshold. Finally, simulations validate our findings and demonstrate the benefits of incorporating vertex data as side information in community detection.

Figures (7)

• Figure 1: Example of a DBM
• Figure 2: $1-\mathrm{ERP}$ versus $a$ at fixed $\alpha$ ($n=1000$, $b=10$, $M=1000$). Vertical guidelines mark $a^\star_{\rm DBM}(b,\alpha)$ and $a^\star_{\rm SBM}(b)$.
• Figure 3: Mean misclassification error (log scale) versus $a$ at fixed $\alpha$ ($n=1000$, $b=10$, $M=1000$).
• Figure 4: Heatmaps of mean accuracy across the $(a,\alpha)$ grid. The DBM threshold curve $a^\star_{\rm DBM}(b,\alpha)=(\sqrt b+\sqrt{2(1-\alpha)})^2$ and the SBM threshold at $a^\star_{\rm SBM}(b)=(\sqrt b+\sqrt2)^2$ are overlaid.
• Figure 5: Heatmaps of ERP across the $(a,\alpha)$ grid, with the same threshold overlays.

Theorems & Definitions (33)

• Definition 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Definition 2.5
• Definition 3.1: Chernoff--Hellinger divergence
• Theorem 3.2: abbe015
• Definition 3.3: Relative entropy
• Definition 3.4: Chernoff information chernoff1952measure
• Proposition 3.5: CH-divergence as Chernoff information asadi2017compressing