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Exact Community Recovery under Side Information: Optimality of Spectral Algorithms

Julia Gaudio, Nirmit Joshi

TL;DR

This work addresses exact community recovery in two-community models under node-attributed side information. It introduces a simple yet optimal spectral algorithm that incorporates side information by a log-likelihood shift and leverages the leading eigenvectors of the observation matrix to mimic genie-aided estimators, enabling exact recovery down to the information-theoretic limit when $I^*>1$. The analysis builds on entrywise eigenvector techniques to show that a carefully weighted combination of top eigenvectors, plus side-information terms, approximates the genie scores with $\ell_\infty$-accuracy $o(\log n)$, thereby achieving the IT threshold in ROS and SBM across Gaussian and Bernoulli observation regimes. The results unify several exact-recovery settings (SBM, Submatrix Localization, $\mathbb{Z}_2$-Synchronization) under a single spectral framework and provide nearly linear-time algorithms with strong performance guarantees. This has practical implications for scalable community detection in networks with attributes, offering provable optimality and efficiency without multi-matrix or multi-stage refinements.

Abstract

We study the problem of exact community recovery in general, two-community block models, in the presence of node-attributed $side$ $information$. We allow for a very general side information channel for node attributes, and for pairwise (edge) observations, consider both Bernoulli and Gaussian matrix models, capturing the Stochastic Block Model, Submatrix Localization, and $\mathbb{Z}_2$-Synchronization as special cases. A recent work of Dreveton et al. 2024 characterized the information-theoretic limit of a very general exact recovery problem with side information. In this paper, we show algorithmic achievability in the above important cases by designing a simple but optimal spectral algorithm that incorporates side information (when present) along with the eigenvectors of the pairwise observation matrix. Using the powerful tool of entrywise eigenvector analysis of Abbe et al. 2020, we show that our spectral algorithm can mimic the so called $genie$-$aided$ $estimators$, where the $i^{\mathrm{th}}$ genie-aided estimator optimally computes the estimate of the $i^{\mathrm{th}}$ label, when all remaining labels are revealed by a genie. This perspective provides a unified understanding of the optimality of spectral algorithms for various exact recovery problems in a recent line of work.

Exact Community Recovery under Side Information: Optimality of Spectral Algorithms

TL;DR

This work addresses exact community recovery in two-community models under node-attributed side information. It introduces a simple yet optimal spectral algorithm that incorporates side information by a log-likelihood shift and leverages the leading eigenvectors of the observation matrix to mimic genie-aided estimators, enabling exact recovery down to the information-theoretic limit when . The analysis builds on entrywise eigenvector techniques to show that a carefully weighted combination of top eigenvectors, plus side-information terms, approximates the genie scores with -accuracy , thereby achieving the IT threshold in ROS and SBM across Gaussian and Bernoulli observation regimes. The results unify several exact-recovery settings (SBM, Submatrix Localization, -Synchronization) under a single spectral framework and provide nearly linear-time algorithms with strong performance guarantees. This has practical implications for scalable community detection in networks with attributes, offering provable optimality and efficiency without multi-matrix or multi-stage refinements.

Abstract

We study the problem of exact community recovery in general, two-community block models, in the presence of node-attributed . We allow for a very general side information channel for node attributes, and for pairwise (edge) observations, consider both Bernoulli and Gaussian matrix models, capturing the Stochastic Block Model, Submatrix Localization, and -Synchronization as special cases. A recent work of Dreveton et al. 2024 characterized the information-theoretic limit of a very general exact recovery problem with side information. In this paper, we show algorithmic achievability in the above important cases by designing a simple but optimal spectral algorithm that incorporates side information (when present) along with the eigenvectors of the pairwise observation matrix. Using the powerful tool of entrywise eigenvector analysis of Abbe et al. 2020, we show that our spectral algorithm can mimic the so called - , where the genie-aided estimator optimally computes the estimate of the label, when all remaining labels are revealed by a genie. This perspective provides a unified understanding of the optimality of spectral algorithms for various exact recovery problems in a recent line of work.
Paper Structure (52 sections, 35 theorems, 158 equations, 6 figures, 1 table)

This paper contains 52 sections, 35 theorems, 158 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Spectral Algorithms.
  4. Models without Side Information.
  5. Unsupervised vs Semi-supervised Learning.
  6. Organization.
  7. Preliminaries
  8. Models
  9. Exact Recovery.
  10. Information Theoretic Threshold from [\ref{['bib-dreveton2024exact']}]
  11. Main Results: Algorithmic Achievability for , and ,
  12. Genie-Aided Estimators
  13. Genie scores.
  14. Motivation behind our Spectral Strategy.
  15. Genie Success with Margin above the IT Limit:
  16. ...and 37 more sections

Key Result

Proposition 2.10

Let $\rho \in (0,1)$ and $(\mathcal{P}_{+},\mathcal{P}_{-}, \mathcal{Q})$ be probability laws. Let $(A,\sigma^*) \sim \mathsf{GBM}_n(\rho, \mathcal{P}_{+}, \mathcal{P}_{-}, \mathcal{Q})$ and we observe $A$. Optionally, for side information laws $(\mathcal{S}_{+}, \mathcal{S}_{-})$, we observe $y \si

Figures (6)

  • Figure 1: Visualization of $\mathsf{BSC}$ and $\mathsf{BEC}$ side information. The red-colored, blue-colored, and uncolored vertices have side information labels of $+1,-1,$ and $0$ respectively.
  • Figure 2: We consider the $\mathbb{Z}_2$-Synchronization model which is $\mathsf{ROS}_n(1/2,a,-a)$. With $n=300$, for each type of side information channel of "strength" $\beta$ (see \ref{['subsec:IT-limits-interpretation']}), we validate the performance of the spectral algorithm over $N=50$ trails over a grid of values for $(\beta,a)$. Lighter pixels correspond to higher rate of success. The blue and red curves are theoretical thresholds with and without side information respectively.
  • Figure 3: Summary of our unified proof framework. Our spectral algorithm is designed such that $(a)$ holds. Note that: $(b)$ follows from the optimality of $\hat{\sigma}_{\textnormal{\tiny MAP}}$ and $(c)$ follows from Lemma \ref{['lem: genie-fails-MAP-fails']}. Finally, we get the threshold collapse phenomenon because below the IT threshold, the event in the fourth block happens with probability $o(1)$, and above the IT threshold, the event in the first block happens with high probability by \ref{['eq:genie-success-with-margi']}.
  • Figure 4: ($\mathsf{BEC}$ Channel). For the symmetric SBM, i.e. $\mathsf{SBM}_n(1/2,a,b)$. We let $n=300$ and consider the $\mathsf{BEC}$ side information channel with $\epsilon=n^{-\beta}$ for $\beta \in \{0,5,0.7\}$ in Figures \ref{['fig:subfig1-BEC']} and \ref{['fig:subfig2-BEC']} respectively. We run the spectral algorithm (\ref{['alg:spectral-sbm-rank2']}) over $N=50$ independent trials and plot its success rate for the exact recovery. Lighter pixels correspond to higher probability of success. Finally, the blue and red curves are the theoretical information-theoretic thresholds.
  • Figure 5: ($\mathsf{BSC}$ Channel). The setup is identical to the one in \ref{['fig:BEC']}, but now with the $\mathsf{BSC}$ channel with a flipping probability $\alpha=1/(n^{\beta}+1)=\Theta(n^{-\beta})$ for $\beta\in \{0,5,0.7\}$.
  • ...and 1 more figures

Theorems & Definitions (80)

  • Definition 2.1: General Two Community Block Model (GBM)
  • Definition 2.2: Rank One Spike (ROS)
  • Definition 2.3: Stochastic Block Model (SBM)
  • Definition 2.4: Side Information (SI)
  • Definition 2.5: Gaussian Features (GF)
  • Definition 2.6: Binary Erasure Channel (BEC)
  • Definition 2.7: Binary Symmetric Channel (BSC)
  • Definition 2.8: Exact Recovery
  • Definition 2.9: MAP Estimator
  • Proposition 2.10: dreveton2024exact
  • ...and 70 more