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Community Recovery on Noisy Stochastic Block Models

Washieu Anan, Gwyneth Liu

TL;DR

This work addresses robust community recovery when networks are influenced by latent geometric structure. It introduces two synergistic components, MASO and GeoDe, to combat geometry-induced noise: MASO uses random-walk based multi-hop PPMI embeddings with triangle-motif–enhanced attention to stabilize spectral clustering, while GeoDe iteratively reweights edges through C- and G-steps to progressively align the graph with latent block structure. Theoretical results establish sharp exact and weak recovery thresholds under latent-kernel SBMs, matching classical SBM theory, and empirical experiments on synthetic data and the Amazon metadata network show substantial gains over baselines and strong denoising effects that improve belief propagation. Together, MASO and GeoDe offer a principled, scalable toolkit for accurate community detection in noisy, geometry-driven networks, with potential extensions to multi-community and dynamic graphs.

Abstract

We study the problem of community recovery in geometrically-noised stochastic block models (SBM). This work presents two primary contributions: (1) Motif--Attention Spectral Operator (MASO), an attention-based spectral operator that improves upon traditional spectral methods; and (2) Iterative Geometric Denoising (GeoDe), a configurable denoising algorithm that boosts spectral clustering performance. We demonstrate that the fusion of GeoDe+MASO significantly outperforms existing community detection methods on noisy SBMs. Furthermore, we show that using GeoDe+MASO as a denoising step improves belief propagation's community recovery by 79.7% on the Amazon Metadata dataset.

Community Recovery on Noisy Stochastic Block Models

TL;DR

This work addresses robust community recovery when networks are influenced by latent geometric structure. It introduces two synergistic components, MASO and GeoDe, to combat geometry-induced noise: MASO uses random-walk based multi-hop PPMI embeddings with triangle-motif–enhanced attention to stabilize spectral clustering, while GeoDe iteratively reweights edges through C- and G-steps to progressively align the graph with latent block structure. Theoretical results establish sharp exact and weak recovery thresholds under latent-kernel SBMs, matching classical SBM theory, and empirical experiments on synthetic data and the Amazon metadata network show substantial gains over baselines and strong denoising effects that improve belief propagation. Together, MASO and GeoDe offer a principled, scalable toolkit for accurate community detection in noisy, geometry-driven networks, with potential extensions to multi-community and dynamic graphs.

Abstract

We study the problem of community recovery in geometrically-noised stochastic block models (SBM). This work presents two primary contributions: (1) Motif--Attention Spectral Operator (MASO), an attention-based spectral operator that improves upon traditional spectral methods; and (2) Iterative Geometric Denoising (GeoDe), a configurable denoising algorithm that boosts spectral clustering performance. We demonstrate that the fusion of GeoDe+MASO significantly outperforms existing community detection methods on noisy SBMs. Furthermore, we show that using GeoDe+MASO as a denoising step improves belief propagation's community recovery by 79.7% on the Amazon Metadata dataset.
Paper Structure (63 sections, 14 theorems, 65 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 63 sections, 14 theorems, 65 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Frameworks
  3. Latent-Kernel SBM
  4. Weak and Exact Recovery
  5. Motif Attention Spectral Operator (MASO)
  6. Motivation
  7. Description of MASO
  8. 1. Random‐walk co‐occurrence and PPMI embeddings.
  9. 2. Triangle‐motif–enhanced attention Laplacian.
  10. Guarantee of Exact Recovery
  11. Limitations of MASO
  12. GeoDe: Iterative Geometric Denoising
  13. Community step (C-step).
  14. Geometry step (G-step).
  15. Edge-reweighting justification.
  16. ...and 48 more sections

Key Result

Theorem 3.1

Let $G$ be drawn from the latent–kernel SBM (Definition def:lksbm) with parameters $a>b>0$, bandwidth $\sigma>0$, and where $c(\sigma)=\mathbb E_{x,y\sim\mathrm{Unif}([0,1]^d)}\bigl[e^{-\|x-y\|^2/(2\sigma^2)}\bigr].$ Fix normalized embeddings $z_i\in\mathbb R^d$ with $\langle z_i,z_j\rangle=\rho_{\rm in}$ if $z_i=z_j$, and $\rho_{\rm out}$ otherwise, $0<\rho_{\rm out}<\rho_{\rm in}<1$. Form $\wid

Figures (3)

  • Figure 1: Left: Average clustering accuracy of four spectral operators as a function of the geometric--noise parameter $\sigma$;“MASO” denotes our Motif--Attention Spectral Operator; “Non--Backtracking (Control)” uses the non--backtracking matrix; “Bethe--Hessian (Control)” is the classical Bethe--Hessian operator; and “Motif--Laplacian (Control)” is the unmodified motif Laplacian. Right: End--to--end community--recovery accuracy of the GeoDe pipeline instantiated with either the MASO backbone or the Bethe--Hessian backbone, compared against three baselines: belief propagation, motif counting, and the graph neural network GCN--MixHop.
  • Figure 2: (a) Ablation on MASO+GeoDe vs. $\sigma$. (b) We show that GeoDe adjusts weights so to capture geometric properties. We calculate a noise metric $\mathcal{N}^k$ via fitting a linear model to predict edge weights from distance and finding the mean-squared residual. Our noise metric captures how the edge weights in GeoDe capture geometric features (see Appendix \ref{['sec:noise-metric']} for details).
  • Figure 3: Empirical Recovery Threshold Validation

Theorems & Definitions (31)

  • Definition 2.1: Latent-Kernel SBM
  • Theorem 3.1: Exact recovery via motif–attention spectral clustering
  • proof : Proof Sketch
  • Theorem 4.1: Convergence of GeoDe
  • proof : Proof sketch
  • Definition C.1: Edge Probability
  • Definition C.2: Average Kernel
  • Theorem C.1: Convergence to a Rescaled SBM
  • proof
  • Lemma C.2: Expectation of mixed weights
  • ...and 21 more