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Stochastic Block Model for Hypergraphs: Statistical limits and a semidefinite programming approach

Chiheon Kim, Afonso S. Bandeira, Michel X. Goemans

TL;DR

The paper establishes a sharp information-theoretic threshold for exact recovery in the stochastic block model on $k$-uniform hypergraphs, showing that recovery is possible via ML when $I( ext{alpha}, ext{beta})>1$ and impossible when $I( ext{alpha}, ext{beta})<1$. It introduces a tractable truncate-and-relax SDP algorithm and proves exact recovery under a related, parameter-dependent threshold $I_{ ext{sdp}}( ext{alpha}, ext{beta})>1$, while also giving a complementary lower bound $I_2( ext{alpha}, ext{beta})$ governing the algorithm’s failure; simulations indicate the truncation threshold aligns with $I_2( ext{alpha}, ext{beta})=1$, suggesting a gap between statistical limits and SDP performance for larger $k$. The results generalize two-community graph SBM phase transitions to hypergraphs and quantify the trade-off between statistical limits and computationally efficient recovery methods. Overall, the work clarifies when convex relaxations can attain the information-theoretic limits and where computational gaps persist, thereby guiding future algorithm design for higher-order relational data.

Abstract

We study the problem of community detection in a random hypergraph model which we call the stochastic block model for $k$-uniform hypergraphs ($k$-SBM). We investigate the exact recovery problem in $k$-SBM and show that a sharp phase transition occurs around a threshold: below the threshold it is impossible to recover the communities with non-vanishing probability, yet above the threshold there is an estimator which recovers the communities almost asymptotically surely. We also consider a simple, efficient algorithm for the exact recovery problem which is based on a semidefinite relaxation technique.

Stochastic Block Model for Hypergraphs: Statistical limits and a semidefinite programming approach

TL;DR

The paper establishes a sharp information-theoretic threshold for exact recovery in the stochastic block model on -uniform hypergraphs, showing that recovery is possible via ML when and impossible when . It introduces a tractable truncate-and-relax SDP algorithm and proves exact recovery under a related, parameter-dependent threshold , while also giving a complementary lower bound governing the algorithm’s failure; simulations indicate the truncation threshold aligns with , suggesting a gap between statistical limits and SDP performance for larger . The results generalize two-community graph SBM phase transitions to hypergraphs and quantify the trade-off between statistical limits and computationally efficient recovery methods. Overall, the work clarifies when convex relaxations can attain the information-theoretic limits and where computational gaps persist, thereby guiding future algorithm design for higher-order relational data.

Abstract

We study the problem of community detection in a random hypergraph model which we call the stochastic block model for -uniform hypergraphs (-SBM). We investigate the exact recovery problem in -SBM and show that a sharp phase transition occurs around a threshold: below the threshold it is impossible to recover the communities with non-vanishing probability, yet above the threshold there is an estimator which recovers the communities almost asymptotically surely. We also consider a simple, efficient algorithm for the exact recovery problem which is based on a semidefinite relaxation technique.

Paper Structure

This paper contains 21 sections, 19 theorems, 182 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Stochastic Block Model for graphs: An overview
  3. The Stochastic Block Model for hypergraphs
  4. Main results
  5. Maximum-likelihood estimator
  6. Sharp phase transition in $\mathsf{HSBM}(n,p,q;k)$
  7. Lower bound
  8. Upper bound
  9. Truncate-and-relax algorithm
  10. Laplacian of the adjacency matrix
  11. Semidefinite relaxation and its dual
  12. Performance of the algorithm
  13. Limitation of the algorithm
  14. Discussion
  15. Tail probability of weighted sum of binomial variables
  16. ...and 6 more sections

Key Result

Theorem 1

Exact recovery in $\mathsf{HSBM}(n,p,q;k)$ is possible if $I(\alpha,\beta)>1$, and impossible if $I(\alpha,\beta)<1$ where $I(\alpha,\beta) = \frac{1}{2^{k-1}}(\sqrt{\alpha}-\sqrt{\beta})^2$.

Figures (2)

  • Figure 1: Visualization of $I$, $I_2$, $I_{sdp}$ when $k=6$: (a) the solid line represents $I(\alpha,\beta)=1$, (b) the circled line represents $I_2(\alpha,\beta)=1$, and (c) the x-marked line represents $I_{sdp}(\alpha,\beta)=1$. The dashed black line is the graph of $\alpha = \beta$.
  • Figure 2: Result of simulation of the truncate-and-relax algorithm for $k=6$ and $n=500$. Each gray-scale block corresponds to a pair $(\alpha,\beta)$, and its color denotes the success rate over 30 trials (black corresponds to 0 success, and brighter color correspond to higher success rate). The solid line represents $I(\alpha,\beta)=1$, the circled line represents $I_2(\alpha,\beta)=1$, and the x-marked line represents $I_{sdp}(\alpha,\beta)=1$.

Theorems & Definitions (30)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Conjecture 1.2
  • Proposition 1
  • Theorem 4
  • proof
  • Lemma 1
  • Lemma 2
  • ...and 20 more