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Sharp exact recovery threshold for two-community Euclidean random graphs

Julia Gaudio, Charlie K. Guan

TL;DR

The paper addresses exact label recovery in the Geometric Hidden Community Model (GHCM) with two communities, establishing that the information-theoretic threshold $\lambda \nu_d \min_{i \neq j} D_+(\theta_i,\theta_j) = 1$ is sharp even without the distinctness-of-distributions assumption. It introduces a two-phase, linear-time algorithm that first achieves almost exact recovery via a data-driven exploration of the visibility graph (Phase I) and then refines the labeling to exact recovery (Phase II). The approach removes a key prior restriction and demonstrates that exact recovery is achievable above the IT threshold, with the method extending to geometric formulations of problems like planted dense subgraph and submatrix localization. The results bridge information-theoretic limits and efficient algorithms in spatial inference, and they open avenues for handling more general $k$-community GHCMs in future work.

Abstract

This paper considers the problem of label recovery in random graphs and matrices. Motivated by transitive behavior in real-world networks (i.e., ``the friend of my friend is my friend''), a recent line of work considers spatially-embedded networks, which exhibit transitive behavior. In particular, the Geometric Hidden Community Model (GHCM), introduced by Gaudio, Guan, Niu, and Wei, models a network as a labeled Poisson point process where every pair of vertices is associated with a pairwise observation whose distribution depends on the labels and positions of the vertices. The GHCM is in turn a generalization of the Geometric SBM (proposed by Baccelli and Sankararaman). Gaudio et al. provided a threshold below which exact recovery is information-theoretically impossible. Above the threshold, they provided a linear-time algorithm that succeeds in exact recovery under a certain ``distinctness-of-distributions'' assumption, which they conjectured to be unnecessary. In this paper, we partially resolve the conjecture by showing that the threshold is indeed tight for the two-community GHCM. We provide a two-phase, linear-time algorithm that explores the spatial graph in a data-driven manner in Phase I to yield an almost exact labeling, which is refined to achieve exact recovery in Phase II. Our results extend achievability to geometric formulations of well-known inference problems, such as the planted dense subgraph problem and submatrix localization, in which the distinctness-of-distributions assumption does not hold.

Sharp exact recovery threshold for two-community Euclidean random graphs

TL;DR

The paper addresses exact label recovery in the Geometric Hidden Community Model (GHCM) with two communities, establishing that the information-theoretic threshold is sharp even without the distinctness-of-distributions assumption. It introduces a two-phase, linear-time algorithm that first achieves almost exact recovery via a data-driven exploration of the visibility graph (Phase I) and then refines the labeling to exact recovery (Phase II). The approach removes a key prior restriction and demonstrates that exact recovery is achievable above the IT threshold, with the method extending to geometric formulations of problems like planted dense subgraph and submatrix localization. The results bridge information-theoretic limits and efficient algorithms in spatial inference, and they open avenues for handling more general -community GHCMs in future work.

Abstract

This paper considers the problem of label recovery in random graphs and matrices. Motivated by transitive behavior in real-world networks (i.e., ``the friend of my friend is my friend''), a recent line of work considers spatially-embedded networks, which exhibit transitive behavior. In particular, the Geometric Hidden Community Model (GHCM), introduced by Gaudio, Guan, Niu, and Wei, models a network as a labeled Poisson point process where every pair of vertices is associated with a pairwise observation whose distribution depends on the labels and positions of the vertices. The GHCM is in turn a generalization of the Geometric SBM (proposed by Baccelli and Sankararaman). Gaudio et al. provided a threshold below which exact recovery is information-theoretically impossible. Above the threshold, they provided a linear-time algorithm that succeeds in exact recovery under a certain ``distinctness-of-distributions'' assumption, which they conjectured to be unnecessary. In this paper, we partially resolve the conjecture by showing that the threshold is indeed tight for the two-community GHCM. We provide a two-phase, linear-time algorithm that explores the spatial graph in a data-driven manner in Phase I to yield an almost exact labeling, which is refined to achieve exact recovery in Phase II. Our results extend achievability to geometric formulations of well-known inference problems, such as the planted dense subgraph problem and submatrix localization, in which the distinctness-of-distributions assumption does not hold.
Paper Structure (9 sections, 4 theorems, 10 equations, 2 algorithms)

This paper contains 9 sections, 4 theorems, 10 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model and Main Results
  3. Exact Recovery Algorithm for the general 2-community GHCM
  4. Phase I
  5. Phase II
  6. Proof of Almost exact recovery
  7. Proof of Proposition \ref{['prop:cstar']}
  8. Propagation
  9. Conclusion

Key Result

Theorem 2.1

Let $G\sim \text{GHCM}(\lambda, n, \pi, P, d)$, $k=2, d\geq 2$, such that (i) $\log(p_{12}(y_{u_{ij}v})/p_{11}(y_{u_{ij}v}))<\eta$, or (ii) $P_{ij}$ are all Gaussians. Then, there exists an efficient algorithm that achieves exact recovery when $\lambda \nu_d \min_{i \neq j} D_+(\theta_i, \theta_j)

Theorems & Definitions (11)

  • Definition 2.1: Geometric Hidden Community Model
  • Definition 2.2: Permissible relabeling
  • Theorem 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3: Visibility graph
  • Proposition 3.1
  • Theorem 3.2
  • proof
  • Lemma 4.1
  • ...and 1 more