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Exact recovery in Gaussian weighted stochastic block model and planted dense subgraphs: Statistical and algorithmic thresholds

Aaradhya Pandey, Sanjeev Kulkarni

TL;DR

This paper provides the information-theoretic threshold in terms of the signal-to-noise ratio (SNR) of the Gaussian weighted version of the Stochastic block model with two symmetric communities and proves that when SNR $<1$, no statistical estimator can exactly recover the community structure with probability bounded away from zero, and shows that when SNR $>1$, the Maximum likelihood estimator itself succeeds in exactly recovering the community structure with probability approaching one.

Abstract

In this paper, we study the exact recovery problem in the Gaussian weighted version of the Stochastic block model with two symmetric communities. We provide the information-theoretic threshold in terms of the signal-to-noise ratio (SNR) of the model and prove that when SNR $<1$, no statistical estimator can exactly recover the community structure with probability bounded away from zero. On the other hand, we show that when SNR $>1$, the Maximum likelihood estimator itself succeeds in exactly recovering the community structure with probability approaching one. Then, we provide two algorithms for achieving exact recovery. The Semi-definite relaxation as well as the spectral relaxation of the Maximum likelihood estimator can recover the community structure down to the threshold value of 1, establishing the absence of an information-computation gap for this model. Next, we compare the problem of community detection with the problem of recovering a planted densely weighted community within a graph and prove that the exact recovery of two symmetric communities is a strictly easier problem than recovering a planted dense subgraph of size half the total number of nodes, by establishing that when the same SNR$< 3/2$, no statistical estimator can exactly recover the planted community with probability bounded away from zero. More precisely, when $1 <$ SNR $< 3/2$ exact recovery of community detection is possible, both statistically and algorithmically, but it is impossible to exactly recover the planted community, even statistically, in the Gaussian weighted model. Finally, we show that when SNR $>2$, the Maximum likelihood estimator itself succeeds in exactly recovering the planted community with probability approaching one. We also prove that the Semi-definite relaxation of the Maximum likelihood estimator can recover the planted community structure down to the threshold value of 2.

Exact recovery in Gaussian weighted stochastic block model and planted dense subgraphs: Statistical and algorithmic thresholds

TL;DR

This paper provides the information-theoretic threshold in terms of the signal-to-noise ratio (SNR) of the Gaussian weighted version of the Stochastic block model with two symmetric communities and proves that when SNR , no statistical estimator can exactly recover the community structure with probability bounded away from zero, and shows that when SNR , the Maximum likelihood estimator itself succeeds in exactly recovering the community structure with probability approaching one.

Abstract

In this paper, we study the exact recovery problem in the Gaussian weighted version of the Stochastic block model with two symmetric communities. We provide the information-theoretic threshold in terms of the signal-to-noise ratio (SNR) of the model and prove that when SNR , no statistical estimator can exactly recover the community structure with probability bounded away from zero. On the other hand, we show that when SNR , the Maximum likelihood estimator itself succeeds in exactly recovering the community structure with probability approaching one. Then, we provide two algorithms for achieving exact recovery. The Semi-definite relaxation as well as the spectral relaxation of the Maximum likelihood estimator can recover the community structure down to the threshold value of 1, establishing the absence of an information-computation gap for this model. Next, we compare the problem of community detection with the problem of recovering a planted densely weighted community within a graph and prove that the exact recovery of two symmetric communities is a strictly easier problem than recovering a planted dense subgraph of size half the total number of nodes, by establishing that when the same SNR, no statistical estimator can exactly recover the planted community with probability bounded away from zero. More precisely, when SNR exact recovery of community detection is possible, both statistically and algorithmically, but it is impossible to exactly recover the planted community, even statistically, in the Gaussian weighted model. Finally, we show that when SNR , the Maximum likelihood estimator itself succeeds in exactly recovering the planted community with probability approaching one. We also prove that the Semi-definite relaxation of the Maximum likelihood estimator can recover the planted community structure down to the threshold value of 2.
Paper Structure (39 sections, 55 theorems, 370 equations)

This paper contains 39 sections, 55 theorems, 370 equations.

Table of Contents

  1. Introduction and related works
  2. Gaussian weighted stochastic block model: description
  3. Gaussian weighted stochastic block model: results and proof sketch
  4. Gaussian weighted planted dense subgraph model: description
  5. Gaussian weighted planted dense subgraphs: results and proof sketch
  6. A comparison of exact recovery in two models and conclusion
  7. Organization
  8. Gaussian weighted stochastic block model
  9. Gaussian weighted planted dense subgraph model
  10. Gaussian weighted stochastic block model
  11. Model description
  12. Maximum likelihood estimation
  13. Statistical impossibility
  14. The statement
  15. First step -- the first moment method
  16. ...and 24 more sections

Key Result

Theorem 1

Let $G_n \sim$ SBM$(n,\mu_1, \mu_2, \tau^2)$ with $\mu_1, \mu_2$ in the critical scaling regime def:CritGWSBM0, and $\alpha > \beta$, $\tau >0$ be constants. If SNR$<1$ then MLE fails in recovering the community label $\sigma^*$ with high probability.

Theorems & Definitions (90)

  • Theorem 1: Statistical impossibility of exact recovery
  • Lemma 2
  • Definition 3
  • Lemma 4
  • Definition 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Theorem 10: Statistical possibility of exact recovery
  • ...and 80 more