[Paper Review] Mixing Time of Metropolis-Hastings for Bayesian Community Detection

Bumeng Zhuo, Chao Gao|arXiv (Cornell University)|Nov 6, 2018

Complex Network Analysis Techniques30 references4 citations

TL;DR

This paper establishes posterior strong consistency for Bayesian community detection under the optimal signal-to-noise ratio condition and proves that a simple Metropolis-Hastings algorithm with single-flip proposals mixes rapidly when the expected degree is of order log n, achieving a mixing time of O(n³ log n). The analysis relies on canonical path arguments and posterior ratio bounds to control spectral gap and ensure efficient sampling from the posterior distribution.

ABSTRACT

We study the computational complexity of a Metropolis-Hastings algorithm for Bayesian community detection. We first establish a posterior strong consistency result for a natural prior distribution on stochastic block models under the optimal signal-to-noise ratio condition in the literature. We then give a set of conditions that guarantee rapid mixing of a simple Metropolis-Hastings algorithm. The mixing time analysis is based on a careful study of posterior ratios and a canonical path argument to control the spectral gap of the Markov chain.

Motivation & Objective

• To analyze the computational efficiency of Metropolis-Hastings algorithms for Bayesian community detection.
• To establish posterior strong consistency under the optimal signal-to-noise ratio condition, specifically when expected degree is Θ(log n).
• To provide theoretical guarantees on mixing time for a simple Metropolis-Hastings algorithm with single-flip proposals.
• To demonstrate that rapid mixing is achievable despite the exponential state space, challenging the belief that mixing time scales exponentially.

Proposed method

• Uses a uniform prior over community label assignments and a Beta prior over edge probabilities in a stochastic block model.
• Employs a Metropolis-Hastings algorithm with single-flip proposals and acceptance probability proportional to the posterior ratio scaled by a temperature parameter.
• Applies the canonical path method to bound the spectral gap and establish rapid mixing by controlling edge congestion in the Markov chain state graph.
• Analyzes posterior ratios carefully to derive quantitative bounds on transition probabilities and chain convergence.
• Leverages concentration inequalities and entropy-based bounds to control deviations in observed and expected edge counts.
• Uses a frequentist framework to assume the network is generated from a true underlying stochastic block model.

Experimental results

Research questions

• RQ1 Under what conditions is the posterior distribution strongly consistent for community detection in stochastic block models?
• RQ2 Can a simple Metropolis-Hastings algorithm with single-flip proposals achieve rapid mixing for Bayesian community detection?
• RQ3 What is the mixing time of the Metropolis-Hastings chain when the expected degree is Θ(log n)?
• RQ4 Is it possible to achieve polynomial mixing time despite the exponential state space of community assignments?
• RQ5 How does the choice of prior (uniform on labels, Beta on edge probabilities) affect posterior concentration and mixing time?

Key findings

• Posterior strong consistency holds when the expected degree is Θ(log n), which is optimal and improves upon the previous Θ(log²n) condition.
• The Metropolis-Hastings algorithm with single-flip proposals mixes in O(n³ log n) time under the optimal signal-to-noise ratio condition.
• The canonical path argument successfully controls congestion and establishes a polynomial upper bound on mixing time.
• The posterior ratio analysis shows that typical configurations are well-separated from atypical ones, enabling efficient exploration.
• The method achieves rapid mixing even in high-dimensional settings where naive intuition suggests exponential mixing time.
• The theoretical bounds are validated through numerical experiments on simulated networks, showing competitive performance.