Spectral properties of the stochastic block model and their application to hitting times of random walks

TL;DR

This work analyzes hitting times for a simple random walk on stochastic block model graphs by exploiting spectral representations of the hitting times through the normalized adjacency and Laplacian. It establishes a law of large numbers for the average target hitting time H_w and the average starting hitting time H^v, showing H^v ∼ N and H_w ∼ (N/M)(∑_m γ_m)/γ_{B(w)} under natural connectivity conditions, with a central limit theorem for H_w under further spectral assumptions. The authors derive precise eigenvalue and spectral gap bounds for the expected and centered adjacency matrices, including the rescaled and symmetric normalized matrices, and they analyze the case of identical block probabilities p_m, where the eigenstructure is explicit. The results illuminate how SBM spectral properties govern hitting times and their fluctuations, extending LLNs and CLTs known for Erdős-Rényi graphs to a multi-block setting with assortativity quantified by a κ parameter.

Abstract

We analyze hitting times of simple random walk on realizations of the stochastic block model. We show that under some natural assumptions the hitting time averaged over the target vertex asymptotically almost surely given by $N(1+o(1))$. On the other hand, the hitting time averaged over the starting vertex asymptotically almost surely depends on expected degrees in the block the target vertex is in. We also show a central limit theorem for the hitting time averaged over the starting vertex. Our main techniques are a spectral decomposition of these hitting times, a spectral analysis of the adjacency matrix and the graph Laplacian.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 3.1
• proof