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Hitting times in the stochastic block model

Andrea Ottolini

Abstract

Given a large connected graph $G=(V,E)$, and two vertices $w,\neq v$, let $T_{w,v}$ be the first hitting time to $v$ starting from $w$ for the simple random walk on $G$. We prove a general theorem that guarantees, under some assumptions on $G$, to approximate $\mathbb E[T_{w,v}]$ up to $o(1)$ terms. As a corollary, we derive explicit formulas for the stochastic block model with two communities and connectivity parameters $p$ and $q$, and show that the average hitting times, for fixed $v$ and as $w$ varies, concentrates around four possible values. The proof is purely probabilistic and uses a coupling argument.

Hitting times in the stochastic block model

Abstract

Given a large connected graph , and two vertices , let be the first hitting time to starting from for the simple random walk on . We prove a general theorem that guarantees, under some assumptions on , to approximate up to terms. As a corollary, we derive explicit formulas for the stochastic block model with two communities and connectivity parameters and , and show that the average hitting times, for fixed and as varies, concentrates around four possible values. The proof is purely probabilistic and uses a coupling argument.
Paper Structure (13 sections, 5 theorems, 81 equations, 2 figures)

This paper contains 13 sections, 5 theorems, 81 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Erdős-Rényi case
  3. Stochastic block model
  4. Main result
  5. Set-up
  6. Main result
  7. Main ingredients
  8. A useful lemma
  9. Relationship between the stationary distributions
  10. Construction of the coupling
  11. Proofs
  12. Proof of Theorem \ref{['thm:main']}
  13. Proofs of Corollaries \ref{['thm:sbm']} and \ref{['thm:er']}

Key Result

Corollary 1

Let $G=(V,E)$ be an Erdős-Rényi random graph with parameters $n$ and $p=p_n$ where $\log^5 n\cdot n^{-1}\cdot p^{-8}\rightarrow 0.$ Then, with high probability, we have that for every pair $w\neq v$ the hitting time $T_{w,v}$ satisfies where, conditional on $\deg v$, the random variables $X$ and $Y$ are independent and

Figures (2)

  • Figure 1: Average hitting times for an instance of the stochastic block model from different starting points. They cluster around four values that are predicted by our result (Theorem \ref{['thm:sbm']}). The clusters reflect the adjacency and community structure of the graph.
  • Figure 2: The average hitting times to a given vertex $v$ for an instance of the stochastic block model on three communities of a $1000$ individuals each. The hitting times cluster around six values.

Theorems & Definitions (10)

  • Corollary 1
  • Corollary 2
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof
  • proof : Proof of Corollary \ref{['thm:sbm']}
  • proof : Proof of Corollary \ref{['thm:er']}