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Self-switching random walks on Erdös-Rényi random graphs feel the phase transition

Giulio Iacobelli, Guilherme Ost, Daniel Y. Takahashi

TL;DR

This work analyzes self-switching random walks on Erdős-Rényi graphs, where the environment is resampled at each return to the origin according to a prior $\mu$. By computing the asymptotics of the expected return time $\mathbb{E}_{n,p}[\tau]$ in both dense ($p$ fixed) and sparse ($p=\lambda/n$) regimes, the authors derive the limiting occupation measures for the edge-probability parameter and show a phase-transition-detecting effect in the sparse regime via a density $f(\lambda)$ tied to a Poisson branching process survival probability. The dense regime yields $\mu_{n,\infty}\to\mu$, while the sparse regime yields a nontrivial limit supported on $\lambda>1$, highlighting how self-switching dynamics reveal underlying graph phase transitions. These results connect random-walk return times, renewal theory, and branching-process extinction to the study of dynamically evolving random environments, with potential implications for modeling search strategies and homing behaviors in complex networks.

Abstract

We study random walks on Erdös-Rényi random graphs in which, every time the random walk returns to the starting point, first an edge probability is independently sampled according to a priori measure $μ$, and then an Erdös-Rényi random graph is sampled according to that edge probability. When the edge probability $p$ does not depend on the size of the graph $n$ (dense case), we show that the proportion of time the random walk spends on different values of $p$ -- {\it occupation measure} -- converges to the a priori measure $μ$ as $n$ goes to infinity. More interestingly, when $p=λ/n$ (sparse case), we show that the occupation measure converges to a limiting measure with a density that is a function of the survival probability of a Poisson branching process. This limiting measure is supported on the supercritial values for the Erdös-Rényi random graphs, showing that self-witching random walks can detect the phase transition.

Self-switching random walks on Erdös-Rényi random graphs feel the phase transition

TL;DR

This work analyzes self-switching random walks on Erdős-Rényi graphs, where the environment is resampled at each return to the origin according to a prior . By computing the asymptotics of the expected return time in both dense ( fixed) and sparse () regimes, the authors derive the limiting occupation measures for the edge-probability parameter and show a phase-transition-detecting effect in the sparse regime via a density tied to a Poisson branching process survival probability. The dense regime yields , while the sparse regime yields a nontrivial limit supported on , highlighting how self-switching dynamics reveal underlying graph phase transitions. These results connect random-walk return times, renewal theory, and branching-process extinction to the study of dynamically evolving random environments, with potential implications for modeling search strategies and homing behaviors in complex networks.

Abstract

We study random walks on Erdös-Rényi random graphs in which, every time the random walk returns to the starting point, first an edge probability is independently sampled according to a priori measure , and then an Erdös-Rényi random graph is sampled according to that edge probability. When the edge probability does not depend on the size of the graph (dense case), we show that the proportion of time the random walk spends on different values of -- {\it occupation measure} -- converges to the a priori measure as goes to infinity. More interestingly, when (sparse case), we show that the occupation measure converges to a limiting measure with a density that is a function of the survival probability of a Poisson branching process. This limiting measure is supported on the supercritial values for the Erdös-Rényi random graphs, showing that self-witching random walks can detect the phase transition.
Paper Structure (10 sections, 13 theorems, 92 equations)

This paper contains 10 sections, 13 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Notations and Main Results
  3. Dense case
  4. Sparse case
  5. Proof of the main results
  6. Proof of Theorem \ref{['thm:p-constant']} (dense case)
  7. Proof of Theorem \ref{['thm:p-lambda_over_n']} (sparse case)
  8. Auxiliary results
  9. Proof of Proposition \ref{['prop:LLN_for_the_empirical_occupation_measure']} and \ref{['prop:CLT_for_the_empirical_occupation_measure']}
  10. Proof of Proposition \ref{['prop:P-conditioned']}

Key Result

Proposition 1

For every $A\in\mathcal{F}$, it holds that, almost surely as $T\to\infty$, where, for all $p\in (0,1]$, $\mathbb{E}_{n,p}[\tau]$ is defined in def_undicional_exp_of_return_time_to_1.

Theorems & Definitions (28)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 18 more