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Mixing time for a noisy SIS model on graphs

Wasiur R. KhudaBukhsh, Yangrui Xiang

TL;DR

This work analyzes the mixing time of a noisy SIS process on graphs, where external infection drives the dynamics in addition to local infection. Using coupling techniques and the path-coupling framework, it proves that in the strong external-infection regime the mixing time scales as $O(n\log n)$, with matching lower bounds up to constants, on general graphs and on several random graph families. The results extend to Erdős–Rényi graphs, regular multigraphs, and Galton–Watson trees, with high-probability bounds under model-specific constraints on the infection parameters. Methodologically, the paper combines contraction arguments, coalescent couplings, and concentration inequalities (e.g., McDiarmid, Chernoff) to establish sharp, model-robust mixing-time bounds, shedding light on how external noise accelerates convergence in epidemic-like networks.

Abstract

We study the mixing time of the noisy SIS (Susceptible-Infected-Susceptible) model on graphs. The noisy SIS model is a variant of the standard SIS model, which allows individuals to become infected not just due to contacts with infected individuals but also due to external noise. We show that, under strong external noise, the mixing time is of order $O(n \log n)$. Additionally, we demonstrate that the mixing time on random graphs, namely Erdös--Rényi graphs, regular multigraphs, and Galton--Watson trees, is also of order $O(n \log n)$ with high probability.

Mixing time for a noisy SIS model on graphs

TL;DR

This work analyzes the mixing time of a noisy SIS process on graphs, where external infection drives the dynamics in addition to local infection. Using coupling techniques and the path-coupling framework, it proves that in the strong external-infection regime the mixing time scales as , with matching lower bounds up to constants, on general graphs and on several random graph families. The results extend to Erdős–Rényi graphs, regular multigraphs, and Galton–Watson trees, with high-probability bounds under model-specific constraints on the infection parameters. Methodologically, the paper combines contraction arguments, coalescent couplings, and concentration inequalities (e.g., McDiarmid, Chernoff) to establish sharp, model-robust mixing-time bounds, shedding light on how external noise accelerates convergence in epidemic-like networks.

Abstract

We study the mixing time of the noisy SIS (Susceptible-Infected-Susceptible) model on graphs. The noisy SIS model is a variant of the standard SIS model, which allows individuals to become infected not just due to contacts with infected individuals but also due to external noise. We show that, under strong external noise, the mixing time is of order . Additionally, we demonstrate that the mixing time on random graphs, namely Erdös--Rényi graphs, regular multigraphs, and Galton--Watson trees, is also of order with high probability.
Paper Structure (14 sections, 8 theorems, 97 equations)

This paper contains 14 sections, 8 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Mixing properties
  3. Mathematical contributions
  4. The noisy SIS model on graphs
  5. Definition and dynamics of the model
  6. Mixing time
  7. Upper bound on the mixing time
  8. Lower bound on the mixing time.
  9. Mixing properties on random graphs.
  10. Mixing property on Erdös--Rényi graphs.
  11. Mixing property on regular multigraphs.
  12. Mixing property on Galton--Watson trees
  13. Binomial offspring
  14. Poisson offspring

Key Result

Lemma 3.1

For the coupled Markov chains $(\sigma_{\sigma_0}(t);t\geq 0)$ and $(\eta_{\eta_0}(t);t\geq 0)$, after one step of update, we have that with $\gamma\coloneqq (1-\kappa)a+(1-a)\kappa-2(n-1)\kappa(1-\kappa)$ and $\gamma>0$.

Theorems & Definitions (20)

  • Lemma 3.1: Contraction property for strong external infection regime
  • proof : Proof of Lemma \ref{['contraction property']}
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['thm:upper_strong']}
  • Remark 3.1
  • Claim 4.1
  • proof : Proof of \ref{['second moment']}
  • Theorem 4.1
  • proof : Proof of Theorem \ref{['thm:lower_strong']}
  • Remark 4.1
  • ...and 10 more