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Functional central limit theorem for the subgraph count of the voter model on dynamic random graphs

Simone Baldassarri, Nikolai Kriukov

TL;DR

The main result of this paper is a functional central limit theorem that proves that, under a proper centering and scaling, the joint functional of the vector of subgraph counts converges to a specific multidimensional Gaussian process.

Abstract

In this paper we consider two-opinion voter models on dynamic random graphs, in which the joint dynamics of opinions and graphs acts as one-way feedback, i.e., edges appear and disappear over time depending on the opinions of the two connected vertices, while the opinion dynamics does not depend on the edge process. Our goal is to investigate the joint evolution of the entries of a voter subgraph count vector, i.e., vector of subgraphs where each vertex has a specific opinion, in the regime that the number of vertices grows large. The main result of this paper is a functional central limit theorem. In particular, we prove that, under a proper centering and scaling, the joint functional of the vector of subgraph counts converges to a specific multidimensional Gaussian process.

Functional central limit theorem for the subgraph count of the voter model on dynamic random graphs

TL;DR

The main result of this paper is a functional central limit theorem that proves that, under a proper centering and scaling, the joint functional of the vector of subgraph counts converges to a specific multidimensional Gaussian process.

Abstract

In this paper we consider two-opinion voter models on dynamic random graphs, in which the joint dynamics of opinions and graphs acts as one-way feedback, i.e., edges appear and disappear over time depending on the opinions of the two connected vertices, while the opinion dynamics does not depend on the edge process. Our goal is to investigate the joint evolution of the entries of a voter subgraph count vector, i.e., vector of subgraphs where each vertex has a specific opinion, in the regime that the number of vertices grows large. The main result of this paper is a functional central limit theorem. In particular, we prove that, under a proper centering and scaling, the joint functional of the vector of subgraph counts converges to a specific multidimensional Gaussian process.

Paper Structure

This paper contains 12 sections, 5 theorems, 103 equations, 1 table.

Table of Contents

  1. Introduction
  2. One-way feedback model: definition and main result
  3. Definition of the model
  4. Graph notations
  5. Main result
  6. Discussion
  7. Proof of the main result
  8. Proof of Lemma \ref{['exp_and_cov']}
  9. Proof of Lemma \ref{['asymptotic_normality']}
  10. Proof of Lemma \ref{['lem:tightness']}
  11. Proof of Proposition \ref{['prop:variance']}
  12. Two-way feedback model: some insights

Key Result

Theorem 2.2

Let $G_n(t)$ for $t\in[0,T]$ be a one-way feedback voter model on a dynamic random graph with $n$ vertices (as described in Section sec:defmodel), and let $H_1,\ldots,H_m$ be deterministic voter graphs for some $m\in\mathbb{N}$. Then, as $n\to\infty$, the process $\boldsymbol{X}_n(\cdot)$ defined in in distribution in $\mathbb{D}([0,T],\mathbb{R}^m)$, where $\boldsymbol{X}(\cdot)$ is a centered ve

Theorems & Definitions (5)

  • Theorem 2.2
  • Proposition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3