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Hydrodynamic limit for repeated averages on the complete graph

Alberto M. Campos, Tertuliano Franco, Markus Heydenreich, Marcel Schrocke

TL;DR

This paper analyzes the hydrodynamic limit of a repeated averaging process on the complete graph, where pairs of vertices are updated to their mean and the process evolves on a timescale of order $N$. Using the entropy method and martingale techniques, the authors prove that the empirical distribution of opinions converges to a unique limit described by a measure-valued differential equation, and, when the initial distribution has a density, to a density $\rho(t,u)$ solving the nonlocal equation $\partial_t \rho(t,u)=4(\rho*\rho)(2u)-2\rho(t,u)$. This non-diffusive, coagulation-like dynamics contrasts with the heat-equation limit observed on lattices, reflecting the strong non-local interactions of the complete graph. The results establish existence, uniqueness, and absolute continuity under suitable initial data and connect the evolution to a Smoluchowski-like coagulation framework via a density that concentrates mass according to a convolution term evaluated at $2u$.

Abstract

We establish a hydrodynamical limit for the averaging process on the complete graph with N vertices, showing that, after a timescale of order N, the empirical distribution of opinions converges to a unique measure. Moreover, if the initial distribution is absolutely continuous concerning the Lebesgue measure, the limiting measure remains absolutely continuous and its density satisfies a non-diffusive differential equation, that resembles the Smoluchowski coagulation equation.

Hydrodynamic limit for repeated averages on the complete graph

TL;DR

This paper analyzes the hydrodynamic limit of a repeated averaging process on the complete graph, where pairs of vertices are updated to their mean and the process evolves on a timescale of order . Using the entropy method and martingale techniques, the authors prove that the empirical distribution of opinions converges to a unique limit described by a measure-valued differential equation, and, when the initial distribution has a density, to a density solving the nonlocal equation . This non-diffusive, coagulation-like dynamics contrasts with the heat-equation limit observed on lattices, reflecting the strong non-local interactions of the complete graph. The results establish existence, uniqueness, and absolute continuity under suitable initial data and connect the evolution to a Smoluchowski-like coagulation framework via a density that concentrates mass according to a convolution term evaluated at .

Abstract

We establish a hydrodynamical limit for the averaging process on the complete graph with N vertices, showing that, after a timescale of order N, the empirical distribution of opinions converges to a unique measure. Moreover, if the initial distribution is absolutely continuous concerning the Lebesgue measure, the limiting measure remains absolutely continuous and its density satisfies a non-diffusive differential equation, that resembles the Smoluchowski coagulation equation.

Paper Structure

This paper contains 2 sections, 5 theorems, 51 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:HydrodinamiclimitClick']}

Key Result

Theorem 1

Assuming that $\omega_0 = (\nu_i)_{i \in [N]} \in \Omega_N$, where $\nu_i$ are i.i.d. random variables with distribution $\nu$ with compact support in the interval $[-M,M]$. For any time $T>0$, the sequence of trajectories of random measures $\{\pi_t^N : 0 \leqslant t \leqslant T\}$ converges weakly

Figures (3)

  • Figure 1: Simulation of the evolution of the opinion density, starting with an initial distribution $\nu \sim \mathrm{Ber}(1/2)$ and considering $N = 3000$ opinions.
  • Figure 2: Simulation of the evolution of the opinion density, starting with an initial absolutely continuous distribution $\nu$ such that $\mathbb{P}\left(\nu<t\right)=\int_0^t 2x dx$, for $t\in [0,1]$, and considering $N=10^6$ opinions.
  • Figure 3: Side by side, the initial distribution of opinions at time zero and at time $t$. The set $\mathcal{I}$ at time $t$ is related to the set $\widehat{\mathcal{I}}$ at time zero. In this case, intervals are mapped to intervals, and using perturbation estimates, one can derive bounds for the Lebesgue measure of the set $\widehat{\mathcal{I}}$.

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • proof
  • Lemma 4
  • proof
  • Proposition 5
  • proof