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Multiple colour interacting urns on complete graphs

Benito Pires, Rafael A. Rosales

TL;DR

The paper studies a multi-colour interacting Pólya urn model on a complete graph with a general reinforcement rule defined by a smooth function $\varphi$ and parameter $\alpha$. Using stochastic approximation and a Hopfield-inspired strict Lyapunov function, it proves convergence of the colour-proportion vector $X(n)$ to fixed points of the induced map $\pi$, with convergence to a single fixed point when Fix$(\pi)$ is finite. For small $|\alpha|$ the unique equilibrium is the uniform distribution $x^*=(1/d,\dots,1/d)$, while large $|\alpha|$ can yield multiple stable fixed points, characterized by the Jacobian $J\pi(x)$. An explicit triangle example demonstrates multiple stable fixed points and validates the theoretical framework through eigenvalue analysis. Overall, the work provides a rigorous route to understanding long-run equilibria in non-gradient flows arising from interactions on graphs and reinforces the role of Hopfield-like energy in analysing complex reinforcement dynamics.

Abstract

We present a multiple colour generalisation of the model of graph interacting urns studied by Benaim et. al., Random Struct. Alg., 46: 614-634, 2015. We show that for complete graphs and for a broad class of reinforcement functions governing the addition of balls in the urns, the process of colour proportions at each urn converges almost surely to the fixed points of the reinforcement function.

Multiple colour interacting urns on complete graphs

TL;DR

The paper studies a multi-colour interacting Pólya urn model on a complete graph with a general reinforcement rule defined by a smooth function and parameter . Using stochastic approximation and a Hopfield-inspired strict Lyapunov function, it proves convergence of the colour-proportion vector to fixed points of the induced map , with convergence to a single fixed point when Fix is finite. For small the unique equilibrium is the uniform distribution , while large can yield multiple stable fixed points, characterized by the Jacobian . An explicit triangle example demonstrates multiple stable fixed points and validates the theoretical framework through eigenvalue analysis. Overall, the work provides a rigorous route to understanding long-run equilibria in non-gradient flows arising from interactions on graphs and reinforces the role of Hopfield-like energy in analysing complex reinforcement dynamics.

Abstract

We present a multiple colour generalisation of the model of graph interacting urns studied by Benaim et. al., Random Struct. Alg., 46: 614-634, 2015. We show that for complete graphs and for a broad class of reinforcement functions governing the addition of balls in the urns, the process of colour proportions at each urn converges almost surely to the fixed points of the reinforcement function.
Paper Structure (7 sections, 10 theorems, 65 equations, 1 figure)

This paper contains 7 sections, 10 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Proof of Theorem \ref{['theorem1']}
  4. Proof of Corollary \ref{['corollary1']}
  5. Proof of Example \ref{['example1']}
  6. Stochastic approximations
  7. Hopfield's energy function

Key Result

Theorem 1

Let $\rscal{G} = (V, E)$ be a finite complete graph, $X=\{X(n)\}_{n\ge 0}$ be the process of colour proportions at each urn given by process2, and $\pi$ be the map defined in pi. If the set of fixed points of $\pi$ is finite, then $X=\{X(n)\}_{n\ge 0}$ converges almost surely to the fixed points of

Figures (1)

  • Figure 1: Sample paths for a single colour and different reinforcement parameters $\alpha$.

Theorems & Definitions (25)

  • Theorem 1
  • Corollary 1
  • Example 1
  • Definition 1: strict Lyapunov function
  • Theorem 2
  • proof : Proof of Theorem \ref{['theorem2']}.$(i)$
  • proof : Proof of Theorem \ref{['theorem2']}.$(\mathit{ii})$
  • Definition 2: chain-recurrent set
  • Corollary 2
  • proof
  • ...and 15 more