Exploring the adaptive voter model dynamics with a mathematical triple jump

TL;DR

This paper introduces a novel mathematical approach combining heterogeneous moment expansion, generating functions, and PDE theory to analyze the adaptive voter model, a complex network system, providing more accurate insights than previous approximation methods.

Contribution

The paper presents a new methodology that links network science with PDE theory, enabling better analysis of the adaptive voter model and similar network dynamics.

Findings

The approach yields more accurate results than traditional approximation schemes.

It establishes a connection between network dynamics and PDE theory.

Applicable to a wide range of network models with discrete states.

Abstract

Progress in theoretical physics is often made by the investigation of toy models, the model organisms of physics, which provide benchmarks for new methodologies. For complex systems, one such model is the adaptive voter model. Despite its simplicity, the model is hard to analyse. Only inaccurate results are obtained from well-established approximation schemes that work well on closely-related models. We use this model to illustrate a new approach that combines a) the use of a heterogeneous moment expansion to approximate the network model by an infinite system of ordinary differential equations, b) generating functions to map the ordinary differential equation system to a two-dimensional partial differential equation, and c) solution of this partial differential equation by the tools of PDE-theory. Beyond the adaptive voter models, the proposed approach establishes a connection between network science and the theory of partial differential equations and is widely applicable to the dynamics of networks with discrete node-states.