Exploring the adaptive voter model dynamics with a mathematical triple jump

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.

14 pages, 3 figures; paper extended, new section and additional explanations added

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Cited by in corpus (10)

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• Macroscopic description of complex adaptive networks co-evolving with dynamic node states
• Concurrency-induced transitions in epidemic dynamics on temporal networks
• Zealotry Effects on Opinion Dynamics in the Adaptive Voter Model
• Prevention of infectious diseases by public vaccination and individual protection
• Spatially inhomogeneous population dynamics: beyond the mean field approximation
• Meta-food-chains as a many-layer epidemic process on networks
• Fitting In and Breaking Up: A Nonlinear Version of Coevolving Voter Models
• Local Symmetry and Global Structure in Adaptive Voter Models
• Analytic description of adaptive network topologies in steady state