Projects per year
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.