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 analyze. Only inaccurate results are obtained from well-established approximation schemes that work well on closely-related models. We use the adaptive voter 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 (ODEs), (b) generating functions to map the ODE system to a twodimensional partial differential equation (PDE), 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 PDEs and is widely applicable to the dynamics of networks with discrete node-states.

A key problem in the study and design of complex systems is the apparent disconnection between the microscopic and the macroscopic. It is not straightforward to identify the local interactions that give rise to an observed global phenomenon, nor is it simple to design a system that will exhibit some desired global property using only local knowledge. Here we propose a methodology that allows for the identification of local interactions that give rise to a desired global property of a network, the degree distribution. Given a set of observable processes acting on a network, we determine the conditions that must satisfied to generate a desired steady-state degree distribution. We thereby provide a simple example for a class of tasks where a system can be designed to self-organize to a given state.

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