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 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.

Local Symmetry and Global Structure in Adaptive Voter Models

“…1 The tunable interaction between opinion and topology updates generates polarized networks of opinion-based communities. AVMs are therefore often considered "model organisms" [32] of endogenous fragmentation, polarization, and segregation in social and information networks.…”

Analytic description of adaptive network topologies in a steady state

“…A prominent example is the degree distribution, the probability distribution of node degrees. For a wide class of adaptive networks in dynamic equilibrium, the shapes of stationary degree distributions appear to be insensitive to initial conditions in state and topology [4][5][6][7] -not only when taken over the whole network (network degree distributions), but also when describing ensembles consisting only of nodes of same state (ensemble degree distributions).While much work on adaptive networks assumes random connectivity in the form of Poissonian degree distributions [8,9], coevolutionary dynamics can generate highly structured steady-state topologies [7,10]. Analytic expressions for the ensuing degree distributions have been so far lacking, and their investigation has relied on numerical procedures [4][5][6][7].…”

“…Analytic expressions for the ensuing degree distributions have been so far lacking, and their investigation has relied on numerical procedures [4][5][6][7]. As a consequence, the distributions' dependency on system parameters is difficult to infer and small parameter regions with counterintuitive topologies prone to be overlooked.…”

“…A prominent example is the degree distribution, the probability distribution of node degrees. For a wide class of adaptive networks in dynamic equilibrium, the shapes of stationary degree distributions appear to be insensitive to initial conditions in state and topology [4][5][6][7] -not only when taken over the whole network (network degree distributions), but also when describing ensembles consisting only of nodes of same state (ensemble degree distributions).…”

“…15 also holds with swapped indices due to the symmetry. Considering the active phase before fragmentation transition [6,8], [SI] > 0 must hold. We moreover assume that the two end nodes of active links are statistically equivalent, setting P S (k) = P I (k) ≡ P (k).…”