Analytic description of adaptive network topologies in a steady state

Abstract

In many complex systems, states and interaction structure coevolve towards a dynamic equilibrium. For the adaptive contact process, we obtain approximate expressions for the degree distributions that characterize the interaction network in such active steady states. These distributions are shown to agree quantitatively with simulations except when rewiring is much faster than state update, and used to predict and to explain general properties of steady-state topologies. The method generalizes easily to other coevolutionary dynamics. Collective phenomena often feature structured interactions commonly conceptualized with complex networks [1]. In adaptive networks, the interaction structure coevolves with the dynamics it supports, yielding a feedback loop that is common in a variety of complex systems [2,3]. Understanding their asymptotic regimes is a major goal of the study of such systems, and an essential prerequisite for applications. In the particular case of a dynamic equilibrium, each node in the adaptive network undergoes a perpetual change in its state and number of connections to other nodes (its degree), while a comprehensive set of network measures become stationary. 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]. As a consequence, the distributions' dependency on system parameters is difficult to infer and small parameter regions with counterintuitive topologies prone to be overlooked.Here, we revisit the adaptive contact process in dynamic equilibrium [11]. Using a compartmental approach [12], we obtain closed-form ensemble degree distributions dependent on a single external parameter, and show that a coarse-grained understanding of the distributions' shapes can be obtained self-containedly. In particular, the emergence of symmetric ensemble statistics from asymmetric dynamics can be explained. The framework's applicability to static networks as well as to other coevolutionary dynamics is also discussed.Model.-The contact process on an adaptive network models the spreading of a disease in a population without immunity, but with disease awareness [11]. The disease is transmitted along active links that connect infected I-nodes with susceptible S-nodes, letting the susceptible end switch to the Istate with rate p. Moreover, I-nodes recov...