Detecting and describing dynamic equilibria in adaptive networks

Abstract

We review modeling attempts for the paradigmatic contact process (or SIS model) on adaptive networks. Elaborating on one particular proposed mechanism of topology change (rewiring) and its mean field analysis, we obtain a coarse-grained view of coevolving network topology in the stationary active phase of the system. Introducing an alternative framework applicable to a wide class of adaptive networks, active stationary states are detected, and an extended description of the resulting steady-state statistics is given for three different rewiring schemes. We find that slight modifications of the standard rewiring rule can result in either minuscule or drastic change of steady-state network topologies.

Moment-closure approximations for discrete adaptive networks

“…Both of these models are adaptive network models of similar complexity, and, depending on personal taste, either can be considered as the most simple non-trivial adaptive network. However, for the adaptive SIS model, the dynamics can be faithfully captured already by simple approximation schemes [19], with more sophisticated approaches leading expectedly to a further improvement [23,70,71]. By contrast, for the adaptive voter model, simple approximation schemes only provide unsatisfactory results [13,15] and, as we show here, more sophisticated approaches can actually perform worse.…”

Asymmetric coevolutionary voter dynamics

“…In the dynamics described by Eqs. (1) (referred to as asymmetric VM in the following), link aquisition and opinion adoption are tailored to specific node ensembles, so that highlyskewed degree distributions can ensue for a wide range of parameters (see [7,21] as examples for related dynamics). For that reason, instead of the regular random graphs taken in [6,10], we decide for initial Erdős-Rényi (ER) graphs (featuring a wider Poissonian degree distribution) and keep η = 1 throughout, for it has been shown that for even more heterogenous degree distributions, the respective moment closure maintains validity [7,14].…”

Analytic description of adaptive network topologies in a steady state

“…Hence for a < 1, infection outweighs the rewiring bias towards S-nodes, yielding a higher connectivity of I-nodes [10]. From Eq.…”

“…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].…”

“…Finally, it has been shown [5,10] that ensemble degree distributions are linked to stationary distributions that describe other features of the steady state. Obtaining the latter would be another contribution to the study of dynamic equilibria in coevolutionary dynamics.…”