Langevin approach for the dynamics of the contact process on annealed scale-free networks

TL;DR

This paper models the contact process on annealed scale-free networks using a Langevin equation, revealing how topological fluctuations influence critical behavior and survival times.

Contribution

It introduces a Langevin equation framework for the contact process on annealed networks, explicitly incorporating stochastic fluctuations and analyzing their effects on critical phenomena.

Findings

Fluctuations cause anomalous scaling with system size.

Outliers with high connectivity shift the critical point.

Analytical expressions for survival time and active site density.

Abstract

We study the dynamics of the contact-process, one of the simplest nonequilibrium stochastic processes, taking place on a scale-free network. We consider the network topology as annealed, i.e. all links are rewired at each microscopic time step, so that no dynamical correlation can build up. This is a practical implementation of the absence of correlations assumed by mean-field approaches. We present a detailed analysis of the contact process in terms of a Langevin equation, including explicitly the effects of stochastic fluctuations in the number of particles in finite networks. This allows us to determine analytically the survival time for spreading experiments and the density of active sites in surviving runs. The fluctuations in the topological structure induce anomalous scaling effects with respect to the system size when the degree distribution has an "hard" upper bound. When the upper bound is soft, the presence of outliers with huge connectivity perturbs the picture even more, inducing an apparent shift of the critical point. In light of these findings, recent theoretical and numerical results in the literature are critically reviewed.