Resilience and Synchronization Behaviour within Mean Field Models

Abstract

The study of the dynamics of large, complex networks is generally very hard. Analytical solutions are rarely available and numerical solutions require immense computation times. Recently, Gao et al. [1]have proposed a new theoretical approach to analyse the average behaviour of complex networks. In this research we discuss the derivation of the mean field approximation and the resulting one dimensional, so called, ”effective” equation. We offer an alternative derivation. We also give an expression for the error in the form of a differential equation. Applying noise to a model of plants and pollinators uncovers a weak point in the given formalism, the effective equation does not correctly predict the average behaviour of the network. For lower dimensional systems, projecting the synchronization manifold on a phase plot reveals why the mean field approximation works well in the specific case. Finally, the theory is applied to some existing models. First, the Generalized Lotka-Volterra model, where it struggles in specific cases where the network has no stable fixed points but the theory does predict one. Second, the Kuramoto model, where a synchronization state of the oscillators is correctly predicted by the theory, and third, a one dimensional array of Josephson junctions where synchronization is also correctly predicted along with the correct stability criteria.