Stochastic and deterministic dynamics in networks with excitable nodes

Abstract

The analysis of the dynamics of a large class of excitable systems on locally tree-like networks leads to the conclusion that at λ = 1 a continuous phase transition takes place, where λ is the largest eigenvalue of the adjacency matrix of the network. This paper is devoted to evaluate this claim for a more general case where the assumption of the linearity of the dynamical transfer function is violated with a non-linearity parameter β which interpolates between stochastic (β = 0) and deterministic (β → ∞) dynamics. Our model shows a rich phase diagram with an absorbing state and extended critical and oscillatory regimes separated by transition and bifurcation lines which depend on the initial state. We test initial states with (I) only one initial excited node, (II) a fixed fraction (10%) of excited nodes, for all of which the transition is of first order for β > 0 with a hysteresis effect and a gap function. For the case (I) in the thermodynamic limit the absorbing state in the only phase for all λ values and β > 0. We further develop mean-field theories for cases (I) and (II). For case (II) we obtain an analytic one-dimensional map which explains the essential properties of the model, including the hysteresis diagrams and fixed points of the dynamics.

Swarming Transition in Super-Diffusive Self-Propelled Particles

“…This leads to the existence of well-defined phase boundaries, known as phase coexistence [17], and results in a bimodal distribution function for the order parameter. The bimodal distribution can be either spatial or temporal, indicating that the two phases may either spatially coexist or dominate during different time intervals [17,46]. The existence of a bimodal distribution function may directly lead to a gap for φ(η) at the transition point [47], around which the hysteresis effect is observed.…”