Stochastic and deterministic dynamics in networks with excitable nodes

The analysis of the dynamics of a large class of excitable systems on locally tree-like networks leads to the conclusion that at $\lambda=1$ a continuous phase transition takes place, where $\lambda$ 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 $\beta$ which interpolates between stochastic ($\beta=0$) and deterministic ($\beta\rightarrow\infty$) 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 ($\mathbb{I}$) only one initial excited node, ($\mathbb{II}$) a fixed fraction ($10\%$) of excited nodes, for all of which the transition is of first order for $\beta>0$ with a hysteresis effect and a gap function. For the case ($\mathbb{I}$) in the thermodynamic limit the absorbing state in the only phase for all $\lambda$ values and $\beta>0$. We further develop mean-field theories for cases ($\mathbb{I}$) and ($\mathbb{II}$). For case ($\mathbb{II}$) 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.