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The analysis of the dynamics of a large class of excitable systems on locallytree-like networks leads to the conclusion that at $\lambda=1$ a continuousphase transition takes place, where $\lambda$ is the largest eigenvalue of theadjacency matrix of the network. This paper is devoted to evaluate this claimfor a more general case where the assumption of the linearity of the dynamicaltransfer function is violated with a non-linearity parameter $\beta$ whichinterpolates between stochastic ($\beta=0$) and deterministic($\beta\rightarrow\infty$) dynamics. Our model shows a rich phase diagram withan absorbing state and extended critical and oscillatory regimes separated bytransition and bifurcation lines which depend on the initial state. We testinitial states with ($\mathbb{I}$) only one initial excited node,($\mathbb{II}$) a fixed fraction ($10\%$) of excited nodes, for all of whichthe transition is of first order for $\beta>0$ with a hysteresis effect and agap function. For the case ($\mathbb{I}$) in the thermodynamic limit theabsorbing state in the only phase for all $\lambda$ values and $\beta>0$. Wefurther develop mean-field theories for cases ($\mathbb{I}$) and($\mathbb{II}$). For case ($\mathbb{II}$) we obtain an analytic one-dimensionalmap which explains the essential properties of the model, including thehysteresis diagrams and fixed points of the dynamics.

Stochastic and deterministic dynamics in networks with excitable nodes