Inspired by recent observations on active self-organized critical (SOC) systems, we designed an active pile (or ant pile) model with two ingredients: beyond-threshold toppling and under-threshold active motions. By including the latter component, we were able to replace the typical power-law distribution for geometric observables with a stretched exponential fat-tailed distribution, where the exponent and decay rate are dependent on the activity’s strength (ζ). This observation helped us to uncover a hidden connection between active SOC systems and α-stable Levy systems. We demonstrate that one can partially sweep α-stable Levy distributions by changing ζ. The system undergoes a crossover towards Bak–Tang–Weisenfeld (BTW) sandpiles with a power-law behavior (SOC fixed point) below a crossover point ζ<ζ*≈0.1.

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.