# Dynamics on complex networks

Based on a theoretical model for opinion spreading on a network, through avalanches, the effect of external field is now considered, by using methods from non-equilibrium statistical mechanics. The original part contains the implementation that the avalanche is only triggered when a local variable (a so called awareness) reaches and goes above a threshold. The dynamical rules are constrained to be as simple as possible, in order to sort out the basic features, though more elaborated variants are proposed. Several results are obtained for a Erdos-Renyi network and interpreted through simple analytical laws, i.e. Weibull distribution, sometimes reduced to a mere exponential, or logistic function, i.e., (i) the sizes, durations, and number of avalanches, including the respective distributions, (ii) the number of times the external field is applied to one possible node before all nodes are found to be above the threshold, (iii) the number of nodes still below the threshold and the number of hot nodes (close to threshold) at each time step.

M. Ausloos and F. Petroni, Threshold Model for Triggered Avalanches on Networks, in *Stock Markets*, F. Petroni, F. Prattico and G. D’Amico, Eds.

(Nova Sc., New York, 2013) pp.83-101