Cellular Automata for excitable media on a Complex Network: The effect of network disorder in the collective dynamics

Complex adaptive systems can be modelled with Networks and Cellular Automata (CA). In the present work, we study the Greenberg–Hastings (GH) cellular automata running in the Watts–Strogatz (WS) network model. We are interested in finding the conditions under which the system operates near a critical point. We introduce the notion of leverage point in such a simple kind of model: a point in parameter space, at criticality, such that we can change the collective state of the system with a minimal effort. Within our proposed framework, the system's response to changes in disorder is maximal at the leverage point. The GH CA includes a transmission coefficient r that sets the threshold size in the dynamics. We evaluate numerically the critical transmission coefficient r_c as a function of the average coordination number of the network K and of the rewiring probability p, where p controls the fluctuations in the coordination number. There is an interval of values in the transmission coefficient r for which the collective state of the system depends on network disorder. This interval narrows as the average coordination number increases and only within it we can tune for criticality by changing disorder alone. Our results are relevant for systems that operate at criticality in order to increase their dynamic range or to operate under optimal information-processing conditions.