Excitable nodes on random graphs: Relating dynamics to network structure

Rhythmic activity in complex systems is generated and sustained through interactions among the constituent units. In this paper we study the interplay between topology and dynamics of excitable nodes on random networks. The nodal dynamics are discrete, each node being in three possible states: active, refractory, or silent. Loading rules are defined whereby a subset of active nodes may be able to convert a silent node into an active one at the next time step. In the case of simple loading (SL) a silent node becomes active if it receives input from any neighbor. In the majority rules (MR) loading, a silent node fires when the majority of its neighbors are active. We address the question of whether a particular network design pattern confers dynamical advantage for the generation and sustainment of rhythmic activity. We find that the intrinsic properties of a node and the rules for interaction between them determine which structural features of the graph permit sustained activity. With SL the level of activity in the graph increases monotonically with the probability of connections between nodes, while for MR, the level of activity may be either monotonic or nonmonotonic, depending on parameters.