General coevolution of topology and dynamics in networks

We present a general framework for the study of coevolution in dynamical systems. This phenomenon consists of the coexistence of two dynamical processes on networks of interacting elements: node state change and rewiring of links between nodes. The process of rewiring is described in terms of two basic actions: disconnection and reconnection between nodes, both based on a mechanism of comparison of their states. We assume that the process of rewiring and node state change occur with probabilities Pr and Pc, respectively, independent of each other. The collective behavior of a coevolutionary system can be characterized on the space of parameters (Pr,Pc). As an application, for a voter-like node dynamics we find that reconnections between nodes with similar states lead to network fragmentation. The critical boundaries for the onset of fragmentation in networks with different properties are calculated on this space. We show that co-evolution models correspond to curves on this space describing functional relations between Pr and Pc. The occurrence of a one-large-domain phase and a fragmented phase in the network is predicted for diverse models, and agreement is found with some earlier results. The collective behavior of the system is also characterized on the space of parameters for the disconnection and reconnection actions. In a region of this space, we find a behavior where different node states can coexist for very long times on one large, connected network.