Chaos synchronization in networks of coupled maps with time-varying topologies

Complexity of dynamical networks can arise not only from the complexity of the topological structure but also from the time evolution of the topology. In this paper, we study the synchronous motion of coupled maps in time-varying complex networks both analytically and numerically. The temporal variation is rather general and formalized as being driven by a metric dynamical system. Four network models are discussed in detail in which the interconnections between vertices vary through time randomly. These models are: 1) i.i.d. sequences of random graphs with fixed wiring probability, 2) groups of graphs with random switches between the individual graphs, 3) graphs with temporary random failures of nodes, and 4) the meet-for-dinner model where the vertices are randomly grouped. We show that the temporal variation and randomness of the connection topology can enhance synchronizability in many cases; however, there are also instances where they reduce synchronizability. In analytical terms, the Hajnal diameter of the coupling matrix sequence is presented as a measure for the synchronizability of the graph topology. In topological terms, the decisive criterion for synchronization of coupled chaotic maps is that the union of the time-varying graphs contains a spanning tree.