Abstract
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.
Highlights
Synchronization of coupled maps in networks is presently an active research topic [1]
We show that the temporal variation and randomness of the connection topology can enhance synchronizability in many cases; there are instances where they reduce synchronizability
The Hajnal diameter of the coupling matrix sequence is presented as a measure for the synchronizability of the graph topology
Summary
Synchronization of coupled maps in networks is presently an active research topic [1]. J=1 where [Gij(t)]m i,j=1, t ∈ Z+, are stochastic matrices It was proved in reference [11] that the connectivity of the switching graphs plays a key role in the consensus dynamics of multi-agent networks with switching topologies. Θ(t)· represents a metric dynamical system {Ω, F , P, θ(t)}, where Ω is the state space, F is the σalgebra, P is the probability measure, and θ(t) is the semiflow satisfying θ(t+s) = θ(t) ◦θ(s), where θ(0) is the identity map, G(θ(t)ω) = [Gij (θ(t)ω)]m i,j=1 ∈ Rm×m denotes the coupling matrix at time t and is measurable on (Ω, F ), and F (x) = [f (x1), · · · , f (xn)] is a differentiable function. The purpose of this paper is to study the synchronization of the coupled map network (5) with time-varying topology. Further examples indicate that the communication between vertices in the dynamical networks might play an important role in synchronizability
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