Better synchronizability in generalized adaptive networks.

In this paper, to study the interaction between network structure and dynamical property in the context of synchronization, a previously proposed adaptive coupling method is generalized where the coupling strength of a node from its neighbors not only develops adaptively according to the local synchronization property between the node and its neighbors (dynamical part) but also is modulated by its local structure, degree of the node with the form 1/k(i)(alpha) (topological part). We can show both numerically and analytically that the input coupling strength of the network after adaptation displays a power-law dependence on the degree, k(-theta), where the exponent theta is controlled by alpha as theta=(1+alpha)/2. Compared to the original adaptive coupling method, after the addition of modulation, the distribution of the node's intensity is tunable and can be more homogenous with alpha approximately 1, which results in better synchronizability. It is also found that the synchronization time can shrink greatly. Our theoretical work in the context of synchronization provides not only a deeper understanding of the interplay between structure and dynamics in real world systems, such as opinion formation and concensus, but also potential approaches to manipulate the global collective dynamics through local adaptive control.