On Synchronization Control And Structures Of Complex Networks

Complex network modeling has been considered as an important approach for describing and understanding complex systems. A complex system is composed of many interacted individuals, which can be naturally represented by a graph with individuals denoted by nodes and interactions by links. The ultimate goal of studying complex networks is to understand how topological properties affect the dynamical processes taking place on them. However, that should be fist done is to deeply understand how complex networks possess those common topological features and the evolution mechanisms. Synchronization behavior of complex dynamical networks can well explain many common phenomena in the natural world and the field of engineering. Research on synchronization phenomena in networks with complex topologies has become an important topic in the area of complex network research. This dissertation discusses complex dynamic network synchronization issues, concerning the relations between synchronization and complex network topology structure, effects of weight and evolution mechanisms on synchronizability, as well as the pinning control to synchronization of complex network.The main contents and achievements of this dissertation include:To the problem of the neglect of changes of the number of links of networks when investigated the effects of heterogeneity on synchronizability, we presented theconcept of link density to describe the quantity of link, and investigated the correlation of synchronizability, link density and heterogeneity of degree distribution in unweighted and weighted network. Synchronizability and link density are approximately proportional to TypeⅠnetworks under fixed heterogeneity of degree distribution. However, the synchronizability of TypeⅡnetworks is drastically enhanced by enhancing the link density when the link density is small. Additionally, synchronizability increases weakly by enhancing the link density when the link density is large. The heterogeneity of degree distribution weaken the synchronizability of TypeⅡweighted or unweighted network, and enhance the synchronizability of TypeⅠweighted networks under fixed link density. To the problem of network structures with the optimal synchronizability, firstly, we proposed a synchronization-optimal network model based on rewiring of links under fixed numbers of nodes and links. Compared with other networks, it exhibits a stronger synchronizability. For TypeⅠnetworks, the heterogeneous star network has the optimal synchronizability than other topology structure. The more homogeneous of degree distribution and random link, the better synchronizability of TypeⅡnetworks has. Secondly, we enhance the synchronizability based on random interchanging mechanism restricting the degree distribution of network. We investigated the changing of assortative coefficient, clustering coefficients, characteristic path length and maximal betweenness during the course of enhancing synchronizability of network. There is no relativity between maximal betweenness and synchronizability of TypeⅠnetwork. However, maximal betweenness decrease as the increase of synchronizability of TypeⅡnetwork. We find that the assortative, characteristic path length and clustering coefficient are decrease as the increase of synchronizability for both Types network.Motivated by the observations that the weight of a link in realistic networks is strongly correlated with the product of the corresponding degrees, we presented an asymmetrical weighted scheme corresponding the degrees and investigated the effects of weighted scheme on synchronizability based on different network structures. There is more remarkable effect on enhanced synchronizability by the weighted method when the more heterogeneous of degree distribution of network. However, there is little effect on enhanced synchronizability by the weighted method for the homogeneous random network. Our conclusion is consistent with what Motter presented when weight is correlated with the degree of head only. We have shown that, no matter for WS small world networks or BA scale-free networks, the synchronizability is maximum when the total strength of all in-links of every node is equal to one.To the problem of the relations between evolution mechanism and synchronizability in unweighted network, firstly, we proposed Extended HK model with adjustive clustering coefficients and power-law exponent based on HK model. Triad formation mechanism is extended to old nodes compared with HK model. Settle the shortages of HK:there is no link evolution in old nodes and the numbers of links of a new node coming to network is fixed. Extend HK model is a scale-free network. The effects of triad formation on synchronizability in unweighted network are investigated. Triad formation mechanism can weaken the synchronizability of both Types networks. Additionally, HK network has the better synchronizability than EHK network when the parameter p is small.Triad formation, random selection as well as community structure evolution mechanisms of weighted network are introduced based on BBV model. Three evolution models with high clustering coefficient of weighted network are proposed. Numerical calculations show that our model possesses the same characteristics as the power-law degree, weight, strength distribution, but with the high clustering at the same time. The effects of three evolution mechanisms on synchronizability of networks are investigated. The Triad Formation mechanism can enhance the synchronizability of TypeⅠnetworks, however it can weaken the synchronizability of TypeⅡnetworks. The random selection mechanisms can enhance the synchronizability of both Types networks. The community structure mechanism can weaken the synchronizability of TypeⅠnetworks. The synchronizability of TypeⅡnetworks is drastically weaken when one node and community structure are concurrent.To the problems of pinning control of complex networks, we investigated the pinning adaptive synchronization on weighted and directed complex networks, and obtained several criteria for locally and globally exponentially asymptotic synchronization. The number of pinned nodes can be got by calculating the eigenvalue of the minor matrix of extended coulping matrix, which means that our pinning adaptive controllers are very simple. We develop the vertex centrality index of the weighted and directed networks based on unweighted and directed networks. Finally, numerical simulations are provided to illustrate the effectiveness of the proposed synchronization criteria. The nodes are selected randomly and with the largest centrality index, respectively.