Synchronization And Superdiffusion On Duplex Networks

Real networks often interact with other networks of similar or different natures,forming what is known as multiplex networks.By considering a multiplex network,new perspectives in the understanding of classical network phenomena,such as synchronization,diffusion,can be obtained,sometimes with counterintuitive results.Synchronization and diffusion in complex networks have been widely studied,while very few works have investigated synchronization and diffusion in multiplex networks.In this dissertation,by using the master stability function,generating functionology methods,matrix theory,we study synchronization and diffusion on duplex networks from the perspective of topological structures,so as to uncover the potential factors influencing synchronizability and diffusion speed.The main contents and contributions of the dissertation are as follows:1.For sychronizability on duplex regular networks,based on the master stability function framework we study potential factors influencing synchronizability,including the network size,the coupling strength and the inter-layer connection density.From both theoretical and numerical results,we obtain that with the same size,same coupling strength and same inter-layer connection density,duplex fully-connected networks have the best synchronizability,the next is duplex stars,followed by duplex rings,and duplex chain networks have the worst synchronizability.The order of synchronizability on duplex regular networks is similar to that on single networks.We find that synchronizability of duplex regular networks is always worse than that of its isolated layer.Furthermore,for duplex fully-connected or star networks with the fixed coupling strength,we find that the more the inter-layer links are,the better the synchronizability is.However,for duplex ring or chain networks,partial inter-layer links can yield the same synchronizability as that caused by one-to-one inter-layer links.Though these findings are obtained by using duplex regular networks,they will provide insight into understanding synchronizability of general multiplex networks,and facilitate the selection of network parameters for obtaining optimum synchronizability.2.For the occurrence of superdiffusion on duplex networks,we propose a sufficient condition by applying the generating functionology from combinatorial mathematics.Our result indicates that,with fixed values of intra-layer diffusion constants,superdiffusion emerges from a certain combination of different network topologies,which provides a topological mechanism to regulate the whole diffusion process.On a more general ground,we provide two linking strategies to expand the applicability of the above topological mechanism on real-world networks,and construct a class of specific duplex networks on which superdiffusion occurs.Additionally,we investigate diffusion on some duplex regular networks with a fixed intra-layer coupling where at least one layer is a directed graph.In this setting,we find an unexpected phenomenon in a duplex network that is composed of two directed rings of the same direction.Given the ubiquity of diffusion in the real world,our results constitute a significant step toward a better understanding of superdiffusion processes on multiplex networks.3.Consider a typical duplex network with two identical layers,which has a high heterogeneity of node degrees,consisting of only hubs and peripheral nodes.We study synchronizability and diffusion with two different inter-layer correlations.Theoretical and numerical results indicate that synchronization or diffusion on such a duplex network with negative inter-layer correlation is better than that with positive inter-layer correlation.We further obtain that if each the single network has a highly heterogeneous degree distribution,superdiffusion is very likely to occur on the duplex consisting of these two identical networks when the inter-layer correlation is negative.