The Visibility Graph Construction And Local Synchronization Of Interdependent Networks

Since the end of the twentieth century,a new science has emerged in exploring the commonality between complex systems: complex network science.After developing more than 10 years,a single complex network field has made a series of fruitful theoretical research results,which have made great contributions to the human social life,science and technology development.However,with the interdependent relationship between the networks growing more and more today,there are some limitations in the research of a single network.How to construct and identify the structures of the interdependent networks,and how to analyze the function of the networks are key problems for investigating the interdependent networks.Some natural and challenging questions are: what is the backbone structure for an interdependent network? How does the backbone structure affect the dynamics of the network? What is the dynamics of the network? These problems are very critical and have extensive scientific significance and application value of the research.Based on the above mentioned,this paper mainly carried out the following two aspects.(1)With two equal time series,the backbone structure construction of the interdependent networks is studied.Based on the horizontal visibility graph,two new visibility graphs algorithms are proposed: many-to-many directional horizontal visibility graph and one-to-one directional horizontal visibility graph.Using R software,in this paper,the topologies of the interdependent network constructed by these two algorithms are analyzed and evaluated.The results show that both algorithms ensure the simplicity of the backbone structure of the network and inherit the morphological characteristics of the time series.(2)The asymptotically local synchronization for a general interdependent dynamical network with two sub-networks and delays is investigated.The coupling function under consideration in this network is continuous and on behalf of both the intra-network and inter-network.Based on the master stability function method,linearization method and Lyapunov-Krasovskii stability theory combing with the delay differential inequality technique,some elegant strategies are established which guarantee the asymptotically exponential local synchronization for each node in the network.Finally,several numerical simulations show the effectiveness and feasibility of the proposed schemes.