Synchronizability?Optimized And Spectral Coarse Graining Of The Hierarchical Network

In recent years,along with the deepen complex network theory research and its expanding application,and due to the high degree of syncretism between the complex network and other interdisciplinary integration that people start to use complex network theory to study various existing large complex systems in the real world.Therefore,complex network is becoming an important tool to describe and study complex systems.In the theoretical study of complex networks,the relationship between the synchronization capability and network structure of complex networks is one of the hottest issues.In the complex network research,the network model is decomposed by various mathematical models and algorithms,and the repeated modules in the network are identified and the complexity of the complex network is simplified,analysis of the structure and function between the modules.This is called the complex network module analysis method.Because there are modular and hierarchical features exist in plenty of complexity network structure.Therefore,module analysis of complex network can not only be applied to biology,medicine and other fields,but also is one of the important methods to study network structure.The construction method of the hierarchical network makes itself have significant modularity characteristics,which makes the hierarchical network play a very important role in the study of network modularity.In addition,some typical network models,such as random network,small world network,scale-free network and so on,do not have hierarchical module topology,and the hierarchical network has a obvious modular structure,In the study of the synchronization ability of complex networks,the theoretical research of synchronization dynamics is difficult,so the related research results are less.This paper focuses on the three aspects of synchronization,optimization and coarse granulation of hierarchical network.The main contents are summarized as follows:(1)To study the synchronization ability of the hierarchical network.The influence of the note number of initial modules and the number of network layers on the synchronization ability of the hierarchical network is analyzed by simulation.A large number of simulation analyses show that the initial module size of the the hierarchical network synchronization capability are related to the number of layers.If the hierarchical network is a type of network,the number of the initial module nodes of the network is positively related to the network synchronization capability.If the hierarchical network is the type 1 network,the number of the initial module nodes of the network is positively related to the network synchronization capability,the number of the initial module nodes of the network and synchronization capability of the network are the positive correlation relationship.On the contrary,if the hierarchical network is the type 2 network,its synchronization capability decreases with the increase of the number of initial module nodes.For the hierarchical network with the same initial node number,the increase of hierarchical network will reduce the synchronization ability of the network.(2)Using the segmentation method to reduce the maximum node betweenness of hierarchical network,so as to improve the synchronization ability of hierarchical network.McGraw and Menzinger put forward the random cross edge method when studying the relationship between the synchronization ability and the clustering coefficient of the complex network.This method can also be used to reduce the average path length of the network,so as to improve the synchronization ability of the network.However,in the case of hierarchical network,a large number of simulation experiments have found that the random cross edge method has the disadvantage of too low efficiency in the actual calculation process.To this end,we put forward a kind of efficient parallel connection method(PRC Method).The theoretical analysis and a lot of numerical simulation results show that the PRC method in reducing the average path length of the network efficiency is obviously better than the random cross edge method.Besides,we also compare the segmentation method and the PRC method,and give the two methods of advantages and disadvantages.(3)Using the improved spectral coarse-grained algorithm(ISCG algorithm)to study the maintaining condition of synchronization capability of the hierarchical network.It is found that the maintaining condition of synchronization ability of the hierarchical network is better after coarse-grained.In addition,when the hierarchical network layer is fixed,if the hierarchical network is type 1 network,the number of initial module nodes become greater,the hierarchical network based on the ISCG method of spectral coarse-grained is better.If the hierarchical network is the type 2 network,the size of initial module node number does not have regular influence on coarse graining effect.When the number of initial module nodes is comfirm,the more the number of layers is,the better the spectral coarse-grained effect of the hierarchical network is based on the ISCG method.