Research On The Nodes Importance Of Complex Network And Control Of Star Network

Based on synchronization theory,exact controllability theory and control energy theory of complex network,we analyze the nodes importance in network synchronization,the exact controllability of a star-type network,and the control energy of star network.Firstly,based on the node contraction method of a network,we investigate the problem of the nodes importance in network synchronization.According to the graph theory of complex network,we analyze the change of network synchronizability between node contraction network and original network on three regular networks from the angle of theory.In the last,we discover that the synchronizability of three regular networks is enhanced after contracting the node.However,we detect that there are synchronizability stimulative nodes and suppressive nodes in BA scale-free network and WS small-world network at the same time by numerical simulations.Secondly,based on the exact controllability theory of complex network,the exact controllability of a star-type network is investigated.By constructing a chain star network,based on the special marking method and recursive method,we provide the exact expressions of characteristic polynomial of the network in theory.From this polynomial,we can directly get the relevant results on exact controllability of our star-type network: the exact controllability of this network is completely determined by its zero eigenvalue.Finally,based on the control energy theory of complex network,the control energy of undirected star network is investigated by using the exact control input matrix which guarantee network controllable.We draw a common conclusion that for control the whole network,the energy consuming is continuous,but for infinite time,it is reduced gradually and finally tends to a constant value.The numerical simulation results prove the validity of our results.