| The simultaneous study of complex networks has received much attention in recent years,and these studies are of great academic value and practical significance for people to deeply understand and apply complex networks.The structural hole is an important concept in the study of social networks,and has a mediating role in establishing connections between multiple unconnected groups.Research related to structural hole theory has been widely used in many fields such as sociology,economics,computer science and transportation.Structural holes exist in many real networks,and the simultaneous study of complex networks with structural holes can help us to better understand and control real networks.In general,a realistic complex network consists of several sub-networks in which the nodes,the number of nodes and the connections vary to a greater or lesser extent.In this paper we combine the concept of structural holes,choose different chaotic systems as network nodes,and use WS and NW small-world networks or scale-free networks as sub-networks to build a model of heterostructured complex networks with a structural hole to study the synchronisation of networks.The main research works are as follows.I.A heterostructured complex network with a structural hole is modelled.Chen chaotic system and Lorenz chaotic system are selected as network nodes,and a controller is designed based on Lyapunov stability theorem.Theoretical derivation and numerical simulation demonstrate that the heterostructured complex network with a structural hole can be fully synchronised under the action of the controller.Subsequently,noise disturbances are added to the network after synchronisation is achieved to test the immunity of the network synchronisation,and numerical simulations demonstrate the robustness of the synchronisation,providing a feasible method for the synchronisation control of realistic heterostructured networks with a structural hole.Finally,considering that complex networks have diverse topologies,a heterostructured complex network with a structural hole is constructed by increasing the number of nodes and changing the sub-network structure at the same time,and the synchronisation of this network is investigated.The numerical simulation results show that increasing the number of nodes and changing the sub-network structure can still achieve complete synchronization of the network,indicating that the method has certain universality.II.The synchronization problem of heterostructured time-lag complex networks with a structural hole is investigated,and a controller is designed based on Lyapunov stability theorem.Firstly,the synchronisation problem of heterostructured complex networks with a structural hole where coupled time lags exist in the network is investigated.A suitable controller is designed using Lyapunov stability theory by placing a Chen chaotic system at the first node of the complex network with a structural hole and a Lorenz chaotic system at the rest of the nodes in the network,and by applying the controller to the nodes in the network where the Lorenz chaotic system is placed,complete synchronisation of the network is achieved.After that,the effect of different coupling time lags τ on the network synchronisation is further investigated.The numerical simulation results show that the network can eventually be synchronised for different coupling time lags,and the smaller the value of τ the faster the network is synchronised.Secondly,the synchronization of a heterostructured double time lag complex network with a structural hole is investigated by using a time lagged Chen chaotic system and a time lagged Lorenz chaotic system as the nodes of the network and considering the coupling time lag of the network.By applying control to the nodes of the time-lagged Lorenz chaotic system placed in the network,the network achieves full synchronization,and numerical simulations demonstrate that the heterostructured dual time-lagged network with a structural hole achieves full synchronization. |
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