Research On The Controllability Of Multi-layer Networks With High-dimensional Nodes

The controllability of complex networks has always been a research hotspot in network science.In the past,people’s research focus has always been on the controllability of single-layer networks with one-dimensional nodes.In recent years,controllability involving high-dimensional networks and multi-layer networks have emerged in an endless stream,and have gradually attracted people’s attention.In this thesis,from the perspective of the rank of the system matrix and the left eigenvector,the controllability of a multi-layer network with high-dimensional nodes is studied.The network topology is described by multiple connection structures with different coupling modes.The main contents and conclusions of this paper are as follows:First,from the perspective of the rank of the system matrix,some criteria for judging whether a multilayer network is controllable are derived,including sufficient and necessary conditions,necessary conditions,and sufficient conditions.In particular,the relationship between the uncontrollable two-layer network topology and the uncontrollable state of the network is given.The relationship between the node system,multi-layer network topology,multiple coupling modes and external input on the controllability of the high-dimensional multi-layer network are analyzed.In addition,for some multi-layer networks with special topologies in some layers,the corresponding controllability conditions are discussed.The research results show that:(1)The controllability of a high-dimensional multilayer network is far more complex than that of a one-dimensional single-layer network;(2)Even if the network topology of each layer in the network is uncontrollable,the network as a whole can be controllable;(3)Even if every node in the network is under control,the network can be uncontrollable;(4)Complementary coupling modes help the network to achieve a controllable state;(5)The existence of ring connections makes the network It is easier to achieve a controllable state.Second,from the perspective of the left eigenvector of the system matrix,the relationship between the left eigenvector of the network and the controllability of the network is analyzed.First of all,for a network of any network topology,we have obtained a more easily verifiable sufficient and necessary condition for judging whether the network is controllable.Secondly,for two types of networks with special network topologies,including two different network topologies that can be block diagonalized and diagonalized at the same time,more specific sufficient and necessary conditions for judging whether they are controllable are given,and specific examples to be verified.At the same time,we found that there is a certain relationship between the left eigenvector of the system matrix and the left eigenvector of the adjacency matrix of the network.Finally,in view of the situation that different network topologies can be diagonalized at the same time,the controllability criteria of two kind of multi-layer networks whose intra-layer topologies are common topologies are given.