Research On The Controllability Of Directed Complex Networks Under The Adjacency Matrix

The controllability of complex networks is still a core issue in the field of control.In this paper,we mainly study the controllability of directed complex networks under the adjacency matrix model.The main contents and contributions of this paper are list as follows.The conditions of structural controllability are analyzed in detail.The minimum input theorem and the approach to identifying the minimum driver nodes required for structural controllability by maximum matching is introduced.For the network that is not fully controllable,this paper introduces the concept of controllable subspace and the fixed controllable subspace,and the fixed control subspace problem in complex networks is deeply studied through the graph theory.In this paper we prove that the minimum number of driver nodes or external controllers required to achieve full control of complex networks is equal to the maximum geometric multiplicity of all eigenvalues of the system matrix.For two special types of networks,we offer the simplified formulas to evaluate the minimum number of driver nodes that need to be controlled by relying on the PBH rank condition.In addition,through the analysis of the PBH rank condition,the approach to identifying a minimum set of driver nodes required to control the whole network is improved.Studies have shown that the selection of the minimum set of drive nodes is not only related to the eigenvalues with the maximum geometric multiplicity,but to all eigenvalues of the system matrix.The concept and system model of edge dynamics are introduced,and then the traditional structural theory is applied to the study of edge controllability of complex networks.In this paper,we develop the conditions of structural edge controllability and put forward an algorithm based on the edge control capacity to allow the minimum driven edges set to be identified to ensure full control,which makes up for the lack of research in the edge dynamics and enables a comprehensive understanding of the edge controllability of complex networks.Moreover,we address the controllability issue by bridging the PBH rank condition for the edge dynamics,and derive a universal theoretical framework for the selection of driven edges that can be applied to the networks with arbitrary structures.The conclusions in the paper are verified by examples.