Complex Network Structure Controllability Index Optimization Algorithm And Control Scheme Research

The control effectiveness of complex networks in the real world is highly dependent on the choice of control scheme.The controllability index K is a measure of the number of control time steps for the network to reach the desired state,and a smaller controllability index means that all nodes in the network take less time to reach the desired state.The system also obtains different structural controllability indices under the same number of drive nodes because different sets of control nodes are selected,and this property has important implications for practical applications,which require comprehensive consideration of control cost and control time indices to select the network control scheme.In this paper,firstly,based on control theory,we introduce the method of network description from two aspects of matrix and graph theory of networks.The theory of step controllability is introduced through the discussion of graph and algebraic theory of network control.The physical meaning and equivalence of the parameter indexes such as step controllable,structural controllability index ,control time step,and basic control path length of the network are analyzed.And the properties related to the cactus structure and the maximal matching algorithm of the graph are discussed.The maximum matching algorithm provides an effective method for the study of the control scheme in this paper and also increases the structural controllability index of the network.Secondly,the problem of structural controllability exponential optimization of directed networks is studied.Based on the "Cactus" structure of the directed network,the system topology is considered as a complex network composed of several interrelated cacti.The cactus structure can be controlled from the root node to achieve a controllable state,so the set of cactus root nodes is the set of driving nodes of the original system.It is also demonstrated from both graph theory and algebraic perspectives that the structural controllability index of a complex network composed of several independent cacti may be reduced by adding connected edges between cacti to form a new system.On the contrary,the structural controllability index of a complex network topologically partitioned into several cactus structures is an upper bound of the structural controllability index of the original complex network.Based on the graph and algebraic proofs,the structural controllability index optimization algorithm for directed network cactus structures is proposed to obtain a lower structural controllability index ,while balancing the control time step and control cost.The effectiveness of the algorithm is verified by simulating the real network and ER random network data.Thirdly,the problem of structural controllability index optimization of undirected networks is studied.In the undirected network,the concept of "Controllable tree" is proposed from the cactus structure and the research method of directed percolation,and the control node selection problem of undirected network is studied.For the undirected network composed of controllable tree structure,the upper bound of the structural controllability index after dividing the undirected network into several subsystems with uniform topology is proved from two perspectives of matrix and graph theory,which provides the theoretical basis for the algorithm.Based on graph and algebraic proofs,optimization algorithm based on controllable trees for uniform topological partitioning of undirected networks is proposed for the selection of control nodes for undirected networks.So that the undirected network has a lower structural controllability index K,while balancing the control time step and control cost of the undirected network.This study simulates the real data set and ER randomly generated network to verify the effectiveness of the algorithm.Finally,the structural controllability of the undirected network is investigated.In contrast to the controllable tree structure,we propose an alternative unit system for studying undirected networks,the "controllable branch",and give an algebraic proof of its structural controllability.The selection of control schemes for controllable trees under single drive node control is investigated by means of matching in graph theory,and it is concluded that the control scheme of controllable trees driven by a single node is not unique,but is related not only to the number of nodes of "controllable branches" and "controllable flowers",but also to the arrangement of controllable branches and controllable flowers.