The Research On Control Energy Of Edge Dynamics In Complex Networks

The complex networks is an interdiscipline involving mathematics,graph theory,statistical physics,control science and many other disciplines.In recent years,the scientific community has become more and more keen on the study of complex network controllabillity,and the ultimate goal of studying complex networks is to make the network reach the desired state through external control.If an appropriate set of control signals can drive the network from any initial state to any final state within a limited time,the network is considered controllable.Even if the network meets the requirements of controllability in theory,sometimes the control energy consumed by the control signal is too large to meet the actual needs.Therefore,it is necessary to estimate the size of the control energy and adopt reasonable methods to reduce the control energy.In recent years,a lot of research has been conducted on the controllability and control energy of complex network nodal dynamic systems,but the corresponding control energy in network edge dynamic systems has rarely been involved.The most fundamental difference from the former is that the state of the edge dynamic system is defined on the edge,and the switch matrix of the node corresponds to the interaction between the state variables of inbound and outbound edges.Based on the edge dynamic system,this thesis studies the control energy problems of complex network’s complete control and target control.The complex network edge dynamic system is under complete control,that is,all edges in the network are controllable.First,the optimal control theory is used to derive the optimal control energy required for the system to change to a certain final state,and the optimal control input signal required within the control time is calculated.When the final state of the system cannot be determined,we prove that the gap between the optimal control energies for moving unit distances in different directions is very large,where the maximum is defined as the maximum control energy,and its size is equal to the reciprocal of the smallest eigenvalue of the Gramian matrix.Therefore,in order to obtain the maximum control energy,we must first calculate the Gramian matrix of the network.Different from the commonly used method of directly calculating the Gramian matrix in nodal dynamics,we use a novel indirect method to calculate the Gramian matrix and prove its feasibility.In order to explore the factors that can affect the maximum control energy,we conducted a lot of simulations on the model networks and the real networks,and analyzed the number of driven edges,density,heterogeneity,and control time of the network for the maximum control energy and its rate of change.influences.We can divide the edges in the network into three categories: critical edges,redundant edges,and discontinuous edges.When different edges are used as driven edges,the amount of control energy is also different.Target control is an effective method to reduce the control energy.It does not need to control the entire network,and only controls the target edge set.We also use the optimal control theory to derive the calculation formulas for the optimal control signal and maximum control energy and other parameters under target control.At this time,the maximum control energy is equal to the reciprocal of the minimum eigenvalue of the output Gramian matrix,and the output Gramian matrix is the main submatrix of the Gramian matrix.Through the simulation results of the maximum control energy of the model networks and the real networks under target control,a control strategy that can effectively reduce the control energy is proposed.