Research On The Minimum Control Energy Of Time-dependent Network Based On Linear Time-invariant System

As a branch of complex networks,temporal networks is a emerging direction of networks research in recent years.This paper mainly studies the problem of the controllable space and the minimum control energy of temporal networks based on linear time-invariant system model.The main contents and contributions of this paper include:1.Firstly,the models of temporal networks and static networks are given.Secondly,the representation of temporal networks based on graph theory and its difference from static networks are given.Finally,the paper classifies the temporal networks innovatively,and explains the equivalent graphical representation and dynamic evolution of the temporal networks.2.The expressions of the controllable space and reachable space of the temporal networks are derived,and the relationship between the controllability of the temporal networks and the controllability of the static networks is explored.A numerical relation is established between the minimum control energy of temporal networks and the characteristic values of Gram matrix,and the general expression of the minimum control energy of temporal networks is derived.At the same time,the expression of the minimum control energy of the static networks is also given.3.Calculate the lower bound of the minimum control energy of the temporal networks.According to the number of snapshots and the number of driver nodes,the temporal networks is divided into three scenarios.In each scenario,the lower bound of the minimum control energy is discussed when the time window is small and large,and the specific calculation method is given.At the same time,the relations between the lower bound of the minimum control energy and the number of driving nodes,the number of snapshots and the negative definite,semi-negative definite and non-negative definite of the matrix are given and proved.4.The minimum control energy of temporal networks is compared with that of static networks.In this paper,the minimum control energy of a static networks with small time window and large time window are given,and the specific numerical relationship between the minimum control energy of a temporal networks and the minimum control energy of a static networks in a special scenario is given.The parameters of temporal networks and static networks are analyzed.Finally,simulation verifies the effectiveness of the conclusion in this paper.