The Collective Behaviour And The Resilience Of Stochastic Networks

Complex networks are ubiquitous in natural,social and artificial systems,such as gene regulatory networks,social networks,mobile sensor networks and so on.As indispensable research branches of complexity science,multi-agent systems and complex dynamical networks are attracting many experts and scholars to carry out in-depth discussion,such as mathematics,physics,robotics and so on.Since stochastic noise is ubiquitous in artificial systems and real networks,it is of great theoretical value and practical significance to discuss the cooperative control of multi-agent systems and complex dynamical networks under noise disturbance.Based on algebraic graph theory,control theory and stability theory,this paper focuses on the collective behavior of stochastic systems.The main work of this paper is as follows:1)The fixed-time consensus problem of multi-agent system under noise disturbance is investigated.Based on the fixed time stability theory,a class of continuous non-Lipschitz protocols are designed,which contains linear control term,finite-time control term,fixed-time control term and noise coupling term.Sufficient conditions for the system to achieve fixed-time consensus under fixed topology and switched topologies are constructed,respectively.The upper bound of the convergence time is also estimated strictly.The influence of control parameters,network topology and noise strength on the convergence speed is analyzed theoretically and numerically.2)The outer synchronization problem of complex networks by means of a class of controllers with or without pinning control is discussed analytically,which combines advantages of the adaptive control technique and the white-noise-based coupling.We compare the designed stochastically adaptive control with the conventional adaptive control through the following two measurements: the convergence time and the energy cost.Numerical simulations show that the added stochastic term enables the adaptive rule to change the coupling gain more rapidly and randomly and thus easier to approach the required control strength for convergence.The influence of network topologies,control parameters and pinning schemes on the system synchronization is also discussed.3)The finite-time synchronization problem of coupled neural networks under noise disturbance is studied by utilizing a new controller,which switches between linear and finite-time control techniques according to different ranges of synchronization error.Based on the stability theory of stochastic differential equations,sufficient conditions to achieve control are derived.The rigorous estimates of time and energy costs of control process are also obtained.The influence of control parameters and network topology on control cost is analyzed.Numerical simulations are given to validate the effectiveness of the theoretical results.4)The resilience of complex networks under noise disturbance is studied.First,a high-dimensional complex system disturbed by internal noise and external noise is modeled.Then,by using the dimension reduction tool,the high-dimensional complex system is reduced to an effective one-dimensional system characterized by the mean state,the recovery time of the networks under noise perturbation is then calculated.