Research On Dynamic Network Reconstructions With Strong Noise

Nowadays complex network is one of the hottest fields,and people focus on the research development of complex network no matter in academia or in industry.The importance of complex networks is reflected in the fact that it provides a set of powerful tools to solve problems.The combination of complex network and dynamics expands the investigation of complex network.Network reconstruction is a hot topic in the dynamic complex networks.Network reconstruction refers to inferring network structure through the output time series of the network,including the local dynamics of network nodes and the interactions between nodes.In practical systems,the real parameters of the systems are unknown,and inferring them directly is very difficult,even pays a high cost.If the structure of the system can be inferred directly from the output time series of the system,it will greatly study and control the system.Moreover,in the era of big data,data are constantly generated and accumulated,which provides broad space for the research of network reconstruction.In practice,the noise exists everywhere,there is no pure isolated system,the system will always be more or less affected by noise.Noise will change the original trajectory of the system and makes network reconstruction to be very difficult.Moreover,the challenge of reconstructing network lies in the existence of nonlinearity in real systems.Due to the nonlinearity,the real world reflects complexity.In addition,network reconstruction is also limited by means of measurement.For example,all the variables of the system cannot be measured(i.e.,existing hidden variables),and data is measured with low sampling frequency such that the details of dynamics are not depicted.Due to the above three difficulties of noise,nonlinearity and hidden variables,it becomes extremely difficult and challenging to solve the problems of network reconstruction.Focusing on the network reconstruction under strong noise,we have made the following progress:(Ⅰ)We propose a method to reconstruct nonlinear dynamical networks subjected to strong white noise.Since the method is based on variable expansion and least squares approximations,we named it VELSA.The key of VELSA method lies in the following three points:expanding the nonlinear variables of the original system to the variables of the "extended system" linearizes the original system;decoupling the interaction between the variables and noise by calculating the correlation of variables,thereby eliminating the influence of noise;and reducing the requirement of measurement frequency by the integral solution of linear equations.The reconstruction method can infer the nonlinear dynamics of nodes,the connection weights between nodes and the statistical characteristics of Gauss white noise from the time series of all variables.The reconstruction errors of VELSA method are analyzed theoretically and verified by numerical simulation experiments.The numerical simulation results are in good agreement with the theoretical results,and the effectiveness and robustness of the algorithm are fully proved.(Ⅱ)We propose a random variable resetting method to infer the coupling functions of nodes.By randomly resetting the state variables of a target node,the coupling between the response node and the target node is detected.By resetting the variables of all nodes,all connections between these nodes can be reconstructed.Random variable resetting is an active method.Its core idea is to make the dynamics of the target node acting on the response node equivalent to the average coupling function and its fluctuations by randomly resetting the state variables of the target node.When there exist no correlation or very small correlation between the fluctuations and the variables of the target node,the reconstruction can be calculated by VELSA method or high-order correlation computations(HOCC)method.The effect of randomly resetting the derivative of the state variable on the reconstructed results is discussed.The numerical results have fully verified the validity of the theoretical derivation.(Ⅲ)The reconstruction of networks with hidden variables is studied under different known conditions.Hidden variables mean that the variables are not detectable or difficult to be detected in the system,and sometimes people do not even know their existence.If we know the statistical information of hidden variables due to the prior knowledge via development of science and technology,or hidden variables can be approximately linearized,we apply the integral methods to solve the dynamics,so that the reconstruction problem can be solved from the time series of measurable variables.If the information about the hidden variables is unknown,we inject Gaussian white noise into measurable variables to enhance the interaction between measurable variables,and to weaken the interaction between measurable variables and the hidden variables,further obtain the network structure of the sub network composed of measurable variables.We discuss the influence of various factors on the accuracy of network reconstruction through numerical simulations.