Dynamic Network Reconstructions Based On Random State Variable Resetting

Networked complex systems are ubiquitous in various fields among natural and social sciences.The structure determining functions of a networked complex system is a common consensus in the network science community,and interactions between nodes are the foundation of the complex dynamic behaviors and diverse functions of a network.However,the actual system is often a "black box",and we often cannot directly observe the interaction between nodes.Then an alternative idea is using the output data to reconstruct interactions in the dynamical network.In recent years,utilizing output data to reconstruct complex networks has aroused the interest of many scientific researchers.The challenges encountered in dynamic network reconstruction include but are not limited to the following:the complexity of the network structure;the influence of multiple degrees of freedom caused by the huge network scale;the strong nonlinearity of the system;the influence of complex and diverse unknown inputs,such as noises;the existence of hidden variables or hidden nodes.In the actual system,the speed of signal propagation is often limited,so reconstructing interactions among nodes is affected by time delays.In dynamic networks,numerous time delays are another major challenge for reconstruction.In recent years,some methods have been proposed to reconstruct dynamical networks without time delays.Typical statistical methods,such as cross-correlation,Granger causality,mutual information,transfer entropy,etc.,are usually used to define functional connections,which describe the dynamic dependence between nodes from a statistical perspective,and provide limited information.Nevertheless,statistical methods are easy to calculate and have universality,utilizing statistical methods to reconstruct dynamic networks has always been one of the hotspots of research.Researchers have explored the reconstruction of dynamic networks by solving ordinary differential equations(ODEs).For examples,the linearization method can reconstruct the system moving near a stable point,and the methods of expanding basis and expanding variables can reconstruct a general nonlinear dynamic network.These methods not only reveal connections of networks,but also provide information on local dynamics and coupling functions.Solving ODEs requires the output data of the whole network,which is a disadvantage for the network in which some nodes or variables are unavailable.The existence of noises will increase the difficulty of network reconstruction,sometimes,will also enhance the target interactions.Therefore,some scholars have proposed methods for reconstructing using the statistical properties of noises.In addition,there are some proactive methods,such as driving-response controlling,copy-synchronization,random phase resetting.The random phase resetting method is only applicable to the phase oscillator network.Through further research,Dr.Shi et al proposed random state variable resetting(RSVR)method,which can reconstruct a general continuous coupled dynamic network.The method is immune to noise,hidden nodes and variables,influence of structure and size,satisfactory reconstruction results can be obtained when there is a large enough amount of data.However,as a method of actively regulating the network,each regulation of the system requires a certain price,and we may not be able to guarantee sufficient data.In this paper,we extended the RSVR method to time-discrete dynamic network,and studied the factors that affect the reconstruction results of this method under a limited amount of data.The results show that when the amount of data is limited,the reconstruction result is sensitive to the fluctuation of the response node.When the fluctuations are relatively small,the random state variable resetting method requires only a small amount of data to obtain a fairly high reconstruction accuracy;when the fluctuation is relatively large,the reconstruction results of a small amount of data have a large error.In addition,the research work of this paper also discusses the influence of multiple attractors.In order to overcome the influence of time delays,we apply random state variable resetting method to the reconstruction of time-delay coupled dynamic network.Through theoretical analysis,the nearest neighbor correlation(NNC)function under different delays is defined,and the NNC function can be used to determine the real time delay.The error of the estimated delay is not greater than one sampling interval.The numerical results verify the correctness of the theoretical analysis and the feasibility of the method.After the delay is determined,the equivalent coupling function can be reconstructed by high order correlation calculation(HOCC)method.The error of the delay will cause a certain deviation in the coefficients in the equivalent coupling function.Reducing the sampling time interval,that is,reducing the error of the delay,can reduce the deviation of the equivalent coupling function.In addition,a method of reconstructing the time-delay coupled binary network using the information gain ratio is proposed.First define the information gain ratio of each pair of nodes under different delays,and find the maximum value of the information gain ratio of each pair of nodes.The numerical simulation method is used to study the distribution of the maximum information gain ratio of different models(Kirman,SIS and Ising)with different network structures(WS small world network,ER random network and BA scale-free network).Based on the characteristics of the distribution,a method of reconstructing the edges by setting a threshold and a method of reconstructing the network by utilizing the number of connected edges are proposed.For the former,the influence of the threshold on the reconstruction performance is discussed;for the latter method,the influence of the network sparsity on the reconstruction result of the network connection is discussed.