The Equivalent Representation Of Time Series And Complex Networks And Its Application

There are two typical methods to explore the intrinsic dynamic behavior of complex systems,namely the time series method and the complex network method.Both of them have been developed rapidly and improved in their respective fields.Many classical statistics have been proposed to describe the characteristics of complex systems.However,the two paradigms have completely different analytical perspectives,which make it difficult to draw a comprehensive conclusion by analyzing and representing complex systems under a single paradigm.The transformation between the two paradigms has attracted more and more attention in recent years.It is expected that a comprehensive understanding of the system can be achieved by combining the two paradigms to characterize the internal dynamic behavior of the system.In spite of this,the present transformation methods still lack the theoretical basis of equivalence,and the extent to which the transformation between the two paradigms can represent the intrinsic characteristics of the system is still to be discussed.Therefore,it is of great theoretical and practical significance to find a method of equivalence transformation between the two paradigms,establish the basis of equivalence representation of the two paradigms,and expand the application of the complementary or hybrid methods of the two paradigms in the study of complex systems.This paper intends to study the transformation between time series and complex networks both in theory and applications.In theoretical aspect,this paper focuses on the quasi-isometric property of transformation methods from the perspective of metric space.Therefore,it is necessary to first study the quasi-isometric theory of metric space,unify the concept of quasi-isometric in metric space,and explain the relationship between different definitions of quasi-isometric.Then,using the relation between the amplitude difference of time series and the shortest path length of complex network,the direct proof of quasi-isometric of univariate separable time series and corresponding complex network is found.In addition,for the univariate time series which do not satisfy the separability,this paper introduces the auxiliary sequence to ensure the connectivity of corresponding complex network,and then proves the quasi-isometric property of time series and complex network.Furthermore,the transformation and equivalence of multivariate time series and corresponding complex network are studied in this paper.Thus,the quasi-isometric property of any time series and its corresponding complex network is proved theoretically.Inpractical application,the critical point and the coupling relationship of complex systems are studied from the perspective of complex network in univariate and multivariate cases,respectively.This paper is dedicated to solve the critical point detection problem by studying the variation of complex network statistics varying with system parameters.Then,the trends of the complex network statistics that can detect the critical point of the system is fitted into a prediction model for the critical point of the system,and the prediction of the critical point of the complex system is realized from the perspective of the complex network.Considering that edge overlap of multilayer complex network can distinguish the dynamic state of the system,the change of edge overlap with time is analyzed in detail in this paper,with the aim of finding an index that can reflect the coupling between layers.Through the found index,we hope that not only the dynamic state can be identified,but also the coupling direction and coupling strength can be described,which can further guide how to control the complex system.The main research contents are as follows:Aiming at the lack of equivalence theory of transformation method between time series and complex network,this paper analyzes the equivalence of amplitude difference mapping method from time series to complex network.In this paper,two quasi-isometric isomorphism theories of metric space are firstly summarized and their relations are found.Secondly,a method to directly prove the quasi-isometric isomorphism of separable univariate time series and corresponding complex networks by amplitude difference mapping method is proposed.Then,the concept of threshold separable time series is proposed,and the quasi-isometric isomorphism of time series and complex network is extended to threshold separable time series.Finally,an auxiliary sequence method which is dependent on the transformation threshold is proposed to prove the quasi-isometric isomorphism between the general univariate time series and the complex network of amplitude difference mapping method.On the basis of quasi-isometric isomorphism,this paper proposes a method to reproduce the original time series on the basis of the shortest path length sequence,and gives an algorithm to select the reference points of this method,and verifies the quasi-isometric isomorphism of time series and complex networks by numerical experiments.In addition,in order to expand the practicability of the amplitude difference mapping method,this paper extends the amplitude difference mapping method to the multivariate time series,and gives the corresponding theoretical proof of isometric isomorphism relation.In order to enrich the method of detecting the critical points of univariate system and expand the application of the method which converts time series into complex networks,the critical points of the system are detected and predicted from the perspective of complex network.Firstly,the amplitude difference mapping method is used to map the time series into a complex network.Second,for comparison,this paper also inspects the other three methods transforming time series to the complex networks,including horizontal visibility graph method,ordinal pattern transition network method and compression code word transition network method.We verify the feasibility of complex network perspective to analyze complex systems for critical point,and conclude that the complex network by the quasi-isometric mapping method has more critical point indicators.Finally,the curve fitting method is applied to the differences,between the parameter of the system and the parameter at its critical point,and the corresponding complex network statistics.According to the fitting results,the critical point of the system can be predicted by the statistics of the complex network.In order to understand the intrinsic properties of multi-dimensional systems,the coupling relationship between different dimensions of the system is studied from the perspective of complex network.In this paper,after mapping multivariate time series into multilayer complex network by amplitude difference mapping method,the similarity characteristics of the interlayer structure of multilayer complex network are investigated,and the dynamic states of the system are accurately described.On this basis,the concept of interlayer entropy is proposed in this paper,which can not only reflect the coupling strength between layers,but also describe the coupling direction.Then,it is theoretically proved that there is an equivalence relationship between the interlayer entropy of multilayer complex networks and the transfer entropy of multivariate time series,which further validates the quasi-isometric isomorphism of the amplitude difference mapping method - the equivalence relationship between the respective characterization indexes is maintained.