Statistical Index Analysis Of Transformation From Time Series To Complex Networks

Complex network is a technique used in studying the complex system, itstopology represents the individual interaction within the system. The complexnetwork becomes increasingly vital as a basic method which is applied foranalyzing the function and property of the complex system. In the modernsociety, internet plays a more and more important role in the normal life ofhumans, along with the development of the computer technology. Meanwhile,with the constantly developing of traditional basic mathematics, appliedmathematics which is based on the computer software is advancing at the sametime. As the new research tool in21st century, complex network enables peopleto better understand artificial or real exist complex system, as well as promotingthe exploring of the applied mathematics.This dissertation focused on studying the network topology which wasachieved by several deterministic model systems and experimental data thatpossessed certain probability distribution. In addition, through certain convertmethods mapping the several types of stationary time series which had the sameprobability density into final network, furthermore, weighting above networktopology through the method of adding weight value. Finally, this research hadobserved and explored the weight distribution of the network model after addingweight value, which focused on examining the property of weight distributionafter mapping various stationary time series into weight network, as well asverifying and judging weight distribution of the weight network model.This dissertation focuses on analysing statistic index of weighted network,based on the previous study of un-weighted and undirected network statisticindex mapped from certain time series, meanwhile, this study area has beenexpanded from pseudo-periodic time series to stationary time series which followdifferent distributions, it has explored whether there is any regularityrelationship between different types of stationary time series and the distributionof statistic index obtained from the correspondent weighted network.With derivation from the convolution formula on the theory level, it hasbeen found that the weight distribution of the weight network topology whichwas mapped from the time series with Gaussian distribution and expone ntialdistribution possesses the unalterable probability density analytical expression.This conclusion is able to be verified by the convolution formula derived fromLaplace principle and mathematical derivation of inverse Laplace transformation. In terms of the chi-square distribution and the Poisson distribution which can behardly derived by theoretical derivation could be examined by the convolutionformular from Laplace principle and mathematical derivation of inverse Laplacetransformation, so that obtaining the data of weight distribution. Through thehypothesis testing on the experiential data, it is found that time series which hasthe chi-square distribution with certain parameter, its corresponding weightdistribution of the weight network topology is chi-square distribution whichowns a certain parameter, while time series which possesses the Poissondistribution of certain parameter, its corresponding weight distribution of theweight network topology is Poisson distribution which gets the associatedparameter. At last,we found that time series which has the power lawdistribution with certain parameter,its corresponding weight distribution of theweight network topology is scale-free.