非线性时间序列与复杂网络图的相互表征研究

In very recent several years, several types of new mapping methods had been made to bridge the gap between time series and complex network graphs. However, how to apply the network analysis method to depict the characteristics of the original time series' statistics (vice versa the inverse process of such a research) is still unclear. Therefore, it has become an extremely urgent topics nowadays. Our project will carry out the research on dynamics and structures' mutual representation between nonlinear time series and complex network graphs. The main research content includes problems listed as follows:..First, by choosing typical one dimensional or multidimensional nonlinear time series as main research objects, we will address on modelling and analyzing nonlinear time series via complex networks methods, i.e., we will try to establishing the corresponding complex network graph models for related nonlinear time series based on the complex network analysis method. We will try to investigate the relationship between the evolutionary dynamics mechanism hidden in the former and the systematic method for topological geometry description for the latter. And, we will also try to infer the implied fluctuation and bifurcation characteristics of the original nonlinear time series via geometric evolution processes under different time scales. ..Then, through the characteristics calculation of the proposed complex network graphs, the corresponding adjacency matrix spectral analysis, and the community detection methods, etc, we will give some new ways for scientific explanation for nonlinear time series beyond the classical statistical analysis background. ..Finally, research on relationship between complex network’s dynamics and evolution of low or nonlinear time series’ development will be investigated, and possible mathematical inverse map between them will be established. ..This project might be useful for revealing the essence and development trend of low or high-dimensional nonlinear time series, as well as for simplex or multiplex complex networks. This topic will be both helpful for promoting the research on nonlinear time series analysis and complex network graph under local and global viewpoints. It will provide new viewpoints and tools both for research on complexity of nonlinear time series and complexity of complex network graphs.

研究表明：不同类型之时间序列可映射为不同结构之网络图！虽用网络图的度量特征对时间序列的属性区分，中短期趋势外推及极端、突发事件预测有指导，但网络拓扑特性与原序列统计属性间到底存在何种因果关联却仍不甚明了！本项目拟针对非线性时间序列与复杂网络图的相互表征问题开展研究：一、选取典型非线性时间序列为研究对象，开展该时间序列伴生网络图建模；将前者蕴藏动力学演化机制与后者拓扑几何特征演化机制相对应；提出对时间序列进行几何表征的系统化方法、依网络演化规律推断不同时间尺度意义下时间序列蕴涵的波动、分岔特性。二、通过伴生网络图的特征量计算、谱分析、社区结构探测等方法，给出针对动力系统行为的经典时序统计分析背景之外的科学解释；三、通过时间序列幅值演化与网络图结构演化过程间相互表征映射的搭建，揭示非线性时间序列与复杂网络图两类对象间内在关联。本项研究将为时间序列及复杂网络复杂性研究及现实应用提供新视角与工具。

针对非线性时间序列与复杂网络图的相互表征问题,项目组开展以下系列问题地研究：一、选取典型非线性时间序列为研究对象，开展了一维和高维非线性时间序列的几何化刻画算法构建研究，并将相关方法用于规则、随机、半随机（混沌）时间序列的非经典统计特征的对偶性刻画；相关原创算法及改进算法被成功用于EEG等生理数据、水沙时间序列数据、水沙协调性分析数据等时间序列伴生网络图建模与分析研究；将前者蕴藏动力学演化机制与后者拓扑几何特征演化机制相对应，给出了一些原创算法设计、已有算法改进性研究，设计了相关软件用于后续科学研究；二、总结归纳了对时间序列进行几何表征的基本进展，以一类变指数无标度模型为例，讨论了网络演化时Hurst指数的演化规律，有助于推断不同时间尺度意义下时间序列蕴涵的波动、分岔等非线性特性；三、通过伴生网络图的特征量计算、谱分析、社区结构探测等方法，给出针对非线性动力系统行为及真实时间序列（水、沙演化时间序列等）的经典时序统计分析背景之外的科学解释及应用研究；四、通过时间序列幅值演化与网络图结构演化过程间相互表征映射的搭建，讨论了二维、高维时间序列对应的伴生单层与多层网络的构建算法及其应用，相关结果可以用于揭示非线性时间序列与复杂网络图两类对象间内在关联。五、针对一些典型随机动力系统动力学分析、系统生物学中噪声对基因表达的影响分析、在单层与多层复杂网络中的网络同步能力刻画与控制新方法，以及基于蛋白质折叠最优结构粗粒化数值模拟中结构与功能的关系讨论等问题开展了系列前期研究，为进一步基于几何化分析手段刻画噪声的作用、同步能力刻画、网络控制新判据提出等研究奠定了基础。本项目研究的主体结果为非线性时间序列分析、复杂网络的结构与功能关系研究、二者的关联性分析，以及其它相关的现实应用均可提供一些新的分析视角与研究工具支持。