Nonlinear Analysis of Typical Hydrological Time Series Under the Viewpoint of Visibility Graph Methods and Its Two Modification Versions

Abstract

Using visibility graph, horizontal visibility graph, and limited penetrable visibility graph methods, which are all proposed recently to change from time series to network graph, the hydrological time series of Umpqua river are analyzed under the view of complex network. After the establishment of related complex networks of this typical hydrological time series, basic network characteristics, such as degree distribution, average path length, clustering coefficient, graph density, and so on are calculated. Furthermore, community analysis of the three types of related complex networks are carried out. The similarities and differences of the three methods for network analysis are proposed. Finally, basic norms for modeling hydrological time series are proposed with comparison of the merits and drawbacks of the three methods. Results indicate that, for real hydrological time series, the related visibility and limited penetrable visibility networks behave obviously small-world property, while horizontal visibility network has not. Three networks have obviously the degree of distribution characteristics of scale-free networks. Overall, horizontal visibility graph showing scale-free feature most obviously (heavy-tailed is not obvious), visibility graph, and limited penetrable visibility graph networks have obvious bending head (heavy-tailed feature is more obvious). Based on the fitting results of three degree distribution curves, it can be seen that the coefficient absolute value of horizontal visibility network is the largest, which shows the sequence is of high sensitivity. In addition, according to the community analysis results, the limited penetrable visibility graph method is best for classifying withered water period, visibility graph method’s ability is worse, horizontal visibility graph method’s ability is worst, while through horizontal visibility graph, one can find the local minimum of hydrological time series easily, which is also consistent with the result of degree distribution.