Nonlinear Time Series Analysis By Means Of Visibility Graphs

Recently, visibility graph has provided much insight of the dynamics of time series from complex network perspectives. This algorithm is introduced as a mapping between time series and complex networks, while the connection of two nodes is determined by a linear visibility condition. Later, horizontal visibility graph and improved visibility graph have been proposed, which estimates fractality of time series reliably when noise is present.This article investigates various time series, including continuous chaotic systems, fractal and auto-regressive stochastic processes. By applying visibility graph analysis to these processes, degree distributions and global network measures capture some structure of time series. The main conclusions are the following:1. There are significant correlations between two independent variables of the same continuous chaotic system. The bifurcation routes to chaos have been successfully captured by global network measures.2. Degree distributions of the resulted complex networks of auto-regressive processes are characterized by exponential forms, while that of fractional Brownian motions fulfill power-law forms.3. The improved visibility graph preserves some structures of the auto-regressive processes. The visibility graph based on time series with different parameters shows the same convergence to exponential degree distributions.4. The critical exponent of degree distribution of horizontal visibility graph λ= ln(3/2) cannot distinguish randomness from chaos in time series.This article provides a better understanding of visibility graph. This approach can be applied to characterizing other time series from a new point of view.