Identification of topological measures of visibility graphs for analyzing transitions in complex time series

Abstract

In this paper, we investigate signatures of variation in the behavior of correlated time series by analyzing changes in the topological properties of the corresponding visibility graph. Variations in six different network measures: assortativity, average path length, clustering, transitivity, density, and the average of the mean link length, are explored. We construct visibility graphs from the original and the magnitude and sign of its increment series. Both the horizontal and the natural visibility graphs are studied. Through extensive numerical studies on the time series of fractional Brownian motion (fBm), we first identify network measures that can reflect the changes in correlations in the time series. The efficacy of these markers is examined to identify the transitions in two systems, a two-dimensional (2D) Ising spin system and EEG data with seizures. While all the identified network measures capture the change in the thermal equilibrium correlations for the Ising spin system, they have limited success in the case of the time-dependent fluctuations in the EEG data. We identify some markers relevant to detecting seizures in the EEG data set.