Visibility graph-based characterization of extreme values in time series

Abstract

Complex dynamical systems often display extreme fluctuations of an observed variable that constitute significant deviations from the long-term average, and which are often associated with severe impacts on the system. By definition, extreme events are therefore usually explored from time series recordings. In this work, we characterize extreme values in time series using visibility graphs, a method that non-parametrically maps a time series onto a network, whose topological structure is known to inherit important characteristics of the original time series dynamics. Unlike threshold-based approaches, extreme values in this framework can be identified without the need to introduce external parameters and can be applied to time series generated by both stationary and nonstationary processes. For stationary processes, we exploit a known property of visibility graphs in which the degree of a node is monotonically and nonlinearly related to the corresponding data value. This nonlinear amplification enhances the contribution of large values while suppressing noise, while the monotonic relationship enables a direct ranking of data points according to node degree. This procedure identifies global extreme values and locally prominent ones. For nonstationary processes, the degree ranking in the visibility graph still provides a robust indicator of relative importance. We validate our findings with synthetic time series and with real climatological data. Our results show that extreme-value characterization in stationary time series is enhanced when combining standard methods with visibility-graph-based detection, whereas for nonstationary data, where conventional approaches are often ill-posed, visibility graphs provide an effective alternative. We discuss how sub-sampling the time series using only peak values preserves the ability to identify extreme values while reducing computational cost.