Measuring the Topological Time Irreversibility of Time Series With the Degree-Vector-Based Visibility Graph Method

Time-irreversibility of a time series, which can be defined as the variance of properties under the time-reversal transformation, is a cardinal property of non-equilibrium systems and is associated with predictability in the study of financial time series. Recent pieces of literature have proposed the visibility-graph-based approaches that specifically refer to topological properties of the network mapped from a time series, with which one can quantify different degrees of time-irreversibility within the sets of statistically time-asymmetric series. However, all these studies have inadequacies in capturing the time-irreversibility of some important classes of time series. Here, we extend the visibility-graph-based method by introducing a degree-vector associated with network nodes to represent the characteristic patterns of the index motion. The newly proposed method is parameter-free and temporally local. The validation to canonical synthetic time series, in the aspect of time-(ir)reversibility, illustrates that our method can differentiate a non-Markovian additive random walk from an unbiased Markovian walk, as well as a GARCH time series from an unbiased multiplicative random walk. We further apply the method to the real-world financial time series and find that the price motions occasionally equip much higher time-irreversibility than the calibrated GARCH model does.