Sequential motif profile of natural visibility graphs

TL;DR

This paper develops a theoretical framework to compute sequential motif profiles in natural visibility graphs, enabling analysis of time series with exact and numerical methods, and compares them to horizontal visibility graphs.

Contribution

It extends motif profile computation from horizontal to natural visibility graphs, providing exact results for certain processes and a fast numerical method for empirical data.

Findings

Exact motif profiles for deterministic aperiodic and Markov processes.

A linear-time numerical estimation method for empirical time series.

Comparison of robustness between HVG and VG under noise.

Abstract

The concept of sequential visibility graph motifs -subgraphs appearing with characteristic frequencies in the visibility graphs associated to time series- has been advanced recently along with a theoretical framework to compute analytically the motif profiles associated to Horizontal Visibility Graphs (HVGs). Here we develop a theory to compute the profile of sequential visibility graph motifs in the context of Natural Visibility Graphs (VGs). This theory gives exact results for deterministic aperiodic processes with a smooth invariant density or stochastic processes that fulfil the Markov property and have a continuous marginal distribution. The framework also allows for a linear time numerical estimation in the case of empirical time series. A comparison between the HVG and the VG case (including evaluation of their robustness for short series polluted with measurement noise) is also presented.