An Horizontal Visibility Graph (for short, HVG) is defined in association with an ordered set of non-negative reals. HVGs realize a methodology in the analysis of time series, their degree distribution being a good discriminator between randomness and chaos [B. Luque, et al., Phys. Rev. E 80 (2009), 046103]. We prove that a graph is an HVG if and only if outerplanar and has a Hamilton path. Therefore, an HVG is a noncrossing graph, as defined in algebraic combinatorics [P. Flajolet and M. Noy, Discrete Math., 204 (1999) 203-229]. Our characterization of HVGs implies a linear time recognition algorithm. Treating ordered sets as words, we characterize subfamilies of HVGs highlighting various connections with combinatorial statistics and introducing the notion of a visible pair. With this technique we determine asymptotically the average number of edges of HVGs.
In this paper we introduce the horizon visibility graph, a simple extension to the popular horizontal visibility graph representation of a time series, and show that it possesses a rigorous mathematical foundation in computational algebraic topology. This fills a longstanding gap in the literature on the horizontal visibility approach to nonlinear time series analysis which, despite a suite of successful applications across multiple domains, lacks a formal setting in which to prove general properties and develop natural extensions. The main finding is that horizon visibility graphs are dual to…
On the degree distribution of horizontal visibility graphs associated to Markov processes and dynamical systems: diagrammatic and variational approaches
Correlations in multifractal series have been investigated, extensively. Almost all approaches try to find scaling features of a given time series. However, the analysis of such scaling properties has some difficulties such as finding a proper scaling region. On the other hand, such correlation detection methods may be affected by the probability distribution function of the series. In this article, we apply the horizontal visibility graph algorithm to map stochastic time series into networks. By investigating the magnitude and sign of a multifractal time series, we show that one can detect li…
A Horizontal Visibility Graph (HVG) is a simple graph extracted from an ordered sequence of real values, and this mapping has been used to provide a combinatorial encryption of time series for the task of performing network based time series analysis. While some properties of the spectrum of these graphs --such as the largest eigenvalue of the adjacency matrix-- have been routinely used as measures to characterise time series complexity, a theoretic understanding of such properties is lacking. In this work we explore some algebraic and spectral properties of these graphs associated to periodic…
