The visibility algorithm has been recently introduced as a mapping between time series and complex networks. This procedure allows to apply methods of complex network theory for characterizing time series. In this work we present the horizontal visibility algorithm, a geometrically simpler and analytically solvable version of our former algorithm, focusing on the mapping of random series (series of independent identically distributed random variables). After presenting some properties of the algorithm, we present exact results on the topological properties of graphs associated to random series, namely the degree distribution, clustering coefficient, and mean path length. We show that the horizontal visibility algorithm stands as a simple method to discriminate randomness in time series, since any random series maps to a graph with an exponential degree distribution of the shape P(k) = (1/3)(2/3)**(k-2), independently of the probability distribution from which the series was generated. Accordingly, visibility graphs with other P(k) are related to non-random series. Numerical simulations confirm the accuracy of the theorems for finite series. In a second part, we show that the method is able to distinguish chaotic series from i.i.d. theory, studying the following situations: (i) noise-free low-dimensional chaotic series, (ii) low-dimensional noisy chaotic series, even in the presence of large amounts of noise, and (iii) high-dimensional chaotic series (coupled map lattice), without needs for additional techniques such as surrogate data or noise reduction methods. Finally, heuristic arguments are given to explain the topological properties of chaotic series and several sequences which are conjectured to be random are analyzed.
The limited penetrable horizontal visibility algorithm is a new time analysis tool and is a further development of the horizontal visibility algorithm. We present some exact results on the topological properties of the limited penetrable horizontal visibility graph associated with random series. We show that the random series maps on a limited penetrable horizontal visibility graph with exponential degree distribution $P(k)\sim exp[-λ(k-2ρ-2)], λ= ln[(2ρ+3)/(2ρ+2)],ρ=0,1,2,...,k=2ρ+2,2ρ+3,...$, independent of the probability distribution from which the series was generated. We deduce the exact…
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…
We employ the horizontal visibility algorithm to map the velocity and acceleration time series in turbulent flows with different Reynolds numbers, onto complex networks. The universal nature of velocity fluctuations in high Reynolds turbulent Helium flow is found to be inherited in the corresponding network topology. The degree distributions of the acceleration series are shown to have stretched exponential forms with the Reynolds number dependent fitting parameter. Furthermore, for acceleration time series, we find a transitional behavior in terms of the Reynolds number in all network feature…
Horizontal visibility graphs (HVGs) are graphs constructed in correspondence with number sequences that have been introduced and explored recently in the context of graph-theoretical time series analysis. In most of the cases simple measures based on the degree sequence (or functionals of these such as entropies over degree and joint degree distributions) appear to be highly informative features for automatic classification and provide nontrivial information on the associated dynam- ical process, working even better than more sophisticated topological metrics. It is thus an open question why t…
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…
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 pro…
A family of classical mathematical problems considers the visibility properties of geometric figures in the plane, e.g. curves or polygons. In particular, the {\it domination problem} tries to find the minimum number of points that are able to dominate the whole set, the so called, {\it domination number}. Alternatively, other problems try to determine the subsets of points with a given cardinality, that maximize the basin of domination, the {\it partial dominating set}. Since a discrete time series can be viewed as an ordered set of points in the plane, the dominating number and the partial d…
Visibility algorithms transform time series into graphs and encode dynamical information in their topology, paving the way for graph-theoretical time series analysis as well as building a bridge between nonlinear dynamics and network science. In this work we introduce and study the concept of sequential visibility graph motifs, smaller substructures of n consecutive nodes that appear with characteristic frequencies. We develop a theory to compute in an exact way the motif profiles associated to general classes of deterministic and stochastic dynamics. We find that this simple property is indee…
In order to extract correlation information inherited in stochastic time series, the visibility graph algorithm has been recently proposed, by which a time series can be mapped onto a complex network. We demonstrate that the visibility algorithm is not an appropriate one to study the correlation aspects of a time series. We then employ the horizontal visibility algorithm, as a much simpler one, to map fractional processes onto complex networks. The degree distributions are shown to have parabolic exponential forms with Hurst dependent fitting parameter. Further, we take into account other topo…
Visibility algorithms are a family of geometric and ordering criteria by which a real-valued time series of N data is mapped into a graph of N nodes. This graph has been shown to often inherit in its topology non-trivial properties of the series structure, and can thus be seen as a combinatorial representation of a dynamical system. Here we explore in some detail the relation between visibility graphs and symbolic dynamics. To do that, we consider the degree sequence of horizontal visibility graphs generated by the one-parameter logistic map, for a range of values of the parameter for which th…
The family of visibility algorithms were recently introduced as mappings between time series and graphs. Here we extend this method to characterize spatially extended data structures by mapping scalar fields of arbitrary dimension into graphs. After introducing several possible extensions, we provide analytical results on some topological properties of these graphs associated to some types of real-valued matrices, which can be understood as the high and low disorder limits of real-valued scalar fields. In particular, we find a closed expression for the degree distribution of these graphs assoc…
We introduce the concept of time series motifs for time series analysis. Time series motifs consider not only the spatial information of mutual visibility but also the temporal information of relative magnitude between the data points. We study the profiles of the six triadic time series. The six motif occurrence frequencies are derived for uncorrelated time series, which are approximately linear functions of the length of the time series. The corresponding motif profile thus converges to a constant vector $(0.2,0.2,0.1,0.2,0.1,0.2)$. These analytical results have been verified by numerical si…
The relation between time series irreversibility and entropy production has been recently investigated in thermodynamic systems operating away from equilibrium. In this work we explore this concept in the context of financial time series. We make use of visibility algorithms to quantify in graph-theoretical terms time irreversibility of 35 financial indices evolving over the period 1998-2012. We show that this metric is complementary to standard measures based on volatility and exploit it to both classify periods of financial stress and to rank companies accordingly. We then validate this appr…
