Horizontal visibility graphs: Exact results for random time series

Abstract

The visibility algorithm has been recently introduced as a mapping between time series and complex networks. This procedure allows us 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 with random series, namely, the degree distribution, the clustering coefficient, and the 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), independent of the probability distribution from which the series was generated. Accordingly, visibility graphs with other P(k) are related to nonrandom 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 independent and identically distributed (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 that are conjectured to be random are analyzed.

Complex network approach to fractional time series

“…In general, local maxima in the original time series typically have high degree, and local minima correspond to the low-degree nodes in the corresponding visibility graphs 33 . In fractional processes, as α decreases, local maxima (minima) are more likely to contribute to the actual global maximum (minimum) values of the original series, i.e., the probability that high-value data points in the process have high degrees increases.…”

“…It has been shown in 33 that for an uncorrelated time series, the corresponding HVG is a small-world network 34 with mean degree k = 4 and also its degree distribution, p k , is as follows:…”

“…An alternative (and much simpler) algorithm is the horizontal visibility graph (HVG) 33 , in which a connection can be established between two data points i and j, if one can draw a horizontal line in the time series joining them that does not intersect any intermediate data height, x q , by the following geometrical criterion:…”

Approximate entropy of network parameters

“…(x > 0). Also of interest are random networks with exponential degree distributions, which naturally arise as the renormalization group fixed point of visibility graphs associated with random uncorrelated time series, with π (x) = λe −λx (x > 0) [20,34].…”

“…V), where the measure acquires a more dynamical meaning. This latter situation is studied within the context of visibility graphs [19,20]. The measure is finally tested in Sec.…”