Relating the large-scale structure of time series and visibility networks

The structure of time series is usually characterized by means of correlations. A new proposal based on visibility networks has been considered recently. Visibility networks are complex networks mapped from surfaces or time series using visibility properties. The structures of time series and visibility networks are closely related, as shown by means of fractional time series in recent works. In these works, a simple relationship between the Hurst exponent $H$ of fractional time series and the exponent of the distribution of edges $\gamma$ of the corresponding visibility network, which exhibits a power law, is shown. To check and generalize these results, in this paper we delve into this idea of connected structures by defining both structures more properly. In addition to the exponents used before, $H$ and $\gamma$, which take into account local properties, we consider two more exponents that, as we will show, characterize global properties. These are the exponent $\alpha$ for time series, which gives the scaling of the variance with the size as $\text{var}\sim T^{2\alpha }$, and the exponent $\kappa$ of their corresponding network, which gives the scaling of the averaged maximum of the number of edges, $\langle k_M\rangle \sim N^{\kappa }$. With this representation, a more precise connection between the structures of general time series and their associated visibility network is achieved. Similarities and differences are more clearly established, and new scaling forms of complex networks appear in agreement with their respective classes of time series.

Figure 1

Diagram of the classification of time series concerning their large-scale structure, using two exponents characterizing a local, ${\alpha }_{\text{loc}}$, and global, $\alpha$, behavior. Time series are ordered in classes: self-affine (SA), stationary noise (SN), stationary fractal (SF), saturated multiplicative (SM), and $1/f$ (1F). Points in each class represent the cases under study.

Figure 2

(a) Estimation of the exponent $\kappa$ from the scaling $\langle k_M\rangle \sim N^{\kappa }$ for visibility networks corresponding to fBm series with ${\alpha }_{\text{loc}}=H=0.7$ (persistent, left triangle), ${\alpha }_{\text{loc}}=H=0.3$ (antipersistent, up triangle), and fBn series with ${\alpha }_{\text{loc}}={\alpha }_s=-0.5$ (white noise, right triangle). Values are obtained from an average of the maximum degree in 100 samples and series sizes of $2^8,2^{10},2^{12},2^{14},2^{16}$. (b) Values of $\kappa$ obtained with this method for the entire range ${\alpha }_{\text{loc}}\in (-1,1)$ including series in the SN and SA classes. The dashed line corresponds to $\kappa =\alpha$. (c) More precise estimation of $\kappa$ for a series in the SN class (${\alpha }_s=-0.5$). Now the number of samples is 1000 and the range of sizes goes from $2^8$ to $2^{20}$. A small curvature is observed. (d) Local exponents ${\kappa }_N$ vs $N$ obtained in the left-hand figure. The dashed red line is the fitted curve, ${\kappa }_N=\frac{1.79}{\text{log}N}$.

Figure 3

Three distinct scaled plots of the degree distribution to account for behaviors of the smallest degrees, $P_k\sim exp-a{k}^{{\mu }_1}$ [(a)–(c)]; intermediate ones, $P_{k/k_M}\sim {(k/k_M)}^{-{\gamma }^{*}}$ [(d)–(f)]; and highest degrees, $P_{k/k_M}\sim {(k/k_M)}^{-\gamma }exp[-A{(k/k_M)}^{{\mu }_2}]$ [(g)–(i)], for the same three series used in the previous figure: ${\alpha }_{\text{loc}}=-0.5$ [(c), (f), and (i)], ${\alpha }_{\text{loc}}=0.3$ [(b), (e), and (h)], and ${\alpha }_{\text{loc}}=0.7$ [(a), (d), and (g)]. Dashed lines correspond to the fitted curves.

Figure 4

Exponents obtained from the three scaled plots of the previous figure for the entire range ${\alpha }_{\text{loc}}\in (-1,1)$. The larger symbols correspond to true exponents, and the smaller symbols correspond to apparent exponents. Continuous lines represent the models: red, $\gamma =1/\kappa =1/{\alpha }_{\text{loc}}$; black, $\gamma =3-2{\alpha }_{\text{loc}}$.

Figure 5

Series in classes with double spectral scaling, whose spectra are size-dependent, produce visibility networks whose degree distributions are also size-dependent. Right: PSD's for series with three sizes $2^{12},2^{14},2^{16}$ in their normal form (b) and collapsed (d). Left: degree densities of the corresponding visibility networks in their normal form (a) and collapsed (c). Series belong to the multiplicative class with ${\alpha }_s=0.5$, $\alpha =1$. The measured exponents of their corresponding visibility networks are $\theta =0.35$ and $\gamma =2.6$. Spectra and degree densities are averaged using 100 samples.

Figure 6

Scaled relative degree densities of visibility networks corresponding to a multiplicative series with ${\alpha }_s=0.5$, $\alpha =1.5$ showing a crossover between two states with power laws. The exponential behavior observed in the part with the highest degrees of the previous cases is substituted here by a power law.

Figure 7

Scaled relative degree densities of visibility networks corresponding to a series in the 1F class (${\alpha }_s=0$, $\alpha =-0.3$). In the inset, the explicit dependence with $N$ is shown. The dashed lines are guides for the eye to compare true power-law and exponential behavior.

Figure 8

Scaled relative degree densities of visibility networks corresponding to a series in the SF class ($\alpha =0$, ${\alpha }_s=0.3$). In the inset, the lack of dependence with the series size is shown.

Figure 9

Exponents $\gamma$, $\kappa$, and $\theta$ obtained from visibility networks corresponding to series in the SA (a), SM (b), 1F (c), and SF (d) classes in a wide range of cases. Exponents $\gamma$ and $\theta$ are obtained from the scaling of the degree distribution, $P_k\sim N^{\theta }{k}^{-\gamma }$, as in Fig. 5. Exponent $\kappa$ is obtained from the scaling of the mean maximum degree, $\langle k_M\rangle \sim N^{\kappa }$, as in Fig. 2. Black dashed lines are $y=x$ lines to guide the eye. The red dashed line in (c) is the natural barrier calculated from (7).

Figure 10

Diagram of the classification of visibility networks concerning their large-scale structure, using two exponents characterizing a local, $\gamma$, and global, $\kappa$, behavior. Points in the diagram correspond to visibility networks mapped from the time series appearing in Fig. 1. The larger symbols represent true exponents, whereas the smaller symbols represent apparent exponents. The red dashed line is the natural barrier, and continuous lines represent models.