Phase transition in the parametric natural visibility graph

We investigate time series by mapping them to the complex networks using a parametric natural visibility graph (PNVG) algorithm that generates graphs depending on arbitrary continuous parameter—the angle of view. We study the behavior of the relative number of clusters in PNVG near the critical value of the angle of view. Artificial and experimental time series of different nature are used for numerical PNVG investigations to find critical exponents above and below the critical point as well as the exponent in the finite size scaling regime. Altogether, they allow us to find the critical exponent of the correlation length for PNVG. The set of calculated critical exponents satisfies the basic Widom relation. The PNVG is found to demonstrate scaling behavior. Our results reveal the similarity between the behavior of the relative number of clusters in PNVG and the order parameter in the second-order phase transitions theory. We show that the PNVG is another example of a system (in addition to magnetic, percolation, superconductivity, etc.) with observed second-order phase transition

Figure 1

Parametric natural visibility graph mapping algorithm. (a) PNVG selection criterion demonstration. (b) NVG and PNVG diagram.

Figure 2

The relative number of clusters $Q\left(\alpha \right)$ for uniformly distributed time series (3a). In the inset the order parameter $m\left(T\right)$ of the second-order phase transition as a function of temperature $T, T_c$ is the critical temperature.

Figure 3

The relative number of clusters $Q$ upon the proximity to the critical angle $\tau =\left(\alpha -{\alpha }_c\right)/{\alpha }_c$ in a log-log scale. (a) Artificial time series (3a)–(3c). (b) Experimental time series: RR intervals of healthy human cardiac rhythm, solar flares, and seismic activity.

Figure 4

The influence of sampling levels number $M$ on (a) the relative number of clusters $Q\left(\tau \right)$ for the uniform distribution (3a) time series and (b) the size of the area near the critical point $\mathrm{\Delta }$ upon the inverse of sampling levels $1/M$.

Figure 5

The plot of $Q(\tau =0,N)$ upon the length of the time series $N$. (a) Artificial time series (3a)–(3c). (b) RR intervals of healthy human cardiac rhythm.