Multifractal cross-correlation effects in two-variable time series of complex network vertex observables

Abstract

We investigate the scaling of the cross-correlations calculated for two-variable time series containing vertex properties in the context of complex networks. Time series of such observables are obtained by means of stationary, unbiased random walks. We consider three vertex properties that provide, respectively, short-, medium-, and long-range information regarding the topological role of vertices in a given network. In order to reveal the relation between these quantities, we applied the multifractal cross-correlation analysis technique, which provides information about the nonlinear effects in coupling of time series. We show that the considered network models are characterized by unique multifractal properties of the cross-correlation. In particular, it is possible to distinguish between Erdös-Rényi, Barabási-Albert, and Watts-Strogatz networks on the basis of fractal cross-correlation. Moreover, the analysis of protein contact networks reveals characteristics shared with both scale-free and small-world models.

Multifractal analysis of financial markets

“…Another example is to label time stamps for the nodes following the movements of a random walker abiding to specific rules [1028][1029][1030]. Multifractal cross-correlation analysis (MF-CCA) of the time series of degree, clustering coefficient, and closeness centrality can distinguish between Erdös-Rényi, Barabási-Albert, and Watts-Strogatz networks [1031]. Also, the degree of right-sided asymmetry of multifractal spectra of the time series of these local node variables is linked with the degree of small-worldness present in networks [1032].…”

Right-side-stretched multifractal spectra indicate small-worldness in networks

“…Nonetheless, as first observed in [29,34,45], time series S P of vertex properties emitted by such a process might posses a complex organization characterized by a non-trivial correlation structure, which can be quantified by means of the multifractal analysis framework. As in [36], here we consider the following properties: vertex degree (VD), clustering coefficient (VCL), and closeness centrality (VCC). Several different methods have been proposed to analyze the fluctuations of the stationary component of a time series [14,37,44].…”

“…Interesting hybridizations of the two frameworks are becoming popular as well, providing the possibility to study system dynamics observed as time series but analyzed in terms of topological features of associated complex network [6,16,42,46]; analogously, complex networks can be studied in terms of time series analysis. For instance, recent works indicate the possibility to characterize network structures in terms of scaling and related (multi)fractal properties of suitably generated time series [29,34,36,45].…”

Topological properties of the limited penetrable horizontal visibility graph family

“…( 15) is that each time series has an associated LPHVG(ρ) with a maximum mean degree (for a aperiodic series) of k(∞) = 4(ρ + 1), which agrees with the previous result in Eq. (11). In Eq.…”

“…These include constructing a complex network from a pseudoperiodic time series [2], using a visibility graph (VG) algorithm [3], a recurrence network (RN) method [4], a stochastic processes method [5], a coarse geometry theory [6], a nonlinear mutual information method [7], event synchronization [8], and a phase-space coarse-graining method [9]. These methods have been widely used to solve problems in a variety of research fields [10][11][12][13][14][15][16][17][18][19][20]. Among all these time series complex network analysis algorithms, visibility algorithms [3,21,22] are the most efficient when constructing a complex network from a time series.…”