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

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.

Figure 1

Results for ER (a) and BA (b) networks. Results for VCC-VD are comparable. In the BA case, cross-correlations can be computed also for VCC-VCL and VCL-VD for positive $q$.

Figure 2

Graphical representation of a small BA network with 20 vertices. In the left-hand panel, size of vertices is proportional to vertex degree. In the right-hand panel, instead, size of vertices is proportional to vertex clustering coefficient. As it is possible to notice, hubs are characterized by small clustering coefficient. On the other hand, hubs are directly connected with vertices having large clustering coefficient. This does not happen in ER networks, since they are characterized by very low average clustering coefficient.

Figure 3

Results for WS networks with $k=6$ and $p=0.1$ (a), 0.5 (b), and 0.9 (c). We show time series excerpts, $F_{XY}(q,s)$, and ${\lambda }_q$ (circles), $h_{xy}\left(q\right)$ (squares). The empty (green and blue) and full (red and black) symbols refer to the short- and long-range correlations, respectively. The vertical dashed lines in the insets indicate the cross-over between different scaling regimes.

Figure 4

Results for proteins JW0058 (a) and JW0179 (b).