Topological properties and fractal analysis of a recurrence network constructed from fractional Brownian motions

Many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks. In this work, we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional Brownian motions (FBMs). First, our results indicate that the constructed recurrence networks have exponential degree distributions; the average degree exponent $\langle \lambda \rangle$ increases first and then decreases with the increase of Hurst index $H$ of the associated FBMs; the relationship between $H$ and $\langle \lambda \rangle$ can be represented by a cubic polynomial function. We next focus on the motif rank distribution of recurrence networks, so that we can better understand networks at the local structure level. We find the interesting superfamily phenomenon, i.e., the recurrence networks with the same motif rank pattern being grouped into two superfamilies. Last, we numerically analyze the fractal and multifractal properties of recurrence networks. We find that the average fractal dimension $\langle d_B\rangle$ of recurrence networks decreases with the Hurst index $H$ of the associated FBMs, and their dependence approximately satisfies the linear formula $\langle d_B\rangle \approx 2-H$, which means that the fractal dimension of the associated recurrence network is close to that of the graph of the FBM. Moreover, our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks, and the multifractality of these networks becomes stronger at first and then weaker when the Hurst index of the associated time series becomes larger from 0.4 to 0.95. In particular, the recurrence network with the Hurst index $H=0.5$ possesses the strongest multifractality. In addition, the dependence relationships of the average information dimension $\langle D\left(1\right)\rangle$ and the average correlation dimension $\langle D\left(2\right)\rangle$ on the Hurst index $H$ can also be fitted well with linear functions. Our results strongly suggest that the recurrence network inherits the basic characteristic and the fractal nature of the associated FBM series.

Figure 1

Relationship between the recurrence threshold $\varepsilon$ and the connection rate of the recurrence network constructed from fractional Brownian motion series of length $L=2^{12}$ with Hurst index $H=0.6$.

Figure 2

Recurrence network constructed from fractional Brownian motion series of length $L=2^8$ with Hurst index $H=0.6$.

Figure 3

Degree distribution of one recurrence network constructed from fractional Brownian motion with Hurst index $H=0.6$.

Figure 4

Relationship between $H$ of fractional Brownian motion and average degree exponent $\langle \lambda \rangle$ of the associated recurrence networks. Here the average is calculated from 1000 realizations.

Figure 5

All two triples centered on vertex $i$.

Figure 6

Relationship between $H$ of fractional Brownian motion and average clustering coefficient $\langle C\rangle$ of the associated recurrence networks. Here the average is calculated from 1000 realizations.

Figure 7

All six network motifs of size 4 in connected and undirected network. These motifs are labeled $M_1,M_2,M_3,M_4,M_5$, and $M_6$, respectively.

Figure 8

Network motif rank distributions of FBMs with different Hurst indices $H$. (a) Motif rank pattern $M_2{M}_3{M}_1{M}_5{M}_6{M}_4$ for $0.4\le H\le 0.6$. (b) Motif rank pattern $M_2{M}_3{M}_5{M}_6{M}_1{M}_4$ for $0.65\le H\le 0.95$. Here the average is calculated from 100 realizations.

Figure 9

Dependence of average occurrence frequencies of motifs $M_1,M_2,M_3$, $M_4$, $M_5$, and $M_6$ on the Hurst indices $H$. Here the average is calculated from 100 realizations.

Figure 10

Fractal scaling of recurrence networks constructed from FBMs with Hurst indices $H=0.4$ and $0.6$. The fractal dimension is the absolute value of the slope of the linear fit.

Figure 11

Relationship between $H$ of fractional Brownian motion and average fractal dimension $\langle d_B\rangle$ of the associated recurrence networks. Here the average is calculated from 100 realizations.

Figure 12

Linear regressions for calculating the generalized dimensions of the recurrence network.

Figure 13

Average ${\tau }_1\left(q\right)$ curves of the recurrence networks. Here the average is calculated from 100 realizations.

Figure 14

Average $D\left(q\right)$ curves of the recurrence networks; $D\left(0\right)$ is the fractal dimension of the network. Here the average is calculated from 100 realizations.

Figure 15

Dependence of the average information dimension $\langle D\left(1\right)\rangle$ and the average correlation dimension $\langle D\left(2\right)\rangle$ of the associated recurrence networks with respect to the Hurst index $H$ of fractional Brownian motion. Here the average is calculated from 100 realizations.