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

Abstract: 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 relationship between $H$ and $<\lambda>$ 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 $<d_{B}>$ of recurrence networks decreases with the Hurst index $H$ of the associated FBMs, and their dependence approximately satisfies the linear formula $<d_{B}> \approx 2 - H$. 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 $H=0.5$ possess the strongest multifractality. In addition, the dependence relationships of the average information dimension $<D(1)>$ and the average correlation dimension $<D(2)>$ 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.