Topological Properties And Fractal Analysis Of A Recurrence Network Constructed From Fractional Brownian Motions

Recently, complex networks have attracted most attention of researchers from various fields. Many studies have shown that we can be more through and effective to study the time series from the brand-new viewpoint of complex networks. To this end, many algorithms have been proposed by many researchers, which can be used to construct complex networks from time series.In this paper, we study the basic topological properties, fractal dimension and multifractal properties of recurrence networks constructed from fractional Brownian motions (FBMs). First, we investigate the fundamental topological properties of recurrence networks. We find that the constructed recurrence net-works have exponential degree distributions; the average exponent (A) of degree distribution increases first and then decreases with the increase of Hurst index H of the associated FBMs; the relationship between H and (A) can be well fitted by a cubic polynomial function. The clustering coefficient C of recurrence networks increases with the Hurst index H. At the micro level, we study the motif rank dis-tribution of recurrence networks, so that we can better understand the dynamics mechanism of networks at the local structure. We find the interesting superfamily phenomenon, i.e., the recurrence networks with same motif rank pattern being grouped into two superfamilies. Then we numerically calculate the fractal dimen-sion of recurrence networks. Our results show that the average fractal dimension <dB> of recurrence networks decreases with the increase of the Hurst index H from0.4to0.95, and their dependence approximately satisfies the linear formula <db>-2-H. This means that the fractal dimension of the associated recur-rence network is close to that of the graph of the FBM. In addition, we explore the multifractality of recurrence networks. Our numerical results indicate 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 from0.4to0.95. In particular, the recurrence network with the Hurst index H=0.5possesses the strongest mul-tifractality. Meanwhile we also find that the the average information dimension <D(1)> and the average correlation dimension <D(2)> roughly decrease with the increase of the Hurst index H from0.4to0.95, and the dependence relation-ships of the average information dimension <D(1)> and the average correlation dimension <D(2)> on the Hurst index H can also be fitted well by linear func-tions. Our results strongly suggest that the recurrence network inherits the basic characteristic and the fractal nature of the associated FBM series.In the last part of this article, we apply the multifractal analysis of complex network in protein molecular dynamics and find that these proteins we considered possess the multifractality.