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.

arXiv comment25 pages, 1 table, 15 figures. accepted by Phys. Rev. E

References in corpus (25)

• Emergence of scaling in random networks
• Statistical mechanics of complex networks
• The structure and function of complex networks
• Uncovering the overlapping community structure of complex networks in nature and society
• Recurrence Plots for the Analysis of Complex Systems
• From time series to complex networks: the visibility graph
• Self-similarity of complex networks
• Recurrence networks - A novel paradigm for nonlinear time series analysis
• Complex Network Approach for Recurrence Analysis of Time Series
• Nonlinear analysis of bivariate data with cross recurrence plots
• The Visibility Graph: a new method for estimating the Hurst exponent of fractional Brownian motion
• How to calculate the fractal dimension of a complex network: the box covering algorithm
• Skeleton and fractal scaling in complex networks
• The Geometry of Chaotic Dynamics -- A Complex Network Perspective
• Fractality in complex networks: critical and supercritical skeletons
• Ambiguities in recurrence-based complex network representations of time series
• Measure representation and multifractal analysis of complete genomes
• Analytical framework for recurrence-network analysis of time series
• Degree distribution of the visibility graphs mapped from fractional Brownian motions and multifractal random walks
• Multifractality in Time Series
• Horizontal visibility graphs transformed from fractional Brownian motions: Topological properties versus Hurst index
• Exploring self-similarity of complex cellular networks: The edge-covering method with simulated annealing and log-periodic sampling
• Fractal and multifractal properties of a family of fractal networks
• Multifractal analysis of complex networks
• Superfamily classification of nonstationary time series based on DFA scaling exponents

Cited by in corpus (10)

• Complex network approaches to nonlinear time series analysis
• Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems
• Determination of multifractal dimensions of complex networks by means of the sandbox algorithm
• Multifractal analysis of weighted networks by a modified sandbox algorithm
• Fractal and complex network analyses of protein molecular dynamics
• Multifractality and Laplace spectrum of horizontal visibility graphs constructed from fractional Brownian motions
• Analyzing long-term correlated stochastic processes by means of recurrence networks: Potentials and pitfalls
• A computationally-efficient sandbox algorithm for multifractal analysis of large-scale complex networks with tens of millions of nodes
• Network analysis of chaotic dynamics in fixed-precision digital domain
• Homology Groups of Embedded Fractional Brownian Motion