Volume 86, Number 3, May 2009
Article Number 30001
Number of page(s) 5
Section General
Published online 12 May 2009
EPL, 86 (2009) 30001
DOI: 10.1209/0295-5075/86/30001

The visibility graph: A new method for estimating the Hurst exponent of fractional Brownian motion

L. Lacasa1, B. Luque1, J. Luque2 and J. C. Nuño3

1   Departamento de Matemática Aplicada y Estadística, ETSI Aeronáuticos, Universidad Politécnica de Madrid Madrid, Spain, EU
2   Departament de Teoria del Senyal i Comunicacions, Universitat Politècnica de Catalunya - Barcelona, Spain, EU
3   Departamento de Matemática Aplicada a los Recursos Naturales, ETSI Montes, Universidad Politécnica de Madrid Madrid, Spain, EU

received 7 January 2009; accepted in final form 14 April 2009; published May 2009
published online 12 May 2009

Fractional Brownian motion (fBm) has been used as a theoretical framework to study real-time series appearing in diverse scientific fields. Because of its intrinsic nonstationarity and long-range dependence, its characterization via the Hurst parameter, H, requires sophisticated techniques that often yield ambiguous results. In this work we show that fBm series map into a scale-free visibility graph whose degree distribution is a function of H. Concretely, it is shown that the exponent of the power law degree distribution depends linearly on H. This also applies to fractional Gaussian noises (fGn) and generic $f^{-\beta}$ noises. Taking advantage of these facts, we propose a brand new methodology to quantify long-range dependence in these series. Its reliability is confirmed with extensive numerical simulations and analytical developments. Finally, we illustrate this method quantifying the persistent behavior of human gait dynamics.

05.45.Tp - Time series analysis.
05.40.Jc - Brownian motion.
89.75.Hc - Networks and genealogical trees.

© EPLA 2009