Analytic degree distributions of horizontal visibility graphs mapped from unrelated random series and multifractal binomial measures

¹ Department of Finance, East China University of Science and Technology - Shanghai 200237, China
² Research Center for Econophysics, East China University of Science and Technology - Shanghai 200237, China
³ Department of Mathematics, East China University of Science and Technology - Shanghai 200237, China
⁴ School of Business, Sun Yat-Sen University - Guangzhou 510275, Guangdong, China

^(a) zqjiang@ecust.edu.cn
^(b) wxzhou@ecust.edu.cn

Received: 8 July 2017
Accepted: 6 October 2017

Abstract

Complex network is not only a powerful tool for the analysis of complex system, but also a promising way to analyze time series. The algorithm of horizontal visibility graph (HVG) maps time series into graphs, whose degree distributions are numerically and analytically investigated for certain time series. We derive the degree distributions of HVGs through an iterative construction process of HVGs. The degree distributions of the HVG and the directed HVG for random series are derived to be exponential, which confirms the analytical results from other methods. We also obtained the analytical expressions of degree distributions of HVGs and in-degree and out-degree distributions of directed HVGs transformed from multifractal binomial measures, which agree excellently with numerical simulations.