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

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