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

## TL;DR

This paper analytically derives the degree distributions of horizontal visibility graphs mapped from random series and multifractal binomial measures, confirming exponential behavior for random series and providing explicit formulas for multifractal measures.

## Contribution

It introduces an analytical method to derive degree distributions of HVGs from both random and multifractal series, expanding understanding of their structural properties.

## Key findings

- Degree distributions of HVGs from random series are exponential.
- Explicit analytical expressions for degree distributions of multifractal binomial measures are provided.
- Results agree well with numerical simulations.

## 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.