Counting Horizontal Visibility Graphs

TL;DR

This paper explores the properties of horizontal visibility graphs (HVGs) derived from time series data, proving their unique determination by degree sequences and counting them using well-known combinatorial numbers.

Contribution

It extends previous work by proving HVGs without equal entries are uniquely determined by their degree sequences and counts HVGs with and without equal entries using Catalan and Schr"oder numbers.

Findings

HVGs without equal entries are uniquely determined by their degree sequences.

Number of HVGs with equal entries corresponds to large Schr"oder numbers.

Number of HVGs without equal entries corresponds to Catalan numbers.

Abstract

Horizontal visibility graphs (HVGs, for short) are a common tool used in the analysis and classification of time series with applications in many scientific fields. In this article, extending previous work by Lacasa and Luque, we prove that HVGs associated to data sequences without equal entries are completely determined by their ordered degree sequence. Moreover, we show that HVGs for data sequences without and with equal entries are counted by the Catalan numbers and the large Schr\"oder numbers, respectively.