Canonical Horizontal Visibility Graphs are uniquely determined by their degree sequence

TL;DR

This paper proves that under certain conditions, horizontal visibility graphs are uniquely determined by their degree sequences, explaining why simple degree-based measures are highly effective in analyzing time series.

Contribution

It establishes a bijection between HVG adjacency matrices and degree sequences, showing HVGs are unigraphs and fully characterized by degree sequences.

Findings

HVGs are unigraphs under certain conditions.

Degree sequences fully determine HVGs.

Provides an explicit construction of the bijection.

Abstract

Horizontal visibility graphs (HVGs) are graphs constructed in correspondence with number sequences that have been introduced and explored recently in the context of graph-theoretical time series analysis. In most of the cases simple measures based on the degree sequence (or functionals of these such as entropies over degree and joint degree distributions) appear to be highly informative features for automatic classification and provide nontrivial information on the associated dynam- ical process, working even better than more sophisticated topological metrics. It is thus an open question why these seemingly simple measures capture so much information. Here we prove that, under suitable conditions, there exist a bijection between the adjacency matrix of an HVG and its degree sequence, and we give an explicit construction of such bijection. As a consequence, under these conditions HVGs are unigraphs and the degree sequence fully encapsulates all the information of these graphs, thereby giving a plausible reason for its apparently unreasonable effectiveness.