Canonical horizontal visibility graphs are uniquely determined by their degree sequence
1 Department of Applied Mathematics, School of Aeronautics, Technical University of Madrid (UPM), Plaza Cardenal Cisneros, 28040 Madrid, Spain
2 School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E14NS, UK
a e-mail: firstname.lastname@example.org
Received: 17 May 2016
Revised: 20 September 2016
Published online: 6 March 2017
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 dynamical 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.
© The Author(s) 2017
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