A note on a new general family of deterministic hierarchical networks

Abstract

It is known that many networks modeling real-life complex systems are small-word (large local clustering and small diameter) and scale-free (power law of the degree distribution), and very often they are also hierarchical. Although most of the models are based on stochastic methods, some deterministic constructions have been recently proposed, because this allows a better computation of their properties. Here a new deterministic family of hierarchical networks is presented, which generalizes most of the previous proposals, such as the so-called binomial tree. The obtained graphs can be seen as graphs on alphabets (where vertices are labeled with words of a given alphabet, and the edges are defined by a specific rule relating different words). This allows us the characterization of their main distance-related parameters, such as the radius and diameter. Moreover, as a by-product, an efficient shortest-path local algorithm is proposed.

This research has been partially supported by AGAUR from the Catalan Government under project 2017SGR1087 and by MICINN from the Spanish Government under project PGC2018- 095471-B-I00. The research of the first author has also been supported by MICINN from the Spanish Government under project MTM2017-83271-R. The research of the first author has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922.