title: Hyperbolicity measures democracy in real-world networks.
abstract: In this work, we analyze the hyperbolicity of real-world networks, a geometric quantity that measures if a space is negatively curved. We provide two improvements in our understanding of this quantity: first of all, in our interpretation, a hyperbolic network is "aristocratic", since few elements "connect" the system, while a non-hyperbolic network has a more "democratic" structure with a larger number of crucial elements. The second contribution is the introduction of the average hyperbolicity of the neighbors of a given node. Through this definition, we outline an "influence area" for the vertices in the graph. We show that in real networks the influence area of the highest degree vertex is small in what we define "local" networks (i.e., social or peer-to-peer networks), and large in "global" networks (i.e., power grid, metabolic networks, or autonomous system networks).
authors.affiliation: IMT Institute for Advanced Studies, Piazza San Francesco 19, 55100 Lucca, Italy.
authors.forename: Michele
authors.initials: M
authors.lastname: Borassi
authors.affiliation: IMT Institute for Advanced Studies, Piazza San Francesco 19, 55100 Lucca, Italy and Linkalab, Complex Systems Computational Laboratory, 09129 Cagliari, Italy.
authors.forename: Alessandro
authors.initials: A
authors.lastname: Chessa
authors.affiliation: IMT Institute for Advanced Studies, Piazza San Francesco 19, 55100 Lucca, Italy; Istituto dei Sistemi Complessi (ISC), via dei Taurini 19, 00185 Roma, Italy; and London Institute for Mathematical Sciences, 35a South St. Mayfair, W1K 2XF London, United Kingdom.
authors.forename: Guido
authors.initials: G
authors.lastname: Caldarelli