# Hyperbolicity measures democracy in real-world networks

An intuition of the definition of delta.

* Physical Review E * 92, 1 (2015)

M. Borassi, A. Chessa, G. Caldarelli

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).

#### Network valuation in financial systems

P. Barucca, M. Bardoscia, F. Caccioli, M. D’Errico, G. Visentin, G. Caldarelli, S. Battiston

*Mathematical Finance*

#### The space of functions computed by deep layered machines

A. Mozeika, B. Li, D. Saad

Sub. to *Physical Review Letters*

#### Replica analysis of overfitting in generalized linear models

T. Coolen, M. Sheikh, A. Mozeika, F. Aguirre-Lopez, F. Antenucci

Sub. to *Journal of Physics A*

#### Degree-correlations in a bursting dynamic network model

F. Vanni, P. Barucca

*Journal of Economic Interaction and Coordination*

#### Phase transition creates the geometry of the continuum from discrete space

R. Farr, T. Fink

*Physical Review E*

124 / 124 papers