# Hyperbolicity measures democracy in real-world networks

M. Borassi, A. Chessa, G. Caldarelli

* Physical Review E * 92, 1 (2015)

#complexnetworks#geometry#graphtheory

An intuition of the definition of delta.

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).

#### Imaginary replica analysis of loopy regular random graphs

F. Lopez, T. Coolen

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*

#### Intelligently chosen interventions have potential to outperform the diode bridge in power conditioning

F. Liu, Y. Zhang, O. Dahlsten, F. Wang

*Scientific Reports *

123 / 123 papers