Hyperbolicity Measures "Democracy" in Real-World Networks

TL;DR

This paper investigates the hyperbolicity of real-world networks, revealing how it reflects their 'democratic' or 'aristocratic' structure, and introduces the concept of influence area to analyze local and global network properties.

Contribution

It provides a mathematical interpretation of hyperbolicity as a measure of network democracy, introduces the influence area concept, and analyzes real-world networks to distinguish their topologies.

Findings

Networks with small hyperbolicity are 'aristocratic' with few key vertices.

Networks with large hyperbolicity are 'democratic' with many crucial vertices.

Influence area varies significantly between local and global networks.

Abstract

We analyze the hyperbolicity of real-world networks, a geometric quantity that measures if a space is negatively curved. In our interpretation, a network with small hyperbolicity is "aristocratic", because it contains a small set of vertices involved in many shortest paths, so that few elements "connect" the systems, while a network with large hyperbolicity has a more "democratic" structure with a larger number of crucial elements. We prove mathematically the soundness of this interpretation, and we derive its consequences by analyzing a large dataset of real-world networks. We confirm and improve previous results on hyperbolicity, and we analyze them in the light of our interpretation. Moreover, we study (for the first time in our knowledge) the hyperbolicity of the neighborhood of a given vertex. This allows to define an "influence area" for the vertices in the graph. We show that the influence area of the highest degree vertex is small in what we define "local" networks, like most social or peer-to-peer networks. On the other hand, if the network is built in order to reach a "global" goal, as in metabolic networks or autonomous system networks, the influence area is much larger, and it can contain up to half the vertices in the graph. In conclusion, our newly introduced approach allows to distinguish the topology and the structure of various complex networks.