Emergent Complex Network Geometry

TL;DR

This paper introduces a two-parameter model for growing networks that naturally develop complex geometrical properties, capturing features like curvature, small-world effects, and community structure, applicable to real-world biological, social, and technological networks.

Contribution

It presents a novel model that generates complex network geometries with emergent properties, bridging dynamical rules and geometric features in network growth.

Findings

Model produces networks with non-trivial curvature distributions.

Networks exhibit exponential growth, small-world, and community structures.

Properties are observed in real biological, social, and technological networks.

Abstract

Networks are mathematical structures that are universally used to describe a large variety of complex systems such as the brain or the Internet. Characterizing the geometrical properties of these networks has become increasingly relevant for routing problems, inference and data mining. In real growing networks, topological, structural and geometrical properties emerge spontaneously from their dynamical rules. Nevertheless we still miss a model in which networks develop an emergent complex geometry. Here we show that a single two parameter network model, the growing geometrical network, can generate complex network geometries with non-trivial distribution of curvatures, combining exponential growth and small-world properties with finite spectral dimensionality. In one limit, the non-equilibrium dynamical rules of these networks can generate scale-free networks with clustering and communities, in another limit planar random geometries with non-trivial modularity. Finally we find that these properties of the geometrical growing networks are present in a large set of real networks describing biological, social and technological systems.