Emergent Hyperbolic Network Geometry

TL;DR

This paper demonstrates that hyperbolic geometry naturally emerges in growing simplicial complexes, which are higher-dimensional generalizations of networks, and explores how their properties depend on dimensionality and face fitness.

Contribution

It extends the understanding of network geometry to simplicial complexes, showing hyperbolic structures emerge spontaneously from purely combinatorial growth models.

Findings

Hyperbolic geometry emerges in simplicial complexes.

Complex networks exhibit scale-free, small-world, and community structures.

Phase transitions occur with heterogeneous face fitness.

Abstract

A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a discretization of spacetime. Here, by extending our knowledge of growing complex networks to growing simplicial complexes we investigate the nature of the emergent geometry of complex networks and explore whether this geometry is hyperbolic. Specifically we show that an hyperbolic network geometry emerges spontaneously from models of growing simplicial complexes that are purely combinatorial. The statistical and geometrical properties of the growing simplicial complexes strongly depend on their dimensionality and display the major universal properties of real complex networks (scale-free degree distribution, small-world and communities) at the same time. Interestingly, when the network dynamics includes an heterogeneous fitness of the faces, the growing simplicial complex can undergo phase transitions that are reflected by relevant changes in the network geometry.