A geometric entropy detecting the Erd\"os-R\'enyi phase transition

TL;DR

This paper introduces a geometric entropy measure based on a Riemannian manifold associated with discrete systems, successfully detecting the Erd"os-Rényi phase transition characterized by the emergence of a giant component in random graphs.

Contribution

It proposes a novel geometric entropy framework for non-Hamiltonian systems, linking differential geometry with phase transition detection in complex networks.

Findings

The geometric entropy detects the Erd"os-Rényi phase transition.

It correlates the entropy increase with the emergence of the giant component.

The method applies to systems without a Hamiltonian description.

Abstract

We propose a method to associate a differentiable Riemannian manifold to a generic many degrees of freedom discrete system which is not described by a Hamiltonian function. Then, in analogy with classical Statistical Mechanics, we introduce an entropy as the logarithm of the volume of the manifold. The geometric entropy so defined is able to detect a paradigmatic phase transition occurring in random graphs theory: the appearance of the `giant component' according to the Erd\"os-R\'enyi theorem.