# Catching homologies by geometric entropy

## TL;DR

This paper introduces a geometric entropy measure based on Riemannian volume to quantify network complexity and detect topological features like homologies in different network types.

## Contribution

It demonstrates how geometric entropy can identify topological features such as homologies in networks, using analytical and numerical methods for various network classes.

## Key findings

- Geometric entropy can detect homologies in networks.
- It distinguishes between random and scale-free networks.
- The method works for both small and large networks.

## Abstract

A geometric entropy is defined as the Riemannian volume of the parameter space of a statistical manifold associated with a given network. As such it can be a good candidate for measuring networks complexity. Here we investigate its ability to single out topological features of networks proceeding in a bottom-up manner: first we consider small size networks by analytical methods and then large size networks by numerical techniques. Two different classes of networks, the random graphs and the scale--free networks, are investigated computing their Betti numbers and then showing the capability of geometric entropy of detecting homologies.