# Scale-free network clustering in hyperbolic and other random graphs

## TL;DR

This paper introduces a variational principle to explain clustering in scale-free networks, particularly in hyperbolic models, showing that clustering remains significant and representative in large networks.

## Contribution

It presents a novel variational principle for understanding clustering in hyperbolic and classical random graphs, highlighting non-vanishing clustering in large-scale models.

## Key findings

- Clustering in hyperbolic models is non-vanishing and self-averaging.
- The variational principle applies to classical models like preferential attachment.
- Single large samples accurately represent clustering properties.

## Abstract

Random graphs with power-law degrees can model scale-free networks as sparse topologies with strong degree heterogeneity. Mathematical analysis of such random graphs proved successful in explaining scale-free network properties such as resilience, navigability and small distances. We introduce a variational principle to explain how vertices tend to cluster in triangles as a function of their degrees. We apply the variational principle to the hyperbolic model that quickly gains popularity as a model for scale-free networks with latent geometries and clustering. We show that clustering in the hyperbolic model is non-vanishing and self-averaging, so that a single random graph sample is a good representation in the large-network limit. We also demonstrate the variational principle for some classical random graphs including the preferential attachment model and the configuration model.