# Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution

## TL;DR

This paper introduces a mathematically rigorous model for sparse, unbiased, and exchangeable random graphs with power-law degree distributions, addressing key statistical realism requirements in network modeling.

## Contribution

It proves that the hypersoft configuration model (HSCM) is an ensemble of sparse, unbiased, and exchangeable or projective power-law graphs, filling a gap in theoretical network models.

## Key findings

- HSCM satisfies sparsity, unbiasedness, and exchangeability.
- Graphon entropy maximization aligns with Gibbs entropy in large graphs.
- The model accurately captures power-law degree distributions in realistic networks.

## Abstract

Even though power-law or close-to-power-law degree distributions are ubiquitously observed in a great variety of large real networks, the mathematically satisfactory treatment of random power-law graphs satisfying basic statistical requirements of realism is still lacking. These requirements are: sparsity, exchangeability, projectivity, and unbiasedness. The last requirement states that entropy of the graph ensemble must be maximized under the degree distribution constraints. Here we prove that the hypersoft configuration model (HSCM), belonging to the class of random graphs with latent hyperparameters, also known as inhomogeneous random graphs or $W$-random graphs, is an ensemble of random power-law graphs that are sparse, unbiased, and either exchangeable or projective. The proof of their unbiasedness relies on generalized graphons, and on mapping the problem of maximization of the normalized Gibbs entropy of a random graph ensemble, to the graphon entropy maximization problem, showing that the two entropies converge to each other in the large-graph limit.