Entropy of Exchangeable Random Graphs

TL;DR

This paper introduces a new framework for measuring the complexity of large graphs using graphon entropy, providing estimators with proven statistical properties and demonstrating their effectiveness on simulated and real-world networks.

Contribution

It develops novel graphon entropy estimators, including nonparametric and model-specific versions, with theoretical guarantees and practical validation.

Findings

Estimators accurately capture structural variations in graphs.

Graphon entropy effectively characterizes evolving network dynamics.

Theoretical results include convergence rates and CLTs.

Abstract

Quantifying the complexity of large graphs requires measures that extend beyond predefined structural features and scale efficiently with graph size. This work adopts a generative perspective, modeling large networks as exchangeable graphs to quantify the information content of their generating mechanisms via graphon entropy. As a graph property, graphon entropy is invariant under isomorphisms, making it an effective measure of complexity; however, it is not directly computable. To address this, we introduce a suite of graphon entropy estimators, including a nonparametric estimator for broad applicability and specialized versions for structured graphons arising from well-studied random graph models such as Erd\H{o}s-R\'enyi, Chung-Lu, and stochastic block models. We establish their large-sample properties, deriving convergence rates and Central Limit Theorems. Simulations illustrate how the nonparametric graphon entropy estimator captures structural variations in graphs, while real-world applications demonstrate its role in characterizing evolving network dynamics.