🤖 AI Summary
This paper addresses the challenge of quantifying information content in large exchangeable random graphs. It introduces graphon entropy as a theoretically grounded measure of the intrinsic complexity of their generative mechanisms, overcoming the computational intractability and structural non-scalability of conventional graph entropy. First, graphon entropy is formally defined as a statistically inferable invariant of graph complexity. Second, two classes of estimators—nonparametric and model-specific—are developed, accompanied by rigorous large-sample theory, including convergence rates and a central limit theorem. The methodology integrates graph limit theory, nonparametric estimation, and asymptotic statistical inference. Empirical validation is conducted on canonical models—including Erdős–Rényi, Chung–Lu, and stochastic block models—as well as real-world networks (e.g., social and communication networks). Simulation and empirical results demonstrate that the proposed estimator accurately captures structural evolution and dynamic complexity.
📝 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 ErdH{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.