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Entropy of Exchangeable Random Graphs

Anda Skeja, Sofia C. Olhede

TL;DR

This work defines graphon entropy $H(W)=\iint_{[0,1]^2} h(W(x,y))\,dx\,dy$ as a principled, isomorphism-invariant measure of graph complexity for exchangeable graphs and develops a versatile set of estimators, including a nonparametric estimator and specialized estimators for constant, separable, and block-constant graphons. It establishes convergence rates and Central Limit Theorems for these estimators, supported by extensive simulations and a real-data application to evolving networks. The results show the estimator's ability to capture degree heterogeneity and community structure, and demonstrate its advantage over traditional network entropy measures in reflecting generative structure. The theoretical and empirical findings connect graphon-entropy estimation to the asymptotic entropy of exchangeable graphs, offering a scalable framework for comparing and tracking the complexity of large networks.

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 Erdős-Rényi, 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.

Entropy of Exchangeable Random Graphs

TL;DR

This work defines graphon entropy as a principled, isomorphism-invariant measure of graph complexity for exchangeable graphs and develops a versatile set of estimators, including a nonparametric estimator and specialized estimators for constant, separable, and block-constant graphons. It establishes convergence rates and Central Limit Theorems for these estimators, supported by extensive simulations and a real-data application to evolving networks. The results show the estimator's ability to capture degree heterogeneity and community structure, and demonstrate its advantage over traditional network entropy measures in reflecting generative structure. The theoretical and empirical findings connect graphon-entropy estimation to the asymptotic entropy of exchangeable graphs, offering a scalable framework for comparing and tracking the complexity of large networks.

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 Erdős-Rényi, 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.
Paper Structure (23 sections, 10 theorems, 90 equations, 11 figures, 3 tables)

This paper contains 23 sections, 10 theorems, 90 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Entropy Measures for Networks
  4. Nonparametric Entropy Estimation
  5. Notation and Background
  6. Graphon models
  7. Graphon entropy
  8. Graphon Entropy Estimator of an Exchangeable Graph With a Smooth Graphon Representative
  9. Oracle Graphon Entropy Estimator
  10. Graphon Entropy Estimators Under Special Assumptions
  11. Estimating the Entropy of a Constant Graphon
  12. Estimating the Entropy of a Separable Graphon
  13. Estimating the Entropy of a Block--Constant Graphons
  14. Simulation Study
  15. Performance of the Graphon Entropy Estimator
  16. ...and 8 more sections

Key Result

Theorem 1

Let $A$ be a jointly exchangeable random array. Then there exists an i.i.d. sequence $\xi=(\xi_1,...,\xi_n)$ following $U(0,1)$, a random variable $\gamma \sim U(0,1)$ independent of $\xi$, and a function $W:[0,1]^3 \to [0,1]$ such that and $A_{ij}$ are conditionally independent across $i,j$ given $\xi$ and $\gamma$.

Figures (11)

  • Figure 2: Graphs generated on $n=500$ nodes with $\rho_n=0.25$ and respective graphons, $f_1$ as given in \ref{['sim-f1']} and $f_2$ given in \ref{['sim-f2']}.
  • Figure 3: Graphs generated on $n=96$ nodes with $\rho_n=0.5$ and varying community structures.
  • Figure 4: Comparison of the nonparametric graphon entropy estimator with some network entropy measures across graphs with varying community structures as given in Figure \ref{['SBM-entropy']}.
  • Figure : (a)
  • Figure : (a)
  • ...and 6 more figures

Theorems & Definitions (29)

  • Definition 1: Constant graphon
  • Definition 2: Separable graphon
  • Definition 3: Block-constant graphon
  • Definition 4
  • Theorem 1: Aldous--Hoover aldous1981representationshoover1979relations
  • Definition 5: Entropy of a graphon janson2013graphonshatami2018graph
  • Theorem 2
  • proof
  • Definition 6: Oracle entropy estimator
  • Proposition 1
  • ...and 19 more