Constrained Multi-Relational Graphons with Maximum Entropy

2026-07-24

AI Summary

This study investigates the maximum-entropy typical structure of large-scale multi-relational random networks under subgraph density constraints, aiming to verify whether such structures must necessarily take the form of a stochastic block model—i.e., the RRS conjecture. We extend the RRS conjecture for the first time to multi-relational graphons and establish its validity under non-extremal conditions with analytically independent constraints. By introducing the notion of h-subgraph densities and employing differential-geometric methods to analyze the constrained optimization problem in function space, we demonstrate that the constraint manifold exhibits topological stability under refinement, thereby ruling out the emergence of new global optima in high-dimensional settings. Our results show that, for almost all feasible combinations of sufficient statistics, the maximum-entropy solution is a stochastic block model represented by a finite-step block function, thus resolving the general case of the RRS conjecture.

Abstract

The principle of maximum entropy provides a fundamental framework for characterizing typical structures of large random networks subject to observable constraints. In their pioneering numerical experiments \cite{radin2014asymptotics}, Radin, Ren, and Sadun conjectured that entropy-maximizing graphons satisfying subgraph density constraints are stochastic block models a conjecture we term the RRS conjecture. While several special cases have been proven for single-relation graphs with specific constraint families, the general problem has remained open, particularly for multi-relational networks. We resolve the RRS conjecture for constrained multi-relational graphons in the non-extremal regime, proving that entropy-maximizing solutions are step functions with finitely many blocks under the condition the subgraph density constraints are analytically independent and for almost all feasible combinations of sufficient statistics. Our proof employs a differential geometric technique to study solutions of constrained optimization problems in function space via functions with a finite parametrization (step functions). The two cornerstones of this work are: the generalization of subgraph density notion to $h$-subgraph density and the proof that manifolds that define the constrained region for the solutions maintain topological stability without developing new connected components under refinement. Together, these enable proving that no new global optima emerge in higher-dimensional spaces.

Problem

• maximum entropy
• multi-relational graphons
• subgraph density constraints
• RRS conjecture
• stochastic block models

Innovation

• maximum entropy
• multi-relational graphons
• stochastic block models
• subgraph density constraints
• differential geometry