AI 中文总结
研究受约束的多关系图子的RRS猜想，在非极值情况且子图密度约束解析独立等条件下，用微分几何技术证明熵最大化解是有限块阶梯函数，推广子图密度概念及证明流形拓扑稳定性是关键，解决了该猜想。
AI 中文摘要
最大熵原理为刻画受可观测约束的大型随机网络的典型结构提供了一个基本框架。在开创性的数值实验中，Radin、Ren和Sadun推测，满足子图密度约束的熵最大化图子是随机块模型，我们称之为RRS猜想。虽然对于具有特定约束族的单关系图已经证明了几个特殊情况，但一般问题仍然悬而未决，特别是对于多关系网络。我们在非极值情况下解决了约束多关系图子的RRS猜想，证明了在子图密度约束解析独立且对于几乎所有充分统计量的可行组合的条件下，熵最大化解是具有有限多个块的阶梯函数。我们的证明采用微分几何技术，通过具有有限参数化的函数（阶梯函数）来研究函数空间中约束优化问题的解。这项工作的两个基石是：将子图密度概念推广到h-子图密度，以及证明定义解的约束区域的流形在细化下保持拓扑稳定性而不产生新的连通分量。这些共同使得能够证明在高维空间中不会出现新的全局最优解。
英文摘要
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.
Comments53 pages, 3 figures
