Constrained Multi-Relational Hyper-Graphons with Maximum Entropy

Abstract

This work has two contributions. The first one is extending the Large Deviation Principle for uniform hyper-graphons from Lubetzky and Zhao \cite{lubetzky2015replica} to the multi-relational setting where each hyper-graphon can have different arities. This extension enables the formulation of the most typical possible world in Relational Probabilistic Logic with symmetric relational symbols in terms of entropy maximization subjected to constraints of quantum sub-hypergraph densities. The second contribution is to prove the most typical constrained multi-relational hyper-graphons (the most typical possible worlds) are computable by proving the solutions of the maximum entropy subjected by quantum sub-hypergraph densities in the space of multi-relational hyper-graphons are step functions except for in a zero measure set of combinations of quantum hyper-graphs densities with multiple relations. This result proves in a very general context the conjecture formulated by Radin et al.\ \cite{radin2014asymptotics} that states the constrained graphons with maximum entropy are step functions.