We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way. We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers 'black-and-white' graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erd\H{o}s-R\'enyi graphons. In this way, we build new models of graph probabilistic programming from graphons.

翻译：我们研究图上的概率编程语言的语义模型，并建立其与图论和组合学中图极限的联系。我们证明，图概率编程语言的每一个良行为的等式理论都对应一个图极限，反之，每一个图极限也都可以通过这种方式产生。我们提供了三种构造来证明每个图极限都源于一个等式理论。第一种是抽象构造，利用马尔可夫范畴和幺半群不定元；第二种和第三种更为具体。第二种基于传统测度论概率，覆盖"黑白"图极限；第三种基于Gabbay和Pitts名义集合上的概率单子。具体而言，我们使用由图理论诱导的名义集的一种变体，覆盖了Erdős-Rényi图极限。通过这种方式，我们从图极限构建了图概率编程的新模型。