Probabilistic Programming Interfaces for Random Graphs: Markov Categories, Graphons, and Nominal Sets

We study semantic models of probabilistic programming languages over graphs,and establish a connection to graphons from graph theory and combinatorics. Weshow that every well-behaved equational theory for our graph probabilisticprogramming language corresponds to a graphon, and conversely, every graphonarises in this way. We provide three constructions for showing that every graphon arises from anequational theory. The first is an abstract construction, using Markovcategories and monoidal indeterminates. The second and third are more concrete.The second is in terms of traditional measure theoretic probability, whichcovers 'black-and-white' graphons. The third is in terms of probability monadson the nominal sets of Gabbay and Pitts. Specifically, we use a variation ofnominal sets induced by the theory of graphs, which covers Erdős-Rényigraphons. In this way, we build new models of graph probabilistic programmingfrom graphons.