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Probabilistic programming interfaces for random graphs: Markov categories, graphons, and nominal sets

Nathanael L. Ackerman, Cameron E. Freer, Younesse Kaddar, Jacek Karwowski, Sean K. Moss, Daniel M. Roy, Sam Staton, Hongseok Yang

TL;DR

The paper develops a semantic bridge between probabilistic programming over graphs and graphons by showing a bidirectional correspondence between well-behaved equational theories for a graph interface and graphons. It builds this link in three layers: (i) an abstract construction via distributive Markov categories and affine monads that assigns a graphon to any suitable equational theory, (ii) concrete measure-theoretic interpretations that yield black-and-white graphons, and (iii) an internal probability framework on Rado-nominal sets that captures Erdős–Rényi graphons. The contributions include a quotients-based method to realize graphons from theories, two explicit semantic styles for graphon modeling, and a principled categorical perspective on graph-based probabilistic programming. The work advances the theoretical foundations of probabilistic programming for graphs and offers practical modeling paradigms tied to graphon theory, with implications for inference, compositional semantics, and the design of graph-oriented PPLs.

Abstract

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 Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons.

Probabilistic programming interfaces for random graphs: Markov categories, graphons, and nominal sets

TL;DR

The paper develops a semantic bridge between probabilistic programming over graphs and graphons by showing a bidirectional correspondence between well-behaved equational theories for a graph interface and graphons. It builds this link in three layers: (i) an abstract construction via distributive Markov categories and affine monads that assigns a graphon to any suitable equational theory, (ii) concrete measure-theoretic interpretations that yield black-and-white graphons, and (iii) an internal probability framework on Rado-nominal sets that captures Erdős–Rényi graphons. The contributions include a quotients-based method to realize graphons from theories, two explicit semantic styles for graphon modeling, and a principled categorical perspective on graph-based probabilistic programming. The work advances the theoretical foundations of probabilistic programming for graphs and offers practical modeling paradigms tied to graphon theory, with implications for inference, compositional semantics, and the design of graph-oriented PPLs.

Abstract

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 Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons.
Paper Structure (43 sections, 23 theorems, 40 equations, 1 figure)

This paper contains 43 sections, 23 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Example of an Implementation of a Random Graph: Geometric Random Graphs
  3. Implementations Regarded as Equational Theories
  4. From Equational Theories to Graphons
  5. From Graphons to Equational Theories
  6. Connection to Practice
  7. Programming interfaces for random graphs: equational theories and Markov categories
  8. Syntax for a Generic Probabilistic Programming Language
  9. Equational Theories and Markov Categories
  10. Equational Theories and Affine Monads
  11. Distributive Markov Categories from Affine Monads
  12. Example Affine Monad: Distribution Monad
  13. Example Affine Monad: Giry Monad
  14. Affine Monads from Distributive Markov Categories
  15. Bernoulli Bases, Numerals and Observation
  16. ...and 28 more sections

Key Result

Proposition 3

The equational theory induced by the interpretation in a distributive Markov category, with given interpretations of type and term constants, always includes the equations eq:let-assoc-- eq:let-affine, and also cd:let_val whenever $\llbracket v\rrbracket$ is a deterministic morphism.

Figures (1)

  • Figure 1: (a) A graph; (b) an inferred geometric realization of it ($θ \approx π/3$); (c) a generated sample for $θ = π/6$.

Theorems & Definitions (45)

  • Example 1
  • Definition 2
  • Proposition 3: e.g. dario-thesis, §7.1
  • Example 4
  • Example 5
  • Definition 6
  • Definition 7: kock-commjacobs-weakeninglindner-affine
  • Proposition 8
  • Definition 9: e.g. jacobs-coalgebra, §4.1
  • Definition 10
  • ...and 35 more