Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Categorical approach to graph limits

Martin Doležal, Wiesław Kubiś

Abstract

We define and study a natural category of graph limits. The objects are pairs $(π,μ)$, where $π$ (the distribution of vertices) is an abstract probability measure on some abstract measurable space $(X,\mathcal{A})$ and $μ$ (the distribution of edges) is an abstract finite measure on the square $(X,\mathcal{A})^2$. Morphisms are random maps between the underlying measurable spaces which preserve the distribution of vertices as well as the distribution of edges. We also define a convergence notion (inspired by s-convergence) for sequences of graph limits. We apply tools from category theory to prove the compactness of the space of all graph limits.

Categorical approach to graph limits

Abstract

We define and study a natural category of graph limits. The objects are pairs , where (the distribution of vertices) is an abstract probability measure on some abstract measurable space and (the distribution of edges) is an abstract finite measure on the square . Morphisms are random maps between the underlying measurable spaces which preserve the distribution of vertices as well as the distribution of edges. We also define a convergence notion (inspired by s-convergence) for sequences of graph limits. We apply tools from category theory to prove the compactness of the space of all graph limits.

Paper Structure

This paper contains 13 sections, 20 theorems, 200 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Markov kernels
  4. Categories
  5. Pushforward measures
  6. Inverse systems of measures and Kolmogorov Extension Theorem
  7. The category of $\square$-graphon s
  8. Convergence
  9. Existence of limits
  10. Comparison with s-convergence
  11. Final remarks
  12. Conclusion.
  13. Acknowledgments.

Key Result

Lemma 2.1

Let $L$ be a non-empty finite set. Let $(X,\mathcal{A})$ and $(Y_l,\mathcal{B}_l)$, $l\in L$, be measurable spaces. Let $\kappa_l$ be a Markov kernel from $(X,\mathcal{A})$ to $(Y_l,\mathcal{B}_l)$, $l\in L$. Then the map $\prod_{l\in L}\kappa_l$ defined by eq:soucinKap is a Markov kernel from $(X,\

Figures (1)

  • Figure 1: The red arrow represents the convergence which we want to prove. Black arrows represent morphisms in the category of all $\square$-graphon s. The orange arrow represents convergence in the sense of \ref{['eq:limitniMiry1']} and \ref{['eq:limitniMiry2']}. Blue arrows represent convergence in the space of all $\square$-graphon s on $\mathcal{F}_k$.

Theorems & Definitions (63)

  • Lemma 2.1
  • proof
  • Example 2.2
  • Lemma 2.3
  • proof
  • Example 2.4
  • Example 2.5
  • Lemma 2.6
  • proof
  • Example 2.7
  • ...and 53 more