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arXiv · 2412.00371

Categorical approach to graph limits

Martin Doležal·Wiesław Kubiś
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Abstract

We define and study a natural category of graph limits. The objects are pairs $(\pi,\mu)$, where $\pi$ (the distribution of vertices) is an abstract probability measure on some abstract measurable space $(X,\mathcal{A})$ and $\mu$ (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.

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Martin Doležal, Wiesław Kubiś. 2024-11-30. Categorical approach to graph limits. https://arxiv.org/abs/2412.00371

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