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Pré-Publication, Document De Travail Année : 2023

PROBABILITY-GRAPHONS: LIMITS OF LARGE DENSE WEIGHTED GRAPHS

Résumé

We introduce probability-graphons which are probability kernels that generalize graphons to the case of weighted graphs. Probability-graphons appear as the limit objects to study sequences of large weighted graphs whose distribution of subgraph sampling converge. The edge-weights are taken from a general Polish space, which also covers the case of decorated graphs. Here, graphs can be either directed or undirected. Starting from a distance $d_m$ inducing the weak topology on measures, we define a cut distance on probability-graphons, making it a Polish space, and study the properties of this cut distance. In particular, we exhibit a tightness criterion for probability-graphons related to relative compactness in the cut distance. We also prove that under some conditions on the distance $d_m$, which are satisfied for some well-know distances like the Prohorov distance, and the Fortet-Mourier and Kantorovitch-Rubinstein norms, the topology induced by the cut distance on the space of probability-graphons is independent from the choice of $d_m$. Eventually, we prove that this topology coincides with the topology induced by the convergence in distribution of the sampled subgraphs.
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Dates et versions

hal-04361443 , version 1 (22-12-2023)

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  • HAL Id : hal-04361443 , version 1

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Romain Abraham, Jean-François Delmas, Julien Weibel. PROBABILITY-GRAPHONS: LIMITS OF LARGE DENSE WEIGHTED GRAPHS. 2023. ⟨hal-04361443⟩
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