Probability-graphons: Limits of large dense weighted graphs

Romain Abraham, Jean-François Delmas, Julien Weibel

Abstract: 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 $d_{\mathrm{m}}$ 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 $d_{\mathrm{m}}$, which are satisfied for some well-know distances like the Lévy–Prokhorov distance, and the Fortet–Mourier and Kantorovitch–Rubinshtein norms, the topology induced by the cut distance on the space of probability-graphons is independent from the choice of $d_{\mathrm{m}}$. Eventually, we prove that this topology coincides with the topology induced by the convergence in distribution of the sampled subgraphs.