Jan Hladky: Relating the cut distance and the weak* topology for graphons and hypergraphons

Speaker: Jan Hladky

Title: Relating the cut distance and the weak* topology for graphons and hypergraphons

Abstract: The theory of graphons is ultimately connected with the so-called cut norm. We approach the cut norm topology via the weak{*} topology. We prove that a sequence $W_{1},W_{2},W_{3}, of graphons converges in the cut distance if and only if we have equality of the sets of weak{*} accumulation points and of weak{*} limit points of all sequences of graphons $W_{1}',W_{2}',W_{3}', that are weakly isomorphic to $W_{1},W_{2},W_{3}, We further give a short descriptive set theoretic argument that each sequence of graphons contains a subsequence with the property above. This in particular provides an alternative proof of the theorem of Lovasz and Szegedy about compactness of graphons. This is joint work with Martin Dolezal, Jan Grebik, Israel Rocha and Vaclav Rozhon (arXiv:1806.07368). We also find ways how to pinpoint cut distance limits in the space of weak* limits as minimizers or maximizers of various graphon parameters. The case of entropy-like parameters is joint work with Martin Dolezal (arXiv:1705.09160), and the general case is joint work with Martin Dolezal, Jan Grebik, Israel Rocha and Vaclav Rozhon (arXiv:1809.03797). With Frederik Garbe, Jon Noel, Diana Piguet, Israel Rocha, Maria Saumell we have a similar program for hypergraphons.