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Relating the cut distance and the weak* topology for graphons

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    0536782 - MÚ 2022 RIV US eng J - Journal Article
    Doležal, Martin - Grebík, Jan - Hladký, Jan - Rocha, Israel - Rozhoň, Václav
    Relating the cut distance and the weak* topology for graphons.
    Journal of Combinatorial Theory. B. Roč. 147, March (2021), s. 252-298. ISSN 0095-8956. E-ISSN 1096-0902
    R&D Projects: GA ČR(CZ) GA17-27844S; GA ČR GF17-33849L; GA ČR(CZ) GJ18-01472Y; GA ČR GJ16-07822Y
    Institutional support: RVO:67985840 ; RVO:67985807
    Keywords : graphon * compactness
    OECD category: Pure mathematics; Pure mathematics (UIVT-O)
    Impact factor: 1.317, year: 2020
    Method of publishing: Limited access

    The theory of graphons is ultimately connected with the so-called cut norm. In this paper, we approach the cut norm topology via the weak* topology (when considering a predual of L1-functions). We prove that a sequence W1, W2, W3, ... 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 W1, W2, W3, ... that are weakly isomorphic to W1, W2, W3, ... . 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 Lovász and Szegedy about compactness of the space of graphons. We connect these results to 'multiway cut' characterization of cut distance convergence from [Ann. of Math. (2) 176 (2012), no. 1, 151-219]. These results are more naturally phrased in the Vietoris hyperspace K over graphons with the weak* topology.
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