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Relating the cut distance and the weak* topology for graphons

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    SYSNO ASEP0536782
    Document TypeJ - Journal Article
    R&D Document TypeJournal Article
    Subsidiary JČlánek ve WOS
    TitleRelating the cut distance and the weak* topology for graphons
    Author(s) Doležal, Martin (MU-W) RID, SAI, ORCID
    Grebík, Jan (MU-W) SAI, RID
    Hladký, Jan (MU-W) RID, SAI, ORCID
    Rocha, Israel (UIVT-O) RID, SAI, ORCID
    Rozhoň, Václav (UIVT-O)
    Source TitleJournal of Combinatorial Theory. B. - : Academic Press - ISSN 0095-8956
    Roč. 147, March (2021), s. 252-298
    Number of pages47 s.
    Languageeng - English
    CountryUS - United States
    Keywordsgraphon ; compactness
    Subject RIVBA - General Mathematics
    OECD categoryPure mathematics
    Subject RIV - cooperationInstitute of Computer Science - General Mathematics
    R&D ProjectsGA17-27844S GA ČR - Czech Science Foundation (CSF)
    GF17-33849L GA ČR - Czech Science Foundation (CSF)
    GJ18-01472Y GA ČR - Czech Science Foundation (CSF)
    GJ16-07822Y GA ČR - Czech Science Foundation (CSF)
    Method of publishingLimited access
    Institutional supportMU-W - RVO:67985840 ; UIVT-O - RVO:67985807
    UT WOS000605462900011
    EID SCOPUS85084518436
    DOI10.1016/j.jctb.2020.04.003
    AnnotationThe 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.
    WorkplaceMathematical Institute
    ContactJarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757
    Year of Publishing2022
    Electronic addresshttps://doi.org/10.1016/j.jctb.2020.04.003
Number of the records: 1  

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