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Relating the cut distance and the weak* topology for graphons
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SYSNO ASEP 0536782 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Relating 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 Title Journal of Combinatorial Theory. B. - : Academic Press - ISSN 0095-8956
Roč. 147, March (2021), s. 252-298Number of pages 47 s. Language eng - English Country US - United States Keywords graphon ; compactness Subject RIV BA - General Mathematics OECD category Pure mathematics Subject RIV - cooperation Institute of Computer Science - General Mathematics R&D Projects GA17-27844S 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 publishing Limited access Institutional support MU-W - RVO:67985840 ; UIVT-O - RVO:67985807 UT WOS 000605462900011 EID SCOPUS 85084518436 DOI 10.1016/j.jctb.2020.04.003 Annotation 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. Workplace Mathematical Institute Contact Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Year of Publishing 2022 Electronic address https://doi.org/10.1016/j.jctb.2020.04.003
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