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Relating the cut distance and the weak* topology for graphons
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Doležal, Martin, Grebik, Jan, Hladký, Jan, Rocha, Israel and Rozhoň, Václav (2021) Relating the cut distance and the weak* topology for graphons. Journal of Combinatorial Theory, Series B, 147 . pp. 252-298. doi:10.1016/j.jctb.2020.04.003 ISSN 0095-8956.
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Official URL: http://doi.org/10.1016/j.jctb.2020.04.003
Abstract
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 -functions). We prove that a sequence 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 that are weakly isomorphic to
. 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 Borgs et al. (2012) [5].
Item Type: | Journal Article | |||||||||||||||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||||||||||||||
Library of Congress Subject Headings (LCSH): | Graph theory, Measure theory, Probabilities, Convergence, Mathematical statistics | |||||||||||||||||||||
Journal or Publication Title: | Journal of Combinatorial Theory, Series B | |||||||||||||||||||||
Publisher: | Academic Press | |||||||||||||||||||||
ISSN: | 0095-8956 | |||||||||||||||||||||
Official Date: | March 2021 | |||||||||||||||||||||
Dates: |
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Volume: | 147 | |||||||||||||||||||||
Page Range: | pp. 252-298 | |||||||||||||||||||||
DOI: | 10.1016/j.jctb.2020.04.003 | |||||||||||||||||||||
Status: | Peer Reviewed | |||||||||||||||||||||
Publication Status: | Published | |||||||||||||||||||||
Access rights to Published version: | Restricted or Subscription Access | |||||||||||||||||||||
Date of first compliant deposit: | 4 April 2023 | |||||||||||||||||||||
Date of first compliant Open Access: | 5 April 2023 | |||||||||||||||||||||
RIOXX Funder/Project Grant: |
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