Can smooth graphons in several dimensions be represented by smooth graphons on [0,1]?
A graphon that is defined on [0,1]dand is Hölder(α)continuous for some d⩾2and α∈(0,1]can be represented by a graphon on [0,1]that is Hölder(α/d)continuous. We give examples that show that this reduction in smoothness to α/dis the best possible, for any d and α; for α=1, the example is a dot product...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Elsevier
2021-11-01
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| Series: | Examples and Counterexamples |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666657X21000070 |
| _version_ | 1819096996763402240 |
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| author | Svante Janson Sofia Olhede |
| author_facet | Svante Janson Sofia Olhede |
| author_sort | Svante Janson |
| collection | DOAJ |
| description | A graphon that is defined on [0,1]dand is Hölder(α)continuous for some d⩾2and α∈(0,1]can be represented by a graphon on [0,1]that is Hölder(α/d)continuous. We give examples that show that this reduction in smoothness to α/dis the best possible, for any d and α; for α=1, the example is a dot product graphon and shows that the reduction is the best possible even for graphons that are polynomials.A motivation for studying the smoothness of graphon functions is that this represents a key assumption in non-parametric statistical network analysis. Our examples show that making a smoothness assumption in a particular dimension is not equivalent to making it in any other latent dimension. |
| first_indexed | 2024-12-22T00:08:04Z |
| format | Article |
| id | doaj.art-c46f048455124791b2b18e3f1412dfea |
| institution | Directory Open Access Journal |
| issn | 2666-657X |
| language | English |
| last_indexed | 2024-12-22T00:08:04Z |
| publishDate | 2021-11-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Examples and Counterexamples |
| spelling | doaj.art-c46f048455124791b2b18e3f1412dfea2022-12-21T18:45:30ZengElsevierExamples and Counterexamples2666-657X2021-11-011100011Can smooth graphons in several dimensions be represented by smooth graphons on [0,1]?Svante Janson0Sofia Olhede1Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden; Corresponding author.Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne, Lausanne, SwitzerlandA graphon that is defined on [0,1]dand is Hölder(α)continuous for some d⩾2and α∈(0,1]can be represented by a graphon on [0,1]that is Hölder(α/d)continuous. We give examples that show that this reduction in smoothness to α/dis the best possible, for any d and α; for α=1, the example is a dot product graphon and shows that the reduction is the best possible even for graphons that are polynomials.A motivation for studying the smoothness of graphon functions is that this represents a key assumption in non-parametric statistical network analysis. Our examples show that making a smoothness assumption in a particular dimension is not equivalent to making it in any other latent dimension.http://www.sciencedirect.com/science/article/pii/S2666657X21000070GraphonSmooth graphonNon-parametric statistical network inference |
| spellingShingle | Svante Janson Sofia Olhede Can smooth graphons in several dimensions be represented by smooth graphons on [0,1]? Examples and Counterexamples Graphon Smooth graphon Non-parametric statistical network inference |
| title | Can smooth graphons in several dimensions be represented by smooth graphons on [0,1]? |
| title_full | Can smooth graphons in several dimensions be represented by smooth graphons on [0,1]? |
| title_fullStr | Can smooth graphons in several dimensions be represented by smooth graphons on [0,1]? |
| title_full_unstemmed | Can smooth graphons in several dimensions be represented by smooth graphons on [0,1]? |
| title_short | Can smooth graphons in several dimensions be represented by smooth graphons on [0,1]? |
| title_sort | can smooth graphons in several dimensions be represented by smooth graphons on 0 1 |
| topic | Graphon Smooth graphon Non-parametric statistical network inference |
| url | http://www.sciencedirect.com/science/article/pii/S2666657X21000070 |
| work_keys_str_mv | AT svantejanson cansmoothgraphonsinseveraldimensionsberepresentedbysmoothgraphonson01 AT sofiaolhede cansmoothgraphonsinseveraldimensionsberepresentedbysmoothgraphonson01 |