Journal article
Nonlocal-interaction equation on graphs: Gradient flow structure and continuum limit
- Abstract:
- We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL2IE). We develop the existence theory for the solutions of the NL2IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL2IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
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(Preview, Version of record, pdf, 1.8MB, Terms of use)
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- Publisher copy:
- 10.1007/s00205-021-01631-w
Authors
- Publisher:
- Springer
- Journal:
- Archive for Rational Mechanics and Analysis More from this journal
- Volume:
- 240
- Issue:
- 2
- Pages:
- 699-760
- Publication date:
- 2021-03-15
- Acceptance date:
- 2021-01-29
- DOI:
- EISSN:
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1432-0673
- ISSN:
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0003-9527
- Language:
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English
- Keywords:
- Pubs id:
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1262765
- Local pid:
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pubs:1262765
- Deposit date:
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2022-06-09
- ARK identifier:
Terms of use
- Copyright holder:
- Esposito et al
- Copyright date:
- 2021
- Rights statement:
- © The Author(s) (2021). Open Access: This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
- Licence:
- CC Attribution (CC BY)
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