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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.00422 |
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| _version_ | 1866918191220916224 |
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| author | Zhang, Edith Scott, James Du, Qiang Porter, Mason A. |
| author_facet | Zhang, Edith Scott, James Du, Qiang Porter, Mason A. |
| contents | Ginzburg--Landau (GL) functionals on graphs, which are relaxations of graph-cut functionals on graphs, have yielded a variety of insights in image segmentation and graph clustering. In this paper, we study large-graph limits of GL functionals by taking a functional-analytic view of graphs as nonlocal kernels. For a graph $W_n$ with $n$ nodes, the corresponding graph GL functional $\GL^{W_n}_\ep$ is an energy for functions on $W_n$. We minimize GL functionals on sequences of growing graphs that converge to functions called graphons. For such sequences of graphs, we show that the graph GL functional $Γ$-converges to a continuous and nonlocal functional that we call the \emph{graphon GL functional}. We also investigate the sharp-interface limits of the graph GL and graphon GL functionals, and we relate these limits to a nonlocal total-variation (TV) functional. We express the limiting GL functional in terms of Young measures and thereby obtain a probabilistic interpretation of the variational problem in the large-graph limit. Finally, to develop intuition about the graphon GL functional, we determine the GL minimizer for several example families of graphons. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00422 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ginzburg--Landau Functionals in the Large-Graph Limit Zhang, Edith Scott, James Du, Qiang Porter, Mason A. Functional Analysis Social and Information Networks Combinatorics Probability Ginzburg--Landau (GL) functionals on graphs, which are relaxations of graph-cut functionals on graphs, have yielded a variety of insights in image segmentation and graph clustering. In this paper, we study large-graph limits of GL functionals by taking a functional-analytic view of graphs as nonlocal kernels. For a graph $W_n$ with $n$ nodes, the corresponding graph GL functional $\GL^{W_n}_\ep$ is an energy for functions on $W_n$. We minimize GL functionals on sequences of growing graphs that converge to functions called graphons. For such sequences of graphs, we show that the graph GL functional $Γ$-converges to a continuous and nonlocal functional that we call the \emph{graphon GL functional}. We also investigate the sharp-interface limits of the graph GL and graphon GL functionals, and we relate these limits to a nonlocal total-variation (TV) functional. We express the limiting GL functional in terms of Young measures and thereby obtain a probabilistic interpretation of the variational problem in the large-graph limit. Finally, to develop intuition about the graphon GL functional, we determine the GL minimizer for several example families of graphons. |
| title | Ginzburg--Landau Functionals in the Large-Graph Limit |
| topic | Functional Analysis Social and Information Networks Combinatorics Probability |
| url | https://arxiv.org/abs/2408.00422 |