Ginzburg--Landau Functionals in the Large-Graph Limit
<|reference_start|>Ginzburg--Landau Functionals in the Large-Graph Limit: 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 $\Gamma$-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. 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 compute the GL minimizer for several example families of graphons.<|reference_end|>
@article{zhang2024ginzburg--landau,
    title={Ginzburg--Landau Functionals in the Large-Graph Limit},
    author={Edith Zhang, James Scott, Qiang Du, Mason A. Porter},
    journal={arXiv preprint arXiv:2408.00422},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.00422},
    primaryClass={math.FA cs.SI math.CO math.PR}
}