Ginzburg--Landau Functionals in the Large-Graph Limit

TL;DR

This paper investigates the large-graph limit of Ginzburg--Landau functionals, showing their convergence to a nonlocal functional on graphons, and explores the associated variational problems and minimizers.

Contribution

It establishes the $ ext{Gamma}$-convergence of graph Ginzburg--Landau functionals to a nonlocal graphon functional, providing a new analytical framework for large-graph limits.

Findings

Graph Ginzburg--Landau functionals $ ext{Gamma}$-converge to a nonlocal graphon functional.

The sharp-interface limits relate to nonlocal total variation.

Explicit minimizers are characterized for specific graphon examples.

Abstract

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 (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.