"Ginzburg–Landau Functionals in the Large-Graph Limit"

Another of my papers has appeared in final form. Here are some details.

Title: Ginzburg–Landau Functionals in the Large-Graph Limit

Authors: Edith J. Zhang, James Scott, Qiang Du, and Mason A. Porter

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_ϵ^{W_n} 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 graphon GL functional. We 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 minimization problem in the large-graph limit. Finally, to develop intuition about graphon GL functionals, we determine the GL minimizer for several example families of graphons.