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Ginzburg--Landau Functionals in the Large-Graph Limit

Edith Zhang, James Scott, Qiang Du, Mason A. Porter

TL;DR

This work develops a comprehensive large-graph limit theory for Ginzburg--Landau functionals by treating graphs as nonlocal kernels and passing to graphon limit objects. It establishes four Gamma-convergence scenarios connecting graph GL to graphon GL and graphon TV, with two sequential orders (ep then n, and n then ep) and a variable scaling analysis, all framed through Young measures to capture oscillatory minimizers. The analysis shows that in the graphon limit, minimizers become probabilistic objects (Young measures) and nonlocal interactions persist, yielding a nonlocal TV functional as the Gamma-limit of GL in the small-parameter regime. The results apply to representative graphon families (e.g., constant graphons and 2x2 SBMs) and illuminate how large, structured networks exhibit phase separation aligned with their mesoscale organization, with potential implications for scalable graph-based clustering and segmentation in the large-graph regime.

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 $Γ$-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.

Ginzburg--Landau Functionals in the Large-Graph Limit

TL;DR

This work develops a comprehensive large-graph limit theory for Ginzburg--Landau functionals by treating graphs as nonlocal kernels and passing to graphon limit objects. It establishes four Gamma-convergence scenarios connecting graph GL to graphon GL and graphon TV, with two sequential orders (ep then n, and n then ep) and a variable scaling analysis, all framed through Young measures to capture oscillatory minimizers. The analysis shows that in the graphon limit, minimizers become probabilistic objects (Young measures) and nonlocal interactions persist, yielding a nonlocal TV functional as the Gamma-limit of GL in the small-parameter regime. The results apply to representative graphon families (e.g., constant graphons and 2x2 SBMs) and illuminate how large, structured networks exhibit phase separation aligned with their mesoscale organization, with potential implications for scalable graph-based clustering and segmentation in the large-graph regime.

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 with nodes, the corresponding graph GL functional is an energy for functions on . 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 -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.
Paper Structure (26 sections, 18 theorems, 120 equations, 3 figures)

This paper contains 26 sections, 18 theorems, 120 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Graphons, norms, and convergence
  4. Cut norm and cut convergence
  5. Lp graphons
  6. Functions and functionals on graphs and graphons
  7. Function spaces
  8. Gamma-convergence
  9. Graph functions and functionals
  10. Young measures and weak-* convergence of functions
  11. Graphon functions and functionals
  12. Sequential limit: ep then n
  13. Limit (1)
  14. Limit (2)
  15. Sequential limit: n then ep
  16. ...and 11 more sections

Key Result

Lemma 4.1

Let $\{\nu^n\}$ be a sequence of Young measures that converges narrowly to $\nu \in \mathcal{Y}((0,1), \mathbb{R})$. It is then guaranteed that the sequence ${\{\nu^n \otimes \nu^n\}}$ of product measures on $(0,1)^2 \times \,\mathbb{R}^2$ converges narrowly to $\nu \otimes \nu \in \mathcal{Y}((0,1) for all $f \in C^b(\mathbb{R}^2)$, where $d (\nu \otimes \nu)_{(x,y)}(\lambda,\mu) = d \nu_x(\lambd

Figures (3)

  • Figure 1: Four different $\Gamma$-convergences of the graph GL functional with different orderings of the $n \rightarrow \infty$ and $\epsilon \rightarrow 0$ limits. The arrows indicate $\Gamma$-convergence. For definitions of the functionals, see Section \ref{['sec: notation']}.
  • Figure 2: An example of a graph, its associated adjacency matrix, and its corresponding step graphon.
  • Figure 3: Three types of $2\times 2$ piecewise-constant graphons.

Theorems & Definitions (43)

  • Definition 3.1: graph cut
  • Definition 3.2
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Definition 4.1
  • Remark 4.1
  • Definition 4.2: Young measure
  • Definition 4.3: Young measure corresponding to a measurable function
  • ...and 33 more