Ginzburg-Landau Functionals in the Large-Graph Limit

2024-08-01

arXiv.org

AI Summary

This study investigates the asymptotic behavior of the Ginzburg–Landau (GL) functional on large graphs under the graphon topology: specifically, when a sequence of graphs converges to a graphon, the discrete GL functional Γ-converges to a nonlocal continuum GL functional—establishing the first rigorous Γ-convergence analysis of graph-based GL functionals within the graphon convergence framework. Methodologically, the analysis integrates nonlocal calculus, Γ-convergence theory, Young measures, and variational techniques to characterize the sharp-interface limit and uncover its intrinsic connection to the nonlocal total variation (TV) functional; Young measures further yield a probabilistic variational interpretation. Key contributions include: (i) a rigorous proof of Γ-convergence; (ii) derivation of a nonlocal TV characterization for the sharp-interface limit; and (iii) explicit computation of GL minimizers for several canonical graphons. These results provide a novel paradigm for continuum approximation and phase-transition modeling of graph-based systems.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Web measurements
• Security and Privacy: Large-scale security measurements

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.

Problem

• Studying large-graph limits of Ginzburg-Landau functionals on graphs
• Analyzing Γ-convergence to continuous nonlocal graphon functionals
• Investigating sharp-interface limits and probabilistic interpretations

Innovation

• Gamma-convergence of graph GL functionals
• Graphon-based continuous nonlocal functional analysis
• Sharp-interface limits related to nonlocal TV