2025 Theses Doctoral

Ginzburg–Landau Functionals and Reaction–diffusion Equations in the Large-Graph Limit

Zhang, Edith Jin

Graphons are a notion of infinite graphs that are spatially continuous rather than discrete (as are the nodes of a graph). They have emerged as a mathematical formalism that attempts to preserve the essential properties of graphs as their sizes approach infinity. In this dissertation, we consider two problems on graphs, and we let the size of the graphs approach infinity. We define graphon limits of these two graph problems, and we show that the limiting problems on graphons are consistent with their graph counterparts in the sense that the solutions of the graph problems converge to the solutions of the limiting graphon problems.

In Chapters 3–6, we consider the problem of minimizing graph Ginzburg–Landau (GL) functionals. Graph GL functionals are relaxations of graph-cut functionals on graphs, and they have yielded a variety of insights in image segmentation and graph clustering. For sequences of graphs 𝑊_𝑛 that converge to a limiting graphon 𝑊, we show that the graph GL functional Γ-converges to a continuous and nonlocal functional that we call the 𝘨𝘳𝘢𝘱𝘩𝘰𝘯 𝘎𝘓 𝘧𝘶𝘯𝘤𝘵𝘪𝘰𝘯𝘢𝘭. We also show that graphon GL functionals Γ-converge, in a sharp-interface limit, to a graphon total variation (TV) functional. Using the Γ-convergence results, we conclude that the minimizers of the Γ-converging sequences of functionals converge to the minimizers of their limiting functionals. To obtain our convergence results, we require an extension of the underlying function spaces; we discuss the need for a probabilistic interpretation (via Young measures) of the variational problems in the graphon limit.

In Chapters 7–8, we study graph RD equations, which are certain systems of differential equa-tions that are defined on the nodes of a graph. Consider a sequence of graphs that converges to a limiting graphon. We show that the solutions of the sequence of graph RD equations converge to the solution of a limiting graphon RD equation, which we define to be a continuous RD equation that has a nonlocal diffusion term. Furthermore, we show that a sequence of stochastic particle processes (that consist of a random walk and a birth-death process) on the sequence of graphs converges to the solution of the graphon RD equation.

The graphon limits of graph problems can be viewed as continuous and nonlocal counterparts of discrete graph problems, which tend to be large systems of coupled equations. Our results establish precedent and intuition for future work on graphon limits of graph problems, and they build the connection between graphon theory and nonlocal analysis.