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Mean Curvature, Threshold Dynamics, and Phase Field Theory on Finite Graphs

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Research paper

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In the continuum, close connections exist between mean curvature flow, the Allen-Cahn AC partial differential equation, and the Merriman-Bence-Osher MBO threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples, and identify open questions for future study. We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation perimeter. This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow. We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters the time-step and diffuse interface scale, respectively are sufficiently small a phenomenon known as freezing or pinning and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize continuum nonlocal total variation.

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  • Theoretical Mathematics

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