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

Abstract

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 timestep 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

Positively curved graphs

“…We consider the question posed verbally by Stroock and written down by Ollivier and Villani [38], “what is the curvature of the discrete hypercube?” This question seems to be a qualitative one as much as a quantitative one, as Ollivier and Villani [38] address this question using at least three distinct notions of curvature. Proposed notions of positive curvature for graphs include coarse Ricci curvature [36,37], Bakry‐Emery version of Ricci curvature [6], dispersion of heat [25], displacement convexity with midpoints of quasi‐geodesics [12,13], displacement convexity with Gaussian midpoints [28], bootstrap percolation [27], among others. The above survey and the rest of this paper is limited to discrete analogues of positive curvature; there is a separate paper [45] where we consider discrete analogues of negative curvature.…”

An MBO Scheme on Graphs for Classification and Image Processing

“…This curvature, κ w , is also used in [15] as a regularizer in a graph adaptation of the Chan-Vese method. In their work-in-progress [58], van Gennip et al propose a different definition of mean curvature on graphs and prove convergence of the MBO scheme on graphs.…”

Graph MBO on Star Graphs and Regular Trees. With Corrections to DOI 10.1007/s00032-014-0216-8

“…In particular, this paper is both a corrigendum and a direct sequel to [14]. The main correction which this paper provides concerns [14,Theorem 4.8].…”

“…In recent years the graph version of this process and variations thereof have been succesfully applied to data clustering and classifcation problems and other graph based problems, e.g. in [4,5,8,9,10,7,15,3], which in turn has prompted further theoretical study of the MBO scheme on graphs [14,1].…”

“…

The graph Merriman-Bence-Osher scheme produces, starting from an initial node subset, a sequence of node sets obtained by iteratively applying graph diffusion and thresholding to the characteristic (or indicator) function of the node subsets. One result in [14] gives sufficient conditions on the diffusion time to ensure that the set membership of a given node changes in one iteration of the scheme. In particular, these conditions only depend on local information at the node (information about neighbors and neighbors of neighbors of the node in question).

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