Graph merriman-bence-osher as a semidiscrete implicit euler scheme for graph allen-cahn flow

In recent years there has been an emerging interest in PDE-like ows defined on finite graphs, with applications in clustering and image segmentation. In particular for image segmentation and semisupervised learning Bertozzi and Flenner [Multiscale Model. Simul., 10 (2012), pp. 1090-1118] developed an algorithm based on the Allen-Cahn (AC) gradient flow of a graph Ginzburg-Landau functional, and Merkurjev, Kostific, and Bertozzi [SIAM J. Imaging Sci., 6 (2013), pp. 1903-1930] devised a variant algorithm based instead on graph Merriman-Bence-Osher (MBO) dynamics. This work offers rigorous justification for this use of the MBO scheme in place of AC flow. First, we choose the double-obstacle potential for the Ginzburg-Landau functional and derive wellposedness and regularity results for the resulting graph AC flow. Next, we exhibit a "semidiscrete" time-discretization scheme for AC flow of which the MBO scheme is a special case. We investigate the long-time behavior of this scheme and prove its convergence to the AC trajectory as the time-step vanishes. Finally, following a question raised by Van Gennip, Guillen, Osting, and Bertozzi [Milan J. Math., 82 (2014), pp. 3-65], we exhibit results toward proving a link between double-obstacle AC flow and mean curvature flow on graphs. We show some promising Γ-convergence results and translate to the graph setting two comparison principles used by Chen and Elliott [Proc. Math. Phys. Sci., 444 (1994), pp. 429-445] to prove the analogous link in the continuum.