  In recent years there has been an emerging interest in PDE-like flows defined
on finite graphs, with applications in clustering and image segmentation. In
particular for image segmentation and semi-supervised learning Bertozzi and
Flenner (2012) developed an algorithm based on the Allen-Cahn gradient flow of
a graph Ginzburg-Landau functional, and Merkurjev, Kosti\'c and Bertozzi (2013)
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 Allen-Cahn flow. First, we choose the double-obstacle
potential for the Ginzburg-Landau functional, and derive well-posedness and
regularity results for the resulting graph Allen-Cahn flow. Next, we exhibit a
"semi-discrete" time-discretisation scheme for Allen-Cahn flow of which the MBO
scheme is a special case. We investigate the long-time behaviour of this
scheme, and prove its convergence to the Allen-Cahn trajectory as the time-step
vanishes. Finally, following a question raised by Van Gennip, Guillen, Osting
and Bertozzi (2014), we exhibit results towards proving a link between
double-obstacle Allen-Cahn flow and mean curvature flow on graphs. We show some
promising $\Gamma$-convergence results, and translate to the graph setting two
comparison principles used by Chen and Elliott (1994) to prove the analogous
link in the continuum.
