  An emerging technique in image segmentation, semi-supervised learning, and
general classification problems concerns the use of phase-separating flows
defined on finite graphs. This technique was pioneered in Bertozzi and Flenner
(2012), which used the Allen-Cahn flow on a graph, and was then extended in
Merkurjev, Kostic and Bertozzi (2013) using instead the Merriman-Bence-Osher
(MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip
(2019), we gave a theoretical justification for this use of the MBO scheme in
place of Allen-Cahn flow, showing that the MBO scheme is a special case of a
"semi-discrete" numerical scheme for Allen-Cahn flow.
  In this paper, we extend this earlier work, showing that this link via the
semi-discrete scheme is robust to passing to the mass-conserving case. Inspired
by Rubinstein and Sternberg (1992), we define a mass-conserving Allen-Cahn
equation on a graph. Then, with the help of the tools of convex optimisation,
we show that our earlier machinery can be applied to derive the mass-conserving
MBO scheme on a graph as a special case of a semi-discrete scheme for
mass-conserving Allen-Cahn. We give a theoretical analysis of this flow and
scheme, proving various desired properties like existence and uniqueness of the
flow and convergence of the scheme, and also show that the semi-discrete scheme
yields a choice function for solutions to the mass-conserving MBO scheme.
Finally, we exhibit initial work towards extending to the multi-class case,
which in future work we seek to connect to recent work on multi-class MBO in
Jacobs, Merkurjev and Esedoglu (2018).
