  We study a graph based version of the Ohta-Kawasaki functional, which was
originally introduced in a continuum setting to model pattern formation in
diblock copolymer melts and has been studied extensively as a paradigmatic
example of a variational model for pattern formation.
  Graph based problems inspired by partial differential equations (PDEs) and
varational methods have been the subject of many recent papers in the
mathematical literature, because of their applications in areas such as image
processing and data classification. This paper extends the area of PDE inspired
graph based problems to pattern forming models, while continuing in the
tradition of recent papers in the field.
  We introduce a mass conserving Merriman-Bence-Osher (MBO) scheme for
minimizing the graph Ohta-Kawasaki functional with a mass constraint. We
present three main results: (1) the Lyapunov functionals associated with this
MBO scheme $\Gamma$-converge to the Ohta-Kawasaki functional (which includes
the standard graph based MBO scheme and total variation as a special case); (2)
there is a class of graphs on which the Ohta-Kawasaki MBO scheme corresponds to
a standard MBO scheme on a transformed graph and for which generalized
comparison principles hold; (3) this MBO scheme allows for the numerical
computation of (approximate) minimizers of the graph Ohta-Kawasaki functional
with a mass constraint.
