  We consider a nonlocal evolution equation representing the continuum limit of
a large ensemble of interacting particles on graphs forced by noise. The two
principle ingredients of the continuum model are a nonlocal term and Q-Wiener
process describing the interactions among the particles in the network and
stochastic forcing respectively. The network connectivity is given by a square
integrable function called a graphon.
  We prove that the initial value problem for the continuum model is
well-posed. Further, we construct a semidiscrete (discrete in space and
continuous in time) and a fully discrete schemes for the nonlocal model. The
former is obtained by a discontinuous Galerkin method and the latter is based
on further discretizing time using the Euler-Maruyama method. We prove
convergence and estimate the rate of convergence in each case. For the
semidiscrete scheme, the rate of convergence estimate is expressed in terms of
the regularity of the graphon, Q-Wiener process, and the initial data. We work
in generalized Lipschitz spaces, which allows to treat models with data of
lower regularity. This is important for applications as many interesting types
of connectivity including small-world and power-law are expressed by graphons
that are not smooth. The error analysis of the fully discrete scheme, on the
other hand, reveals that for some models common in applied science, one has a
higher speed of convergence than that predicted by the standard estimates for
the Euler-Maruyama method. The rate of convergence analysis is supplemented
with detailed numerical experiments, which are consistent with our analytical
results.
  As a by-product, this work presents a rigorous justification for taking
continuum limit for a large class of interacting dynamical systems on graphs
subject to noise.
