  Stochastic partial differential equations (SPDE) on graphs were introduced by
Cerrai and Freidlin [Ann. Inst. Henri Poincar\'e Probab. Stat. 53 (2017)
865-899]. This class of stochastic equations in infinite dimensions provides a
minimal framework for the study of the effective dynamics of much more complex
systems. However, how they emerge from microscopic individual-based models is
still poorly understood, partly due to complications near vertex singularities.
In this work, motivated by the study of the dynamics and the genealogies of
expanding populations in spatially structured environments, we obtain a new
class of SPDE on graphs of Wright-Fisher type which have nontrivial boundary
conditions on the vertex set. We show that these SPDE arise as scaling limits
of suitably defined biased voter models (BVM), which extends the scaling limits
of Durrett and Fan [Ann. Appl. Probab. 26 (2016) 456-490]. We further obtain a
convergent simulation scheme for each of these SPDE in terms of a system of
It\^o SDEs, which is useful when the size of the BVM is too large for
stochastic simulations. These give the first rigorous connection between SPDE
on graphs and more discrete models, specifically, interacting particle systems
and interacting SDEs. Uniform heat kernel estimates for symmetric random walks
approximating diffusions on graphs are the keys to our proofs.
