  We propose random walks on suitably defined graphs as a framework for
finescale modeling of particle motion in an obstructed environment where the
particle may have interactions with the obstructions and the mean path length
of the particle may not be negligible in comparison to the finescale. This
motivates our study of a periodic, directed, and weighted graph embedded in
${\mathbb R}^d$ and the scaling limit of the associated continuous-time random
walk $Z(t)$ on the graph's nodes, which jumps along the graph's edges with jump
rates given by the edge weights. We show that the scaled process $\varepsilon^2
Z(t/\varepsilon^2)$ converges to a linear drift $\bar{U}t$ and the case of
interest to us is that of null drift $\bar{U}=0$. In this case, we show that
$\varepsilon Z(t/\varepsilon^2)$ converges weakly to a Brownian motion. The
diffusivity of the limiting Brownian motion can be computed by solving a set of
linear algebra problems. As we allow for jump rates to be irreversible, our
framework allows for the modeling of very general forms of interactions such as
attraction, repulsion, and bonding. We provide some sufficient conditions for
null drift that include certain symmetries of the graph. We also provide a
formal asymptotic derivation of the effective diffusivity in analogy with
homogenization theory for PDEs. For the case of reversible jump rates, we
derive an equivalent variational formulation. This derivation involves
developing notions of gradient for functions on the graph's nodes, divergence
for ${\mathbb R}^d$-valued functions on the graph's edges, and a divergence
theorem.
