  We formulate and analyze difference equations on graphs analogous to
time-independent diffusion equations arising in the study of diffuse scattering
in continuous media. Moreover, we show how to construct solutions in the
presence of weak scatterers from the solution to the homogeneous (background
problem) using Born series, providing necessary conditions for convergence and
demonstrating the process through numerous examples. In addition, we outline a
method for finding Green's functions for Cayley graphs for both abelian and
non-abelian groups. Finally, we conclude with a discussion of the effects of
sparsity on our method and results, outlining the simplifications that can be
made provided that the scatterers are weak and well-separated.
