Eigenfunctions of the Laplacian matrix give a notion of how information flows through a graph. Is there an analogue of this for geometric graphs which tells the spatial directions in which information tends to flow?
For example, consider a graph on a 1024-dimensional hypercube where the only edges are between vertices $(u,v)$ where $v_i = u_i + \delta_{i1} + \delta_{i2}$, e.g. the first and second coordinates of $v$ are shifted up by one unit, and the other coordinates match those of $u$. All information in this case should flow along the vector $(1,1,0,...,0)$.
Is there some way to decompose a geometric graph in this manner?