Dirichlet Problem...Solved!...Sort of...

Okay, so an alternate route to finding the probabilities has been discovered. Thanks to Richard, the guy doing his Ph.D.

So, while representing the image as an electric circuit, we can apply Kirchoff's Current Law:

{sum I=0}

at each node, of course, using the edge weights as the conductances of the "resistors".

This results in a system of linear equations with n-equations and n-unknowns...which is easily solvable using an elimination method such as Gaussian elimination or Gauss-Jordan elimination.

Keep in mind that the more known node values we have (i.e., user-defined labels), the fewer number of unknowns we have.