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.
