IR-TST 2004 Problem 5

This problem is generalization of this one. Suppose $G$ is a graph and $S\subset V(G)$ . Suppose we have arbitrarily assign real numbers to each element of $S$ . Prove that we can assign numbers to each vertex in $G\backslash S$ that for each $v\in G\backslash S$ number assigned to $v$ is average of its neighbors.