Lecture 4 - 5 : Effective Resistance and Simple Random Walks

The notion of electrical flows arises naturally when we treat our graph as a resistor network. Given a graph G = (V,E) with weights w(.) on the edges, we replace each edge e with a resistance of resistor 1/w(e). In other words, think of w(e) as the conductance of the edge e. We can then study how the electricity flows in this network. Now, we write two underlying properties of electrical flows. The first one is the flow conservation property. Say we are sending one unit of flow from s to t. The flow conservation property says that for any vertex v 6= s, t, the sum of the flows into v is zero, this sum is +1 for s and −1 for t. For any edge (u, v) let x(e) be the flow along edge e, that is x(e) is non-negative if electricity is going from u to v and it is non-positive otherwise. Also, let δ−(v) be the neighbors u of v where the edge (u, v) is oriented from u to v, and δ(v) be the rest of the neighbors of v. Then, ∑