In this experiment we model max clusters as resistor networks. In this model, graph edges with source/target pairs (x, y) are resistors with resistance R(x, y) = 1 / Ε(u) where Ε(u) is the graph energy Ε(u) = ∑_a~b(u(a) - u(b))².

We set up each resistor as follows. Pick two vertices x and y in the graph and set u(x) = 0 and u(y) = 1. Now, for all z ≠ x, y, u(z) is simply the average of the values of u at each of z's neighbors. In this way we can obtain a system of equations that is used to obtain all u(z) and finally find the resistance R(x, y). These resistance values can now be used to judge a walk recurrent or transient. Given two endpoints, we would expect a walk to be recurrent if the resistance between the endpoints is quite large and transient otherwise. We study this behavior on max clusters by varying the probability p used to produce them, fixing some vertex x in the graph and looking at R(x, y) for all y ≠ x in the graph.

This experiment comes from Wolfgang Woess. The setup is similar to the one in the previous section, except rather than fix a pair (x, y) with u(x) = 0 and u(y) = 1 we instead fix an x with u(x) = 1 and some r ≥ 1. Then for all y ∈ V if d(x, y) ≥ r we set u(y) = 0. d(x, y) can be taken to be the length of the shortest path between x and y or something simpler such as the distance between x and y in the parent graph. We then use a system of equations as above to fill in the values for u and find the resistances 1 / Ε(u) for increasing values of r. In this way, we compute the analogue of the resistance at infinity on these finite graphs.