Cutting Graphs, Personal PageRank and Spilling Paint

You may have figured out by now that I like cutting graphs into pieces. It is how I discover the structure of a graph. When I encounter a new graph and want to understand what it looks like I first try drawing it. If the drawing looks good, I feel like I understand it. If the drawing looks good, I try chopping the graph into pieces without cutting too many edges. That is, I look for cuts of low conductance. The first way I try to find cuts of low conductance is by examining the eigenvector of the secondlargest eigenvalue of the walk matrix. In lecture 9, we observed that lazy walks converge to the stable distribution, and that they do so at a rate that depends on the magnitude of the secondlargest eigenvalue. This was because