We show that whenever there is a sharp drop in the numerical rank defined by a personalized PageRank vector, the location of the drop reveals a cut with small conductance. We then show that for any cut in the graph, and for many starting vertices within that cut, an approximate personalized PageRank vector will have a sharp drop sufficient to produce a cut with conductance nearly as small as the original cut. Using this technique, we produce a nearly linear time local partitioning algorithm whose analysis is simpler than previous algorithms.
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