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Seminar 217, Risk Management: PageRank on directed complex networksSeminar: Risk Seminar  January 25  12:302 p.m.  1011 Evans Hall Speaker: Mariana OlveraCravioto, UC Berkeley Center for Risk Management Research The talk will center around a set of recent results on the analysis of Google’s PageRank algorithm on directed complex networks. In particular, it will focus on the socalled powerlaw hypothesis, which states that the distribution of the ranks produced by PageRank on a scalefree graph (whose indegree distribution follows a powerlaw) also follows a powerlaw with the same tailindex as the indegree. We show that the distribution of PageRank on both the directed configuration model and the inhomogeneous random digraph does indeed follow a powerlaw whenever the indegree does, and we provide explicit asymptotic limits for it. Moreover, our asymptotic expressions exhibit qualitatively different behaviors depending on the level of dependence between the indegree and outdegree of each vertex. On graphs where the indegree and outdegree are close to independent, our main theorem predicts that PageRank will tend to grant high ranks to vertices with large indegrees, but also to vertices who have highlyranked inbound neighbors. However, when the indegree and outdegree are positively correlated, the latter can potentially disappear, strengthening the impact of highdegree vertices on the ranks produced by the algorithm. 

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