In this talk, we’ll discuss the connection between random walks on graphs, graph spectra and geometric features of networks. As part of this discussion, we’ll talk about PageRank, which is a random walk based method originally developed for web-search. We’ll discuss a way of viewing PageRank as a type of parabolic operator, and will talk about how we can use recently developed notions of graph curvature to prove Harnack-like inequalities to understand the relative importance of different webpages. To do this we develop a gradient estimate for PageRank in the vein of the Li-Yau inequality for positive solutions to the heat equation. (More precisely, our inequality is closer to Hamilton’s gradient estimate for positive solutions.

Paul Horn is recently an associate professor at University of Denver. His research interest is in combinatorics， Specifically, he is interested in using ideas from probability, linear algebra, and geometry methods to understand graphs, which are a mathematical abstraction of networks.
He received his PhD in 2009 from the University of California at San Diego, advised by Fan Chung, and joined the faculty of University of Denver in the fall of 2013 after postdoctoral positions at Emory University and Harvard University.