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Unformatted text preview: May 19, 2010 Mary Radcliffe Introduction to the Heat Kernel 1 Introduction In this lecture, we introduce the heat kernel for a graph, the heat kernel PageRank, and the zeta function for a graph. We develop some associated theorems to these functions. 2 Heat Kernel PageRank Recall the definition of the transition probability matrix W = D- 1 A , so that W ( u,v ) = 1 d u u ∼ v else . Let L = I- W = D- 1 / 2 L D 1 / 2 , the tilted Laplacian . Let f : V → R , α ∈ [0 , 1] . Recall that we define PageRank with respect to α,f to be the unique vector pr α,f satisfying pr α,f = αf + (1- α ) pr α,f W . Solving this equation for pr α,f , we obtain pr α,f = f αI I- (1- α ) W = α ∞ X k =0 (1- α ) k fW k . We can use this generating function type structure to give us information about PageRank. How- ever, this is a geometric sum with ratio (1- α ) W , so when the eigenvalues of this matrix are large, the sum may not converge, or may converge very slowly. In order to take advantage of the generating function structure without having to worry about convergence issues, we turn to an exponential generating function, called the heat kernel PageRank. Definition 1. Let f : V → R . For t ≥ , define the heat kernel PageRank to be the function h t,f = e- t X k ≥ t k k !...
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- Fall '08
- Math, Probability theory, Summation, Markov chain, Generating function, ht