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150 7 ⋅ conclusion 7.2.6 PageRank as a function of a matrix It would be nice to be able to compare the PageRank derivative and the RAPr models directly. Each comes from a totally different approach, but per- haps there are some similarities. As luck would have it, there is a framework where we can simultaneously look at both models: PageRank as a function of a matrix. Recall that PageRank is a vector rational function of α. A rational function is a special case of an analytic function and analytic functions of scalars have equivalent matrix functions. Formally, if f (x) is an analytic function of x, then f (A) is well defined for any square matrix A. Thus, we propose the PageRank function of a matrix: x(A). WhenA=[α 1]thenx(A)containsboththePageRankvectoranditsderiva- 0α tive. When A = J, the Jacobi matrix for the orthogonal polynomials on the distribution of a random variable A, then A computes E [x(A)]! While both vectors fit into this model, it does not seem possible to compare them further. Nevertheless, the PageRank function of a matrix is a tantalizing generaliza- tion of the PageRank problem. It is somewhat of an aesthetic generalization because we have no compelling uses for it; although, the equivalence between the linear system formulation of PageRank and the eigensystem and Markov chain interpretation finally disappears. There are many aspects of this thesis that included results for both the PageRank linear system, which tended to be easy to derive, and the PageRank eigensystem, which tended to be difficult to derive. It is refreshing to conclude there are some differences between them.PDF Image | ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION
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