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2.2.3 Other variants Our list here is not exhaustive. Langville and Meyer [2006a, section 8.4] define a bounce-back stochastic matrix from any sub-stochastic matrix where each edge into a 0 column produces a new vertex to return the Markov chain to the previous state. Another correction addresses a theoretical concern with the limit as α → 1 [Vigna, 2005]. Each variant yields a PageRank problem described completely by problem 1. It is for these reasons that the PageRank problem really is (I − αP)x = (1 − α)v. 2.2.4 Historical note Astonishingly, an early paper on ranking the nodes of a social network proposed a method with surprising similarities to PageRank [Katz, 1953]. After renormalization and notation adjustment, the Katz model is (I − αWT )x = αWT e (2.16) with α = 0.5. 2.2.5 Summary of important properties We conclude our discussion of the PageRank problem with a summary of properties, not all of which have been explicitly mentioned so far: the PageRank problem is problem 1 (page 17); the PageRank vector x has unit sum (eT x = 1); the PageRank linear system is (I − αP)x = (1 − α)v; the PageRank eigensystem is Mx = x where M = αP + (1 − α)veT ; the matrix (I − αP) is a nonsingular M-matrix; the PageRank variants are strongly preferential, weakly preferential, and sink preferential; the PageRank vector x can be defined via a Neumann series when α < 1, x=∑∞ (αP)nv; n=0 the maximum eigenvalue of M is 1 and it has a unique eigenvector x; and the second largest eigenvalue of M is no larger than α [Eldén, 2004]. See Langville and Meyer [2006a] for formal derivations of these results. 2.2 ⋅ the pagerank problem 21PDF Image | Instagram Cheat Sheet
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