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7CONCLUSION That it should come to this! Hamlet At the outset of this thesis, we embarked on an exploration of α. Recall the setting. PageRank is a technique to rank the nodes of any graph by their importance. All too often, people introduce PageRank with the idea that “important nodes” connect to other ”important nodes.” Such a definition suggests an importance vector x that satisfies Px = x, where P is a column stochastic matrix describing a flow of importance. Typ- ically, importance flows uniformly along the edges of the graph. But, these introductions ignore α, and it is α that distinguishes PageRank! PageRank needs α because Px = x creates a model where the importance scores, or ranks of the nodes, are not well defined. Instead, better definitions of PageRank begin with α. We suggest a few possibilities. First, “important pages” probably connect to other “important pages.” The value of α arises immediately to quantify the term probably in this definition. Another possibility is to begin outright with α: aparameterbetween0and1toreducetheflowofinfluenceinagraph. Or, perhaps α: theprobabilitythatarandomsurferinthewebfollowsalink. This last definition ties PageRank too closely to web search, however. Begin- ning with all of these definitions quickly leads to PageRank itself: (I − αP)x = (1 − α)v. In the first two definitions, v is any intrinsic measure of node importance. Without other information, a uniform choice is entirely appropriate as the choice of intrinsic importance. In all definitions though, importance flows as αP—and that is the key to PageRank. Given any definition of PageRank, it must involve α, and it is surprising how little attention α received in early discussions. The first five years of PageRank research largely ignored the impact of α. That finally changed. It almost seemed as if α were “in the air,” to borrow a phrase from my late adviser. A stream of papers between 2003 and 2007 attacked α directly. These attacks established that PageRank is a rational function α [Boldi et al., 2005], examined the parametric structure of the so-called Google matrix definition of PageRank [Serra-Capizzano, 2005], and even proposed a theoretically 143PDF Image | ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION
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