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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. 159PDF Image | Instagram Cheat Sheet
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