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4 1 ⋅ introduction 1.2 pagerank on a graph Although PageRank models a random surfer on the web and computes the probability of finding this surfer at any given page, the output of PageRank is a number for each page on the web related to its importance. In the previous section, we explained PageRank where a surfer sits at a current page in the web. The proper mathematical abstraction of the linked nature of the web is a graph. A graph is a set of nodes and connections. For the web, the nodes are web pages and the connections are the links. Figure 1.2 shows this relationship pictorially on a small subset of Wikipedia. This mathematical abstraction is relevant because it means that PageRank exists for any graph and not just the web graph. PageRank on a graph produces an importance score for each node, and this places PageRank amongst a class of network analysis techniques [Brandes and Erlebach, 2005] known as centrality measures or indices [Koschützki et al., 2005]. Instead of looking at a “random surfer” on the web, the non-web PageRank models a random walk on the graph. The behavior of the walk is the same as the random surfer: with probability α the walk continues along an edge of the graph and with probability 1 − α the walk jumps to a random node in the graph. Random walks are a common technique to analyze graphs, with a rich history predating PageRank. Because they apply when looking at PageRank on a general graph, the results of this thesis are not limited to web search. See section 4.8.3 for one example, but do read the background material first. 1.3 variations on the pagerank theme PageRank is a simple model for the random surfer. After hearing about this model, someone invariably approaches and asks: “Why doesn’t the surfer do . . . , instead?” Sometimes, the answer is: “So-and-so looked at that already, they found . . . ” Often, it’s: “That’s a great idea! It hasn’t been looked at yet.” The key thing to remember is that PageRank is just a model. Of course there are potential improvements to the model and there have been many proposed extensions to the model. We cover some of them in the next chapter. An important extension is the personalized PageRank model, in which the surfer does not randomly restart browsing anywhere on the web after choosing not to follow a link. Rather, the surfer in this new model restarts at one of only a few pages. If the pages relate to one person, then the resulting PageRank vector is called a personalized PageRank vector. If the pages are topically related, then the vector may be called a topic-specific PageRank vector. In either case, what the surfer does when not following links on a page determines the type of pages that are important.PDF Image | ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION
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