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8 1 ⋅ introduction network alignment Singh et al. [2007] introduce the IsoRank algo- rithm7 to identify functionally similar nodes between two different graphs. The random surfer analogy does not directly apply in this case, and the “Google Juice” formulation is more appropriate. Instead of working on the graphs in the original problem, IsoRank uses a com- bined graph based on all pairs of vertices. The PageRank of this new graph helps identify which nodes are potential mates. Section 5.6.5 discusses this application further. 1.5 contributions At this point, we have motivated the PageRank problem from its original use as a description of a random web surfer and shown how the same model yields many different applications. The aspect of PageRank we address in this thesis is the sensitivity of the PageRank vector with respect to the parameter α. Some aspects of the sensitivity of PageRank were previously understood. For example, as the value α gets close to 1, the random surfer is typically following links in the graph. The impact of the graph is exaggerated in this case. Also, as α gets close to 1, the vector may change rapidly. We’ll review the relevant background material for sensitivity in chapter 2. A simple approach to investigating the sensitivity of the PageRank vector is to look at the derivative with respect to α, which we explore in chapter 3.8 Algorithms to compute the derivative of PageRank were already known, but we propose a new algorithm that can use any existing PageRank solver without modification. The results from our algorithm (algorithm 1) on the pages from Wikipedia are shown in table 1.2. 7 There is another algorithm called IsoRank based on isotonic regres- sion [Zheng et al., 2008]. α = 0.50 United States C:Living people United Kingdom Race in the US. Census C:Ctgs. by country France England Canada Germany World War II List of sovereign states α = 0.85 C:Main topic classif. C:Contents C:Fundamental C:Wikipedia admin. C:Ctgs. by topic C:Society Por:List of portals C:Articles Por:Contents/Portals C:Ctgs. by location C:Categories α=0.99 C:Main topic classif. C:Contents C:Fundamental C:Wikipedia admin. C:Ctgs. by topic C:Society Por:List of portals C:Articles Por:Contents/Portals C:Ctgs. by location C:Ctgs. by country 8 Although PageRank was described as a random surfer model, it also has a nice expression as a function of the parameter α. Our investi- gation is of the derivative of this function with respect to α. Table 1.2 – Pages in Wikipedia with thelargestderivative. Pagesin Wikipedia with the largest deriva- tive, by value, not by magnitude.PDF Image | Instagram Cheat Sheet
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