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64 4 ⋅ random alpha pagerank Empirically measured browsing patterns on the web show that individual users, unsurprisingly, have different browsing behavior [Huberman et al., 1998; White and Drucker, 2007]. We also confirm this result in section 4.5. If we assume that all users have their own probability αi of teleporting, then the PageRank model suggests we should set α = N1 ∑Ni=1 αi , i.e. the mean of these values. More generally, if A is a random variable with a density function encoding the distribution of teleportation parameters amongst multiple (per- haps infinite) surfers, then the PageRank model suggests α = E [A], where E [⋅] is the expectation operator. The flaw in PageRank is that using α = E [A] still does not yield the correct PageRank vector in light of the surfer values αi . We will justify this statement shortly; intuitively it arises because a single value of α condenses all surfers into a single über-surfer. Instead, we propose to give a small vote to the PageRank vector x(αi) corresponding to each random surfer and create a global metric that amalgamates this information. In other words, we want to examine the random surfer model with “α = A,” where A is a random variable modeling the users’ individual behaviors. Figure 4.1 gives a pictorial view of this change. If A is a random variable, then the PageRank vector x(A) is a random vector, and we can synthesize a new ranking measure from its statistics. We call this measure Random Alpha Pagerank (RAPr), it is pronounced “wrapper.” An earlier work, Constantine and Gleich [2007], introduced a means of handling multiple surfers in PageRank. This chapter extends those ideas by clarifying the presentation, expanding the computational algorithms, and compiling additional results. In particular, the previous paper used the poly- nomial chaos approach to investigate the behavior of multiple surfers algo- rithmically. In Constantine et al. [2009], we showed that the polynomial chaos and quadrature methods are equivalent in the case of PageRank. The → x(E [A]) (a) The PageRank Model Figure 4.1 – Differences between PageRank and the Random Alpha PageRank model. The PageRank model assumes a single random surfer representing an expected user. Our model assumes that each surfer is unique with a different value of α, which we represent as a random variable A. If the function x(⋅) gives the PageRank vector for a deterministic or random α or A, respectively, we then compute the expected PageRank given the distribution for A. → E [x(A)] ⋯ (b) Our random α PageRank modelPDF Image | Instagram Cheat Sheet
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