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Katz [1953] used α = 0.5 in a model closely related to PageRank.8 More recently, two papers suggested α = 0.5 for PageRank. Back in sec- tion 2.6, we discussed Avrachenkov et al. [2007]. They argue that α = 1/2 is the right choice.9 The second paper applies the random surfer model to a graph of literature citations [Chen et al., 2007]. They claim that citation behavior on literature networks contain very short citation paths of average length 2 based on co- citation analysis. This analysis then suggests α = 0.5. 4.3.2 Usage logs and behavior analysis Huberman et al. [1998] studied the behavior of web surfers, even before the original paper on PageRank, and suggested a Markov model for surfing behavior. In contrast with the Brin and Page random surfer, Huberman et al. [1998] empirically measure the probability that surfers follow paths of length l and then compute n = f Pln (4.5) ðl0 for the expected number of surfers on each page after l transitions. They found that fl, the probability of following a path of length l, is approximately an inverse Gaussian; see figure 4.3. This model is strongly related to the path damping models discussed next and in section 4.4.3. An earlier study showed that the average path length of users visiting a site decayed quickly, but did not match the decay to a distribution [Catledge and Pitkow, 1995]. Both of these studies focused on the browsing behavior at a single site and not across the web in general. Subsequently, many papers suggest measuring surfer behavior from usage logs to improve local site search [Wang, 2002; Xue et al., 2003; Farahat et al., 2006]. In the context of web search, a recent study identified two types of surfers: navigators and explorers [White and Drucker, 2007]. Navigators proceed roughly linearly whereas explorers frequently branch their browsing patterns and revisit previous pages before going to new links. The first behavior cor- responds to a larger value of α than the latter. This paper also contains an extensive review of relevant literature. None of these studies directly measures α or the distribution A. A recent patent from Yahoo! [Berkhin et al., 2008] describes a modification of the PageRank equations to build a “user-sensitive PageRank” system by incorporating observed page transitions and user segment modeling.10 The key idea in the patent is to modify the Markov chain transition probabilities to give higher weight to transitions observed and change the teleportation 4.3 ⋅ related work 69 8 Katz’s model was (I − αWT )k = αWT e for an adjacency matrix W. 9 The authors employ graph the- oretic techniques to examine the mass of PageRank in the largest strong component of the underlying graph. As α → 1, the mass in this strong component goes to zero if there are other strong components reachable from the largest strong component. This situation is un- desirable because many important pages exist in the largest strong component. They argue that, conse- quently, α should be far from 1, and they suggest α = 1/2 because α = 0 gives an equally useless ranking. 0 5 10 15 20 L Figure 4.3 – Inverse Gaussian density. The inverse Gaussian distribution has a probability density √2 ρ(L) = λ exp[−λ(L−μ) ] 2πL3 2μ2 L supported on L = (0, ∞). The density plotted here uses μ = 5, λ = 10. 10 A user segment is a group of users related by a common fac- tor. Age, sex, and interests are all possibilities. densityPDF Image | Instagram Cheat Sheet
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