In-Degree and PageRank: Why Do They Follow Similar Power Laws?

Litvak, Nelly; Scheinhardt, Willem R.W.; Volkovich, Yana

doi:10.1080/15427951.2007.10129293

Cited by 53 publications

(70 citation statements)

References 25 publications

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“…Also, by letting N be a Poisson random variable and fixing C i ≡ 1 and Q ≡ 1, (1.1) becomes the recursion that the number of customers in a busy period of an M/G/1 queue satisfies. Recursion (1.2) and its connection to the busy period when the weights D i are equal to a deterministic constant was exploited in [26].…”

Section: Related Processesmentioning

confidence: 99%

Information ranking and power laws on trees

Jelenković

Olvera–Cravioto

2010

Adv. Appl. Probab.

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In this paper we consider the stochastic analysis of information ranking algorithms of large interconnected data sets, e.g. Google's PageRank algorithm for ranking pages on the World Wide Web. The stochastic formulation of the problem results in an equation of the form Rand {C, C i } i≥1 are independent nonnegative random variables, the {C, C i } i≥1 are identically distributed, and the {R i } i≥1 are independent copies of R; ' d =' stands for equality in distribution. We study the asymptotic properties of the distribution of R that, in the context of PageRank, represents the frequencies of highly ranked pages. The preceding equation is interesting in its own right since it belongs to a more general class of weighted branching processes that have been found to be useful in the analysis of many other algorithms. Our first main result shows that if E N E[C α ] = 1, α > 0, and Q, N satisfy additional moment conditions, then R has a power law distribution of index α. This result is obtained using a new approach based on an extension of Goldie's (1991) implicit renewal theorem. Furthermore, when N is regularly varying of index α > 1, E N E[C α ] < 1, and Q, C have higher moments than α, then the distributions of R and N are tail equivalent. The latter result is derived via a novel sample path large deviation method for recursive random sums. Similarly, we characterize the situation when the distribution of R is determined by the tail of Q. The preceding approaches may be of independent interest, as they can be used for analyzing other functionals on trees. We also briefly discuss the engineering implications of our results.

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Section: Related Processesmentioning

confidence: 99%

Information ranking and power laws on trees

Jelenković

Olvera–Cravioto

2010

Adv. Appl. Probab.

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“…In this thesis we characterize the power law behavior of the PageRank using the approach that we developed in our works [74,75,111,112]. In the remainder of the section we briefly describe the main ideas of the approach.…”

Section: In-degree and Pagerankmentioning

confidence: 99%

“…Using probabilistic techniques from Jessen and Mikosch [59], we defined asymptotical properties of R (k) . Finally, we combine techniques from [74,75] and [112] in a generalization of our model for the case of non-uniform PageRank. Thus, in Chapter 2 we define asymptotics of PageRank after each iteration using probabilistic approach as in [112], and in Chapter 3 we justify the power law behavior of the PageRank using an analytical approach similar to [74].…”

Section: In-degree and Pagerankmentioning

confidence: 99%

“…In [26] Capocci et al, and in [25] Buriol et al analyze in-and out-degrees distribution, and distribution of the PageRank for the Wikipedia samples, and also confirm the similarity in the power law behavior of the in-degree and the PageRank. In our works [74,112,113,111,115] this problem was studied for the different Web and Wikipedia samples. We refer for numerical results to Chapter 4.…”

Section: Power Lawsmentioning

confidence: 99%

“…We start our analysis with simplified model in [74,75], where we assume that all pages have constant out-degree, that equals average in-and out-degree. Then, inspired by formula (1.1), the stochastic equation for the PageRank is as follows:…”

Section: In-degree and Pagerankmentioning

confidence: 99%

See 2 more Smart Citations

Stochastic analysis of web page ranking

Volkovich

Self Cite

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Generalized PageRank on directed configuration networks

Chen

Litvak

Olvera–Cravioto

2016

Random Struct. Alg.

104

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This paper studies the distribution of a family of rankings, which includes Google's PageRank, on a directed configuration model. In particular, it is shown that the distribution of the rank of a randomly chosen node in the graph converges in distribution to a finite random variable scriptR* that can be written as a linear combination of i.i.d. copies of the attracting endogenous solution to a stochastic fixed‐point equation of the form R=scriptD∑i=1NscriptCiscriptRi+Q, where (Q,N,{scriptCi}) is a real‐valued vector with N∈{0,1,2,…}, P(|Q|>0)>0, and the {scriptRi} are i.i.d. copies of scriptR, independent of (Q,N,{scriptCi}). Moreover, we provide precise asymptotics for the limit scriptR*, which when the in‐degree distribution in the directed configuration model has a power law imply a power law distribution for scriptR* with the same exponent. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 237–274, 2017

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In-Degree and PageRank: Why Do They Follow Similar Power Laws?

Cited by 53 publications

References 25 publications

Information ranking and power laws on trees

Information ranking and power laws on trees

Stochastic analysis of web page ranking

Generalized PageRank on directed configuration networks

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