Asymptotic analysis for personalized Web search

Volkovich, Yana; Litvak, Nelly

doi:10.1017/s0001867800004201

Cited by 42 publications

(97 citation statements)

References 16 publications

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“…x → ∞, for some constant 0 < H < ∞, when N has a power-law distribution [20,34]. Theorem 3.3 in this paper shows that the same type of representation holds for the family of inhomogeneous random digraphs studied here, provided the in-degree and out-degree of the same vertex are asymptotically independent.…”

Section: Introductionsupporting

confidence: 52%

PageRank on inhomogeneous random digraphs

Lee

Olvera–Cravioto

2020

Stochastic Processes and their Applications

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We study the typical behavior of a generalized version of Google's PageRank algorithm on a large family of inhomogeneous random digraphs. This family includes as special cases directed versions of classical models such as the Erdős-Rényi model, the Chung-Lu model, the Poissonian random graph and the generalized random graph, and is suitable for modeling scale-free directed complex networks where the number of neighbors a vertex has is related to its attributes. In particular, we show that the rank of a randomly chosen node in a graph from this family converges weakly to the attracting endogenous solution to the stochastic fixed-point equation. and independent of (N , Q); D = denotes equality in distribution. This result can then be used to provide further evidence of the powerlaw behavior of PageRank on scale-free graphs.

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Section: Introductionsupporting

confidence: 52%

PageRank on inhomogeneous random digraphs

Lee

Olvera–Cravioto

2020

Stochastic Processes and their Applications

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“…One can generalize further, e.g., allow the probability c to be random as well. The literature [20,40,42,49] usually studies the following graph-normalized equation:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Some progress has been made in [9] for the average PageRank in a Preferential Attachment model. In a series of papers [43,49], the result was proved provided that PageRank satisfies a branching-type of recursion. Then, in fact, a network is modeled as a branching process, with independent labels representing out-degrees.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Local weak convergence for PageRank

Garavaglia

Hofstad

Litvak³

2020

Ann. Appl. Probab.

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PageRank is a well-known algorithm for measuring centrality in networks. It was originally proposed by Google for ranking pages in the World-Wide Web. One of the intriguing empirical properties of PageRank is the so-called 'power-law hypothesis': in a scale-free network the PageRank scores follow a power law with the same exponent as the (in-)degrees. Up to date, this hypothesis has been confirmed empirically and in several specific random graphs models. In contrast, this paper does not focus on one random graph model but investigates the existence of an asymptotic PageRank distribution, when the graph size goes to infinity, using local weak convergence. This may help to identify general network structures in which the power-law hypothesis holds. We start from the definition of local weak convergence for sequences of (random) undirected graphs, and extend this notion to directed graphs. To this end, we define an exploration process in the directed setting that keeps track of in-and out-degrees of vertices. Then we use this to prove the existence of an asymptotic PageRank distribution. As a result, the limiting distribution of PageRank can be computed directly as a function of the limiting object. We apply our results to the directed configuration model and continuous-time branching processes trees, as well as preferential attachment models.

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“…This result covers the case where the weights {C i } are nonnegative and either the limiting in-degree N or the limiting personalization value Q have a regularly varying distribution, which in turn implies the regular variation of R. Then, we deduce the asymptotics of R * using some results for weighted random sums with heavy-tailed summands. The corresponding theorems can be found in [56,68]. We point out that Theorem 6.4 holds for any bi-degree sequence such that the out-degrees have finite 2 + κ moments, and the regular variation in this section is only used to prove the "power law hypothesis" on the scale-free DCM.…”

Section: Asymptotic Behavior Of the Limitmentioning

confidence: 94%

Generalized PageRank on directed configuration networks

Chen

Litvak

Olvera–Cravioto

2016

Random Struct. Alg.

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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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Asymptotic analysis for personalized Web search

Cited by 42 publications

References 16 publications

PageRank on inhomogeneous random digraphs

PageRank on inhomogeneous random digraphs

Local weak convergence for PageRank

Generalized PageRank on directed configuration networks

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