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Asymptotic analysis for personalized Web search. Memorandum 1884
, 2008
"... Personalized PageRank is used in Web search as an importance measure for Web documents. The goal of this paper is to characterize the tail behavior of the PageRank distribution in the Web and other complex networks characterized by power laws. To this end, we model the PageRank as a solution of a s ..."
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Cited by 17 (2 self)
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Personalized PageRank is used in Web search as an importance measure for Web documents. The goal of this paper is to characterize the tail behavior of the PageRank distribution in the Web and other complex networks characterized by power laws. To this end, we model the PageRank as a solution of a stochastic equation R d = ∑N i=1 AiRi + B, where Ri’s are distributed as R. This equation is inspired by the original definition of the PageRank. In particular, N models the number of incoming links of a page, and B stays for the user preference. Assuming that N or B are heavytailed, we employ the theory of regular variation to obtain the asymptotic behavior of R under quite general assumptions on the involved random variables. Our theoretical predictions show a good agreement with experimental data.
Spectral methods for parameterized matrix equations
 SIAM. J. Matrix Anal. Appl
"... Abstract. We apply polynomial approximation methods — known in the numerical PDEs context as spectral methods — to approximate the vectorvalued function that satisfies a linear system of equations where the matrix and the right hand side depend on a parameter. We derive both an interpolatory pseudo ..."
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Cited by 11 (3 self)
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Abstract. We apply polynomial approximation methods — known in the numerical PDEs context as spectral methods — to approximate the vectorvalued function that satisfies a linear system of equations where the matrix and the right hand side depend on a parameter. We derive both an interpolatory pseudospectral method and a residualminimizing Galerkin method, and we show how each can be interpreted as solving a truncated infinite system of equations; the difference between the two methods lies in where the truncation occurs. Using classical theory, we derive asymptotic error estimates related to the region of analyticity of the solution, and we present a practical residual error estimate. We verify the results with two numerical examples. Key words. parameterized systems, spectral methods 1. Introduction. We
Random Alpha PageRank
 INTERNET MATHEMATICS
, 2009
"... We suggest a revision to the PageRank random surfer model that considers the influence of a population of random surfers on the PageRank vector. In the revised model, each member of the population has its own teleportation parameter chosen from a probability distribution, and consequently, the rank ..."
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Cited by 6 (0 self)
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We suggest a revision to the PageRank random surfer model that considers the influence of a population of random surfers on the PageRank vector. In the revised model, each member of the population has its own teleportation parameter chosen from a probability distribution, and consequently, the ranking vector is random. We propose three algorithms for computing the statistics of the random ranking vector based respectively on (i) random sampling, (ii) paths along the links of the underlying graph, and (iii) quadrature formulas. We find that the expectation of the random ranking vector produces similar rankings to its deterministic analogue, but the standard deviation gives uncorrelated information (under a Kendalltau metric) with myriad potential uses. We examine applications of this model to web spam.
Moment based estimation of stochastic Kronecker graph parameters
, 2011
"... Stochastic Kronecker graphs supply a parsimonious model for large sparse real world graphs. They can specify the distribution of a large random graph using only three or four parameters. Those parameters have however proved difficult to choose in specific applications. This article looks at method o ..."
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Cited by 4 (0 self)
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Stochastic Kronecker graphs supply a parsimonious model for large sparse real world graphs. They can specify the distribution of a large random graph using only three or four parameters. Those parameters have however proved difficult to choose in specific applications. This article looks at method of moments estimators that are computationally much simpler than maximum likelihood. The estimators are fast and in our examples, they typically yield Kronecker parameters with expected feature counts closer to a given graph than we get from KronFit. The improvement was especially prominent for the number of triangles in the graph. 1
Stochastic Kronecker Graph Parameters
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Sandia National Labs
"... PageRank computes the importance of each node in a directed graph under a random surfer model governed by a teleportation parameter. Commonly denoted alpha, this parameter models the probability of following an edge inside the graph or, when the graph comes from a network of web pages and links, c ..."
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PageRank computes the importance of each node in a directed graph under a random surfer model governed by a teleportation parameter. Commonly denoted alpha, this parameter models the probability of following an edge inside the graph or, when the graph comes from a network of web pages and links, clicking a link on a web page. We empirically measure the teleportation parameter based on browser toolbar logs and a click trail analysis. For a particular user or machine, such analysis produces a value of alpha. We find that these values nicely fit a Beta distribution with mean edgefollowing probability between 0.3 and 0.7, depending on the site. Using these distributions, we compute PageRank scores where PageRank is computed with respect to a distribution as the teleportation parameter, rather than a constant teleportation parameter. These new metrics are evaluated on the graph of pages in Wikipedia.
Tracking the random surfer: Empirically measured . . .
, 2010
"... PageRank computes the importance of each node in a directed graph under a random surfer model governed by a teleportation parameter. Commonly denoted alpha, this parameter models the probability of following an edge inside the graph or, when the graph comes from a network of web pages and links, cli ..."
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PageRank computes the importance of each node in a directed graph under a random surfer model governed by a teleportation parameter. Commonly denoted alpha, this parameter models the probability of following an edge inside the graph or, when the graph comes from a network of web pages and links, clicking a link on a web page. We empirically measure the teleportation parameter based on browser toolbar logs and a click trail analysis. For a particular user or machine, such analysis produces a value of alpha. We find that these values nicely fit a Beta distribution with mean edgefollowing probability between 0.3 and 0.7, depending on the site. Using these distributions, we compute PageRank scores where PageRank is computed with respect to a distribution as the teleportation parameter, rather than a constant teleportation parameter. These new metrics are evaluated on the graph of pages in Wikipedia.
ITERATIVE GRAPH COMPUTATION IN THE BIG DATA ERA
, 2015
"... Iterative graph computation is a key component in many realworld applications, as the graph data model naturally captures complex relationships between entities. The big data era has seen the rise of several new challenges to this classic computation model. In this dissertation we describe three p ..."
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Iterative graph computation is a key component in many realworld applications, as the graph data model naturally captures complex relationships between entities. The big data era has seen the rise of several new challenges to this classic computation model. In this dissertation we describe three projects that address different aspects of these challenges. First, because of the increasing volume of data, it is increasingly important to scale iterative graph computation to large graphs. We observe that an important class of graph applications performing little computation per vertex scales poorly when running on multiple cores. These computationally light applications are limited by memory access rates, and cannot fully utilize the benefits of multiple cores. We propose a new blockoriented computation model which creates two levels of iterative computation. On each processor, a small block of highly connected vertices is iterated locally, while the blocks are updated iteratively at the global level. We show that blockoriented execution reduces the communicationtocomputation ratio and significantly improves the perfor
EdgeWeighted Personalized PageRank: Breaking A DecadeOld Performance Barrier
"... ABSTRACT Personalized PageRank is a standard tool for finding vertices in a graph that are most relevant to a query or user. To personalize PageRank, one adjusts node weights or edge weights that determine teleport probabilities and transition probabilities in a random surfer model. There are many ..."
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ABSTRACT Personalized PageRank is a standard tool for finding vertices in a graph that are most relevant to a query or user. To personalize PageRank, one adjusts node weights or edge weights that determine teleport probabilities and transition probabilities in a random surfer model. There are many fast methods to approximate PageRank when the node weights are personalized; however, personalization based on edge weights has been an open problem since the dawn of personalized PageRank over a decade ago. In this paper, we describe the first fast algorithm for computing PageRank on general graphs when the edge weights are personalized. Our method, which is based on model reduction, outperforms existing methods by nearly five orders of magnitude. This huge performance gain over previous work allows us for the very first time to solve learningtorank problems for edge weight personalization at interactive speeds, a goal that had not previously been achievable for this class of problems.