Asymptotic analysis for personalized Web search

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 $R\stackrel{d}{=}\sum_{i=1}^NA_iR_i+B$, where $R_i$'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 heavy-tailed, 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

Degree-degree correlations in random graphs with heavy-tailed degrees

Mixing patterns in large self-organizing networks, such as the Internet, the World Wide Web, social and biological networks are often characterized by degree-degree {dependencies} between neighbouring nodes. One of the problems with the commonly used Pearson's correlation coefficient (termed as the assortativity coefficient) is that {in disassortative networks its magnitude decreases} with the network size. This makes it impossible to compare mixing patterns, for example, in two web crawls of different size. We start with a simple model of two heavy-tailed highly correlated random variable $X$ and $Y$, and show that the sample correlation coefficient converges in distribution either to a proper random variable on $[-1,1]$, or to zero, and if $X,Y\ge 0$ then the limit is non-negative. We next show that it is non-negative in the large graph limit when the degree distribution has an infinite third moment. We consider the alternative degree-degree dependency measure, based on the Spearman's rho, and prove that it converges to an appropriate limit under very general conditions. We verify that these conditions hold in common network models, such as configuration model and Preferential Attachment model. We conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns

Moment based estimation of stochastic Kronecker graph parameters

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.Comment: 22 pages, 4 figure

FAST AND OPTIMAL SOLUTION ALGORITHMS FOR PARAMETERIZED PARTIAL DIFFERENTIAL EQUATIONS

This dissertation presents efficient and optimal numerical algorithms for the solution of parameterized partial differential equations (PDEs) in the context of stochastic Galerkin discretization. The stochastic Galerkin method often leads to a large coupled system of algebraic equations, whose solution is computationally expensive to compute using traditional solvers. For efficient computation of such solutions, we present low-rank iterative solvers, which compute low-rank approximations to the solutions of those systems while not losing much accuracy. We first introduce a low-rank iterative solver for linear systems obtained from the stochastic Galerkin discretization of linear elliptic parameterized PDEs. Then we present a low-rank nonlinear iterative solver for efficiently computing approximate solutions of nonlinear parameterized PDEs, the incompressible Navier–Stokes equations. Along with the computational issue, the stochastic Galerkin method suffers from an optimality issue. The method, in general, does not minimize the solution error in any measure. To address this issue, we present an optimal projection method, a least-squares Petrov–Galerkin (LSPG) method. The proposed method is optimal in the sense that it produces the solution that minimizes a weighted l2-norm of the solution error over all solutions in a given finite-dimensional subspace. The method can be adapted to minimize the solution error in different weighted l2-norms by simply choosing a specific weighting function within the least-squares formulation

Structure-oriented prediction in complex networks

Complex systems are extremely hard to predict due to its highly nonlinear interactions and rich emergent properties. Thanks to the rapid development of network science, our understanding of the structure of real complex systems and the dynamics on them has been remarkably deepened, which meanwhile largely stimulates the growth of effective prediction approaches on these systems. In this article, we aim to review different network-related prediction problems, summarize and classify relevant prediction methods, analyze their advantages and disadvantages, and point out the forefront as well as critical challenges of the field

Stochastic analysis of web page ranking

Today, the study of the World Wide Web is one of the most challenging subjects. In this work we consider the Web from a probabilistic point of view. We analyze the relations between various characteristics of the Web. In particular, we are interested in the Web properties that affect the Web page ranking, which is a measure of popularity and importance of a page in the Web. Mainly we restrict our attention on two widely-used algorithms for ranking: the number of references on a page (indegree), and Google’s PageRank. For the majority of self-organizing networks, such as the Web and the Wikipedia, the in-degree and the PageRank are observed to follow power laws. In this thesis we present a new methodology for analyzing the probabilistic behavior of the PageRank distribution and the dependence between various power law parameters of the Web. Our approach is based on the techniques from the theory of regular variations and the extreme value theory. We start Chapter 2 with models for distributions of the number of incoming (indegree) and outgoing (out-degree) links of a page. Next, we define the PageRank as a solution of a stochastic equation R d= PN 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 in-degree of a page, and B stays for the user preference. We use a probabilistic approach to show that the equation has a unique non-trivial solution with fixed finite mean. Our analysis based on a recurrent stochastic model for the power iteration algorithm commonly used in PageRank computations. Further, we obtain that the PageRank asymptotics after each iteration are determined by the asymptotics of the random variable with the heaviest tail among N and B. If the tails of N and B are equally heavy, then in fact we get the sum of two asymptotic expressions. We predict the tail behavior of the limiting distribution of the PageRank as a convergence of the results for iterations. To prove the predicted behavior we use another techniques in Chapter 3. In Chapter 3 we define the tail behavior for the models of the in-degree and the PageRank distribution using Laplace-Stieltjes transforms and the Tauberian theorem. We derive the equation for the Laplace-Stieltjes transforms, that corresponds to the general stochastic equation, and obtain our main result that establishes the tail behavior of the solution of the stochastic equation. In Chapter 4 we perform a number of experiments on the Web and the Wikipedia data sets, and on preferential attachment graphs in order to justify the results obtained in Chapters 2 and 3. The numerical results show a good agreement with our stochastic model for the PageRank distribution. Moreover, in Section 4.1 we also address the problem of evaluating power laws in the real data sets. We define several state of the art techniques from the statistical analysis of heavy tails, and we provide empirical evidence on the asymptotic similarity between in-degree and PageRank. Inspired by the minor effect of the out-degree distribution on the asymptotics of the PageRank, in Section 4.4 we introduce a new ranking scheme, called PAR, which combines features of HITS and PageRank ranking schemes. In Chapter 5 we examine the dependence structure in the power law graphs. First, we analytically define the tail dependencies between in-degree and PageRank of a one particular page by using the stochastic equation of the PageRank. We formally establish the relative importance of the two main factors for high ranking: large in-degree and a high rank of one of the ancestors. Second, we compute the angular measures for in-degrees, out-degrees and PageRank scores in three large data sets. The analysis of extremal dependence leads us to propose a new rank correlation measure which is particularly plausible for power law data. Finally, in Chapter 6 we apply the new rank correlation measure from Chapter 5 to various problems of rank aggregation. From numerical results we conclude that methods that are defined by the angular measure can provide good precision for the top nodes in large data sets, however they can fail in a small data sets