Community Detection on Evolving Graphs

Clustering is a fundamental step in many information-retrieval and data-mining applications. Detecting clusters in graphs is also a key tool for finding the community structure in social and behavioral networks. In many of these applications, the input graph evolves over time in a continual and decentralized manner, and, to maintain a good clustering, the clustering algorithm needs to repeatedly probe the graph. Furthermore, there are often limitations on the frequency of such probes, either imposed explicitly by the online platform (e.g., in the case of crawling proprietary social networks like twitter) or implicitly because of resource limitations (e.g., in the case of crawling the web). In this paper, we study a model of clustering on evolving graphs that captures this aspect of the problem. Our model is based on the classical stochastic block model, which has been used to assess rigorously the quality of various static clustering methods. In our model, the algorithm is supposed to reconstruct the planted clustering, given the ability to query for small pieces of local information about the graph, at a limited rate. We design and analyze clustering algorithms that work in this model, and show asymptotically tight upper and lower bounds on their accuracy. Finally, we perform simulations, which demonstrate that our main asymptotic results hold true also in practice

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