In the study of the spectra of power law graphs, there are basically two competing approaches. One is to prove analogues of Wigner's semicircle law while the other predicts that the eigenvalues follow a power law distributions. Although the semicircle law and the power law have nothing in common, we will show that both approaches are essentially correct if one considers the appropriate matrices. We will show that (under certain conditions) the eigenvalues of the (normalized) Laplacian of a random power law graph follow the semicircle law while the spectrum of the adjacency matrix of a power law graph obeys the power law. Our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants. Of interest are a number of (new) values for the exponent $\beta$ where phase transitions for eigenvalue distributions occur. The spectrum distributions have direct implications to numerous graph algorithms such as randomized algorithms that involve rapidly mixing Markov chains, for example.

We anticipate that future web search techniques will exploit changes in web structure and content. As a first step in this direction, we examine the problem of integrating observed changes in link structure into static hyperlink-based ranking computations

We present a very efficient algorithm to incrementally compute good approximations to Google's PageRank [Brin and Page 98], as links evolve. Our experiments reveal that this algorithm is both fast and yields excellent approximations to PageRank, even in light of large changes to the link structure

Our algorithm derives intuition and partial justification from a rigorous sensitivity analysis of Markov chains. Consider a regular Markov chain with stationary probability $\pi$, and suppose the transition probability into a state $j$ is increased. We prove that this can only cause

$\pi_j$ to incease---adding a link to a site can only cause the stationary probability of the target site to increase;

the rank of $j$ to improve---if the states are ordered according to their stationary probabilities, then adding a link to a site can only cause the rank of the target site to improve.

This analysis formalizes why the intuition that drives Google never fails.

In this paper, we introduce and analyze a new, dynamic generative user model to explain the behavior of file size distributions. Our Recursive Forest File model combines multiplicative models that generate lognormal distributions with recent work on random graph models for the web. Unlike similar previous work, our Recursive Forest File model allows new files to be created and old files to be deleted over time, and our analysis covers problematic issues such as correlation among file sizes. Moreover, our model allows natural variations where files that are copied or modified are more likely to be copied or modified subsequently.

Previous empirical work suggests that file sizes tend to have a lognormal body but a Pareto tail. The Recursive Forest File model explains this behavior, yielding a double Pareto distribution, which has a Pareto tail but close to a lognormal body. We believe the Recursive Forest model may be useful for describing other power law phenomena in computer systems as well as other fields.

This paper serves as a companion or extension to the "Inside PageRank'' paper by Bianchini et al. It is a comprehensive survey of all issues associated with PageRank, covering the basic PageRank model, available and recommended solution methods, storage issues, existence, uniqueness, and convergence properties, possible alterations to the basic model, suggested alternatives to the traditional solution methods, sensitivity and conditioning, and finally the updating problem. We introduce a few new results, provide an extensive reference list, and speculate about exciting areas of future research.