Recent work on modeling the web graph has dwelt on capturing the degree distributions observed on the web. Pointing out that this represents a heavy reliance on “local” properties of the web graph, we study the distribution of PageRank values on the web. Our measurements suggest that PageRank values on the web follow a power law. We then develop generative models for the web graph that explain this observation and moreover remain faithful to previously studied degree distributions. We analyze these models and compare the analysis to both snapshots from the web and to graphs generated by simulations on the new models. To our knowledge this represents the first modeling of the web that goes beyond fitting degree distributions on the web.

We propose a new random model of web graphs in which the degree of a vertex depends on its age. We characterize the degree sequence of this model and study its behaviour near the connectivity threshold.

We introduce a novel bookmark-coloring algorithm (BCA) that computes authority weights over the web pages utilizing the web hyperlink structure. The computed vector (BCV) is similar to the PageRank vector defined for a page-specific teleportation. Meanwhile, BCA is very fast, and BCV is sparse. BCA also has important algebraic properties. If several BCVs corresponding to a set of pages (called hub) are known, they can be leveraged in computing arbitrary BCV via a straightforward algebraic process and hub BCVs can be efficiently computed and encoded.

We consider the problem of computing information-theoretic functions, such as entropy, on a data stream, using sublinear space.

Our first result deals with a measure we call the entropy norm of an input stream: it is closely related to entropy but is structurally similar to the well-studied notion of frequency moments. We give a polylogarithmic-space, one-pass algorithm for estimating this norm under certain conditions on the input stream. We also prove a lower bound that rules out such an algorithm if these conditions do not hold.

Our second group of results is for estimating the empirical entropy of an input stream. We first present a sublinear-space, one-pass algorithm for this problem. For a stream of $m$ items and a given real parameter $\alpha$, our algorithm uses space $\widetilde{O}(m^{2\alpha})$ and provides an approximation of $1/\alpha$ in the worst case and $(1+\eps)$ in "most'' cases. We then present a two-pass, polylogarithmic-space, $(1+\eps)$-approximation algorithm. All our algorithms are quite simple.

We examine a number of generalized and extended versions of concentration inequalities and martingale inequalities. These inequalities are effective for analyzing processes with quite general conditions as illustrated in an example for an infinite Polya process and web graphs.