The spectral and Jordan structures of the web hyperlink matrix $G(c) = cG + (1-c)ev^T$ have been analyzed when $G$ is the basic (stochastic) Google matrix, $c$ is a real parameter such that $0<c<1$, $v$ is a nonnegative probability vector, and $e$ is the all-ones vector. Typical studies have relied heavily on special properties of nonnegative, positive, and stochastic matrices. There is a unique nonnegative vector $y(c)$ such that $% y(c)^TG(c)=y(c)^T$ and $y(c)^T e=1$. This PageRank vector $y(c)$ can be computed effectively by the power method.

We consider a square complex matrix $A$ and nonzero complex vectors $x$ and $% v$ such that $Ax=\lambda x$ and $v^*x=1$. We use standard matrix analytic tools to determine the eigenvalues, the Jordan blocks, and a distinguished left $\lambda$-eigenvector of $A(c)=cA + (1-c)\lambda xv^*$ as a function of a complex variable $c$. If $\lambda$ is a semisimple eigenvalue of $A$, there is a uniquely determined projection $P$ such that $\lim_{c \rightarrow 1}y(c)=Pv$ for every $v$ such that $v^*x=1$; this limit may fail to exist for some $v$ if $\lambda $ is not semisimple. As a special case of our results, we obtain a complex analog of PageRank for the web hyperlink matrix $G(c)$ with a complex parameter $c$. We study regularity, limits, expansions, and conditioning of $% y(c)$, and we propose a complex extrapolation algorithm that may provide an efficient way to compute PageRank.

This paper describes Armil, a meta-search engine that groups the web snippets returned by auxiliary search engines into disjoint labeled clusters. The cluster labels generated by Armil provide the user with a compact guide to assessing the relevance of each cluster to his/her information need. Striking the right balance between running time and cluster well-formedness was a key point in the design of our system. Both the clustering and the labeling tasks are performed on the fly by processing only the snippets provided by the auxiliary search engines, and they use no external sources of knowledge. Clustering is performed by means of a fast version of the furthest-point-first algorithm for metric $k$-center clustering. Cluster labeling is achieved by combining intra-cluster and inter-cluster term extraction based on a variant of the information gain measure. We have tested the clustering effectiveness of Armil against Vivisimo, the de facto industrial standard in web snippet clustering, using as benchmark a comprehensive set of snippets obtained from the Open Directory Project hierarchy. According to two widely accepted ``external'' metrics of clustering quality, Armil achieves better performance levels by $10%$. We also report the results of a thorough user evaluation of both the clustering and the cluster labeling algorithms. On a standard desktop PC (AMD Athlon 1-Ghz Clock with 750 Mbytes RAM), Armil performs clustering and labeling altogether of up to 200 snippets in less than one second.

This paper introduces a family of link-based ranking algorithms that propagate page importance through links. The algorithms include a damping function that decreases with distance, thus a direct link implies greater endorsement than a link via a longer path. PageRank is the most widely known ranking function of this family.

The main objective of this paper is to determine whether this family of ranking techniques is of some interest per se and how different choices for the damping function affect rank quality and convergence speed. Even though our results suggest that PageRank can be approximated with other, more simple forms of rankings that may be computed more efficiently, our focus is more speculative in nature, given that it aims at separating the kernel of PageRank, that is, link-based importance propagation, from the way propagation decays over paths.

We focus on three damping functions that have, respectively, linear, exponential, and hyperbolic decay on the lengths of the paths. The exponential decay corresponds to PageRank, and the other functions are new. The work we carry includes algorithms, analysis, comparisons, and experiments that study their behavior under different parameters in real web graph data.

Amongst other results, we show how to calculate a linear approximation that induces a page ordering that is almost identical to PageRank's using a fixed number of iterations. Comparisons were made using Kendall's $\tau$ on large domain datasets.

Recently, there has been a surge of research activity in the area of link analysis ranking, where hyperlink structures are used to determine the relative authority of webpages. One of the seminal works in this area is that of Kleinberg [Kleinberg 98], who proposed the HITS algorithm. In this paper, we undertake a theoretical analysis of the properties of the HITS algorithm on a broad class of random graphs. Working within the framework of Borodin et al. [Borodin et al. 05], we prove that, under some assumptions, on this class (a) the HITS algorithm is stable with high probability and (b) the HITS algorithm is similar to the InDegree heuristic that assigns to each node weight proportional to the number of incoming links. We demonstrate that our results go through for the case that the expected in-degrees of the graph follow a power law distribution. We also study experimentally the similarity between HITS and InDegree, and we investigate the general conditions under which the two algorithms are\linebreak similar.

The behavior of inlink and outlink distributions appears to be one of the most studied properties of the web structure. The literature agrees that the inlink distribution follows a power law, but no such agreement exists for the outlink distribution. Accurate observations show that in the low-degree region the link distribution fails to fit a power law with a discrepancy larger for outlinks than for inlinks. Moreover, a power law, as well as any continuous function, does not fit the scattered behavior shared by both the link distributions for large-degree values. The linking model we consider here is a mixed one, based on both the preferential attachment strategy and the uniform attachment strategy. A new approximation technique is devised to detect the parameters of the steady state solution that describe a real data set. A stochastic technique is suggested to describe the scattering of the data. With these techniques the model appears to be well suited for describing both inlink and outlink distributions. The experimentation on subsets of the World Wide Web and of Wikipedia shows that our approach produces an approximation more adequate than the power law. This approximation suggests that the two attachment strategies play a different role in the inlink and the outlink cases.