Luca Pretto

A Theoretical Study of a Generalized Version of Kleinberg's HITS Algorithm

Kleinberg’s HITS algorithm (Kleinberg 1999), which was originally developed in a Web context, tries to infer the authoritativeness of a Web page in relation to a specific query using the structure of a subgraph of the Web graph, which is obtained considering this specific query. Recent applications of this algorithm in contexts far removed from that of Web searching (Bacchin, Ferro and Melucci 2002, Ng et al. 2001) inspired us to study the algorithm in the abstract, independently of its particular applications, trying to mathematically illuminate its behaviour. In the present paper we detail this theoretical analysis. The original work starts from the definition of a revised and more general version of the algorithm, which includes the classic one as a particular case. We perform an analysis of the structure of two particular matrices, essential to studying the behaviour of the algorithm, and we prove the convergence of the algorithm in the most general case, finding the analytic expression of the vectors to which it converges. Then we study the symmetry of the algorithm and prove the equivalence between the existence of symmetry and the independence from the order of execution of some basic operations on initial vectors. Finally, we expound some interesting consequences of our theoretical results.

PageRank: when order changes

As PageRank is a ranking algorithm, it is of prime interest to study the order induced by its values on webpages. In this paper a thorough mathematical analysis of PageRank-induced order changes when the damping factor varies is provided. Conditions that do not allow variations in the order are studied, and the mechanisms that make the order change are mathematically investigated. Moreover the influence on the order of a truncation in the actual computation of PageRank through a power series is analysed. Experiments carried out on a large Web digraph to integrate the mathematical analysis show that PageRank -- while working on a real digraph -- tends to hinder variations in the order of large rankings, presenting a high stability in its induced order both in the face of large variations of the damping factor value and in the face of truncations in its computation.

The power of local information in PageRank

Can one assess, by visiting only a small portion of a graph, if a given node has a significantly higher PageRank score than another? We show that the answer strongly depends on the interplay between the required correctness guarantees (is one willing to accept a small probability of error?) and the graph exploration model (can one only visit parents and children of already visited nodes?).

Annotations and digital libraries: designing adequate test-beds

The increasing number of users and the diffusion of Digital Libraries (DLs) has increased the demand for newer and improved systems to give better assistance to the user during the search of resources in collections managed by Digital Library Systems (DLSs). In this perspective, the annotations made on documents offer an interesting possibility for improving both the user experience of the DLS and the retrieval performance of the system itself. However, while different approaches based on annotations have been proposed, they still lack a full experimental evaluation, mainly because an experimental collection with annotation is missing. Therefore, this paper addresses the problem of setting an adequate experimental test-bed for DL search algorithms which exploit annotations, and discusses a flexible strategy for creating test collections with annotated documents.

Brief Announcement: On Approximating PageRank Locally with Sublinear Query Complexity

Can one compute the PageRank score of a single, arbitrary node in a graph, exploring only a vanishing fraction of the graph? We provide a positive answer to this extensively researched open question. We develop the first algorithm that, for any n -node graph, returns a multiplicative

HITS Can Converge Slowly, But Not Too Slowly, in Score and Rank

(1\pmε)

Analysis of Web Link Analysis Algorithms: The Mathematics of Ranking

-approximation of the score of any given node with probability

An approach for the construction of an experimental test collection to evaluate search systems that exploit annotations

(1-δ)

A Theoretical Analysis of Google's PageRank

, using at most

Local computation of PageRank: the ranking side

O\big(n^2/3 łn(n)^1/3 łn(1/δ)^2/3 ε^-2/3 \big) = \tildeO (n^2/3 )