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.