Authority Rankings from HITS, PageRank, and SALSA: Existence, Uniqueness, and Effect of Initialization

Algorithms such as Kleinberg’s HITS algorithm, the PageRank algorithm of Brin and Page, and the SALSA algorithm of Lempel and Moran use the link structure of a network of web pages to assign weights to each page in the network. The weights can then be used to rank the pages as authoritative sources. These algorithms share a common underpinning; they find a dominant eigenvector of a nonnegative matrix that describes the link structure of the given network and use the entries of this eigenvector as the page weights. We use this commonality to give a unified treatment, proving the existence of the required eigenvector for the PageRank, HITS, and SALSA algorithms, the uniqueness of the PageRank eigenvector, and the convergence of the algorithms to these eigenvectors. However, we show that the HITS and SALSA eigenvectors need not be unique. We examine how the initialization of the algorithms affects the final weightings produced. We give examples of networks that lead the HITS and SALSA algorithms to return nonunique or nonintuitive rankings. We characterize all such networks in terms of the connectivity of the related HITS authority graph. We propose a modification, Exponentiated Input to HITS, to the adjacency matrix input to the HITS algorithm. We prove that Exponentiated Input to HITS returns a unique ranking, provided that the network is weakly connected. Our examples also show that SALSA can give inconsistent hub and authority weights, due to nonuniqueness. We also mention a small modification to the SALSA initialization which makes the hub and authority weights consistent.