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Authority rankings from hits, pagerank, and salsa: Existence, uniqueness, and effect of initialization
 SIAM Journal on Scientific Computing
, 2006
"... Abstract. 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 webpages to assign weights to each page in the network. The weights can then be used to rank the pages as authoritative ..."
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Abstract. 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 webpages 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, so long as 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.
Search and Retrieval—Search process
"... PageRank computes the importance of each node in a directed graph under a random surfer model governed by a teleportation parameter. Commonly denoted alpha, this parameter models the probability of following an edge inside the graph or, when the graph comes from a network of web pages and links, cli ..."
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PageRank computes the importance of each node in a directed graph under a random surfer model governed by a teleportation parameter. Commonly denoted alpha, this parameter models the probability of following an edge inside the graph or, when the graph comes from a network of web pages and links, clicking a link on a web page. We empirically measure the teleportation parameter based on browser toolbar logs and a click trail analysis. For a particular user or machine, such analysis produces a value of alpha. We find that these values nicely fit a Beta distribution with mean edgefollowing probability between 0.3 and 0.7, depending on the site. Using these distributions, we compute PageRank scores where PageRank is computed with respect to a distribution as the teleportation parameter, rather than a constant teleportation parameter. These new metrics are evaluated on the graph of pages in Wikipedia.