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Abstract

To what extent can changes in PageRank's damping factor affect node ranking? We prove that, at least on some graphs, the top k nodes assume all possible k! orderings as the damping factor varies, even if it varies within an arbitrarily small interval (e.g. [0.84999,0.85001][0.84999,0.85001]). Thus, the rank of a node for a given (finite set of discrete) damping factor(s) provides very little information about the rank of that node as the damping factor varies over a continuous interval. We bypass this problem introducing lineage analysis and proving that there is a simple condition, with a "natural" interpretation independent of PageRank, that allows one to verify "in one shot" if a node outperforms another simultaneously for all damping factors and all damping variables (informally, time variant damping factors). The novel notions of strong rank and weak rank of a node provide a measure of the fuzziness of the rank of that node, of the objective orderability of a graph's nodes, and of the quality of results returned by different ranking algorithms based on the random surfer model. We deploy our analytical tools on a 41M node snapshot of the .it Web domain and on a 0.7M node snapshot of the CiteSeer citation graph. Among other findings, we show that rank is indeed relatively stable in both graphs; that "classic" PageRank (d=0.85) marginally outperforms Weighted In-degree (d->0), mainly due to its ability to ferret out "niche" items; and that, for both the Web and CiteSeer, the ideal damping factor appears to be 0.8-0.9 to obtain those items of high importance to at least one (model of randomly surfing) user, but only 0.5-0.6 to obtain those items important to every (model of randomly surfing) user.

Content

This paper addresses the fundamental question of how the ranking induced by PageRank can be affected by variations of the damping factor. This introduction briefly reviews the PageRank algorithm (Section 1.1) and the crucial difference between score and rank (Section 1.2) before presenting an overview of our results and the organization of the rest of the paper (Section 1.3). Vgl. auch: doi:10.1016/j.jda.2009.11.001. http://www.sciencedirect.com/science/article/pii/S1570866709000926.