Choose the damping, choose the ranking?

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]). 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.