Martin Farach-Colton Closure Operators and Spam Markets for PageRank Ranking functions, such as standard PageRank and HITS (hubs and authorities), assign values to nodes in a directed graph in order to rank them, e.g. during a web query. Link spam is the attempt to manipulate a ranking function by changing the edges of the graph. We define a model of link spam and spam resistance that explain the observation that HITS and standard PageRank are easy to spam, whereas personalized PageRank is hard to spam. However, personalized PageRank concentrates the ranking values too much to be useful. We show that the space of all possible PageRanks has a rich algebra that allows personalized PageRanks to be combined in a way that inherits much of their spam resistances while mitigating the issue of PageRank concentration.