The first thing that comes to mind is how rankings would be determined. It sounds like all miners would have to agree that a given rank is valid. If there was a deterministic ranking method, a sort of DoS attack could be possible whereby one creates pages that satisfy the ranking criteria fully but have no usable content.
edit:If you'd like to see some empirical evidence of this attack, look up
search engine optimization.
It turns out the economically rational action is to game the ranking algorithm.
What do you mean by "to game the ranking algorithm"?
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I am saying this on the assumption that this network would require a huge amount of data to be stored between all of the nodes, if that is true I think it might be more feasible for each node to have only a subset of the entire rank list. There would be many copies of the list stored by many nodes, when a search is done a consensus is taken from the nodes that respond to the search request as to what the results should be.
The required amount of copies of information about a page in the rank list is the amount of other pages linking to that page. So if there are 10,000 links to a site, then there should be ~10,000 nodes with that sites(and its into links) information. Every time a node responds to a search results and returns a set of nodes, it "releases" that pages information to a new node which then takes in the links, crawls the web and checks them, and then "waits" for searches to them and the cycle continues.
I think this would provide a very small probability that one entity could control all of (or most of) the nodes that contain the current link information for a page. Thus assuming everyone works together to creates good links and page ranks, then it should work out that an attacker would have to control the entire network or a very large amount to effect search results.