Can we measure Google's monopoly like PageRank is measured?

Jeremy Pickens posted an interesting note on his IR new blog:

Is it really true that Google is competing on a click-by-click basis? In the user studies that Google does, which of the following happens more often when the user types in a query to Google, and sees that Google has not succeeded in producing the information that they sought (fails):

1. Does the user reformulate his or her query, and click “Search Google” again (one click)? Or,
2. Does the user leave Google (one click), and try his or her query on Yahoo or Ask or MSN (second click), instead?

His points about actions 1 versus 2 are very astute. I’d guess that #2 happens a LOT on the # 2-10 search engines. Meaning people give that engine a try.. maybe attempt a reformulation.. then abandon that engine and try on Google. And I’m betting that people ‘abandon’ Google at a far less rate than other engines.. ie asymmetry of abandonment.

I’d love to do the following analysis given a browser log of search behavior:

Form a graph where the major search engines are nodes in the graph

For each pair of searches found in the log at time t and time t+1 for a given user, increment the counter on the edge SearchEngine(t) -> SearchEngine(t+1). Once the entire log is processed normalize the weights on all edges leaving a particular node.

We now have a markov chain of engine usage behavior. The directional edges in the graph represent probability of use transference to another engine, self-loops are the probability of sticking with the current engine.

If we calculate the stationary distribution of the adjacency matrix of probabilities, we should have a probability distribution that closely matches the market shares of the major engines. (FYI - this is what PageRank version 1.0 is - the stationary distribution of the link graph of the entire web)

What else can we do? We can analyze it like it’s a random walk and calculate the expected # of searches until a given user of any internet search engine will end up using Google. If the probabilities on the graph are highly asymmetric.. which I think they are.. this is a measure of the monopolistic power of people’s Google habit.

This should also predict the lifetime of a given ‘new’ MSN Live or Ask.com user.. meaning the number of searches they do before abandoning it for some other engine.

Predicted End Result: Google is the near-absorbing state of the graph.. meaning that all other engines are transient states on the route to Google sucking up market share. Of course this is patently obvious unless one of the bigs changes the game.