The mathematics of Schellings segregation model

For those who don't know the开发者_如何学C model. You can read this pdf. I want to find what is the probability that 2 nodes are each others neighbors when the algorithm converges (i.e. when all nodes are happy).

Here's the model in a gist. You have a grid (say 10x10). You have nodes of two kind (red and green) 45 each. So we have 10 empty spaces. We randomly place the nodes on the grid. Now we scan through this grid (Exact order does not matter according to Schelling). Each node wants a specific percentage of people of same kind in its Moore neighborhood (say b = 50% for each red and green). We calculate the happiness of each node (a = Number of neighbors of same kind/Number of neighbors of different kind). If a node is unhappy (a < b) it moves to an empty cell where it knows it will be happy. This movement can change the dynamics of old as well as new neighborhood. Algorithm converges when all nodes are happy.

PS - I am looking for links for any mathematical analysis of the Schelling's model.

There is an account of this model in "Networks, Crowds, and Markets: Reasoning about a Highly Connected World", by Easley and Kleinberg, - see http://www.cs.cornell.edu/home/kleinber/networks-book/ This is a very good book.

However they say "As a final point, we note that while the model is mathematically precise and self-contained, the discussion has been carried out in terms of simulations and qualitative observations. This is because rigorous mathematical analysis of the Schelling model appears to be quite difficult, and is largely an open research question.." They do reference some work by Young, by Mobius and Rosenblat, and by Vinkovic and Kirman.