Thursday, 29 August 2013

The Schelling Model


Lest you think I've forgotten about the Ising model, here is another application (sort of). Below are some maps of cities in America where each dot is a person and the colour represents their race. Taken from here. As you can see the distribution is less than homogeneous. I especially like the little red enclave in Detroit as well at the obvious "domain wall" along 8-Mile Road. New York and Chicago have more types of people. New York has many smaller clusters, possibly helped to form by geography.

New York



An economist named Thomas Schelling heard about these divisions and wanted to know if they were caused by people having strong preferences to be near their own kind, so he created a model to get to the essence of the problem. Like any good model we strip away the inessentials. People move for various reasons; for new jobs, better schools, to avoid meteor strikes, in Schelling's model people move based only on who their neighbours are. Our city is a large grid. At each point on the grid we have an agent of type A, an agent of type B or an empty space. When we examine the grid points we count the total number of neighbours and the number of neighbours of the same type as the central point. Note we are looking at the Moore neighbourhood of A.

In the plot above A has 7 neighbours 3 of which are the same type as him and 4 of which are different. We now pick a number called the intolerance which acts very much like the temperature in the Ising model. If the equation \[ \frac{ \text{Number different neighbours} }{\text{Number of neighbours}} > \text{intolerance} \] is true the agent moves to a vacant space (chosen at random), if not the agent stays put. For example with an intolerance of $30\%$ the agent will be okay with being in the minority as long as at least $30\%$ of its neighbours are the same type as it. Click the box below to see this in action, starting from a random distribution of agents and evolving.

The process we sought to model, racial segregation, has materialised. The original somewhat surprising finding was that for quite tolerant agents the final "diversity" measured by checking how many neighbours of the same type each agent has is relatively low (or the % same = 1 - diversity is relatively high). Much lower than what each individual agent will tolerate. Slightly biased individuals give rise to a very biased society. As usual there are some parameters to play with. Change the intolerance to see a phase transition to a noisy phase where everyone hates being near agents of different type so much that they move all the time. Decrease the occupancy to make the grid more sparsely populated, you can see that the segregated domains are separated by a "wall" of vacant sites. Finally play with the number of agent types to see that even an ethnically diverse city segregates, though in this case the place looks a bit more diverse since there are more, smaller clusters, kind of like the New York map. Interestingly, if the segregation phenomenon occurs then the agents are made extremely tolerant this does nothing to increase diversity, proving mathematically that society can never improve :)

Going back to look at the Ising model the differences are (a) conservation of agents: no one dies during this simulation - there are always the same number of A type and B types. The Ising model doesn't conserve number of up spins for example though it will fluctuate pretty close to its equilibrium value. (b) However this means there is no "conquest" of A agents by B agents. For example if you put the Ising model to a random configuration and rapidly lower the temperature (increase $\beta$) you will see a similar domain structure form. However, wait long enough and one spin will ultimately dominate. I can't decide if this makes a more or less realistic model. Also if you play with the Ising model in this way you sometimes get two solid blocks of up and down separated by a domain wall, Detroit style, which I haven't seen occur in the Schelling model.

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