The Math of Segregation
Four Behavioral Regimes
The BIKK analysis applies only to the case τ=1/2. Recently a second group has filled in details for other values of τ. The authors are George Barmpalias of the Chinese Academy of Sciences and Richard Elwes and Andy Lewis-Pye of the University of Leeds in England (the BEL group). They construct a phase diagram for the one-dimensional Schelling model. The system passes through four main behavioral regimes as τ goes from 0 to 1.
As in the NetLogo model, the smallest values of τ define a region where not much happens. Most agents are “born happy”; they never have cause to leave their initial position, and so the racial structure is little changed from the random distribution. The BEL group shows that this phase of near-indifference to race extends from τ=0 up to a certain constant κ, which has a numerical value of about 0.353. (The derivation of κ is a fascinating bit of number theory and analysis, but too long a detour to present here.)
At values of τ above κ but below 1/2, a new, segregated phase abruptly takes over. Indeed, in this region—where agents tolerate being in a local minority—the system is more segregated than it is at τ=1/2, the threshold where minority status becomes unacceptable. (In this context, “more segregated” means the typical length of a monochromatic block is greater.)
Kleinberg suggests a way of understanding this counterintuitive finding. At τ=1/2, he says, “what prevents segregation from happening on a massive scale is that smaller-scale segregation happens first, and it imposes a stable structure that limits the further spread of monochromatic runs.” In the τ<1/2 region, by contrast, fewer firewall structures provide nuclei for monochromatic runs, with the result that those runs can grow to greater length.
Continuing through the phase diagram, at τ=1/2 we enter the territory already explored by the BIKK group. Then, for even higher τ values, where agents demand not just equality of numbers but an absolute majority, the final phase appears. Here the outcome is total apartheid, as the agents of each color withdraw into a single mass occupying half the ring. Once this configuration is reached, the separation is irreversible.
Many more variations of the Schelling model remain to be investigated. One might ask how the outcome changes when the two subpopulations differ in size. Or one might allow preferences to vary between groups or between individuals with a group. And the biggest question is whether any of the results for linear models can be made to work on the plane. That task will be challenging: In one dimension, the boundary between two regions is always a single point, but borderlines on the plane can have convoluted shapes that affect neighborhood composition.