The emergence of social segregation

Individuals have many preferences. One is that we tend to like to associate with people who have some commonality with us. The commonality could involve hobbies, political views, language, age, wealth, ethnicity, religion, values, ... But some of us also enjoy a certain amount of diversity, at least in certain areas of life. We also have varying amounts of tolerance for difference.
A common social phenomenon is segregation: groups of people clump together in spatial regions (or internet connectivity) with those similar to them. Examples range from ethnic ghettos and teenage cliques to "echo chambers" on the internet.

The figure below shows ethnic/racial segregation in New York City. It is taken from here.

In 1971 Thomas Schelling published a landmark paper in the social sciences. It surprised many because it showed how small individual preferences for similarity can lead to large scale segregation. The motivation for his work was to understand how racially segregated neighbourhoods emerged in cities in the USA.

One version of Schelling's model is the following. Take a square lattice and each lattice point can be black, white or vacant. Fix the relative densities of the three quantities and begin with a random initial distribution. A person is "unhappy" if only 2 or less of their 8 neighbours (nearest and next-nearest neighbours) on the lattice are like them. [They have a 25% threshold for moving]. They then move to a nearby vacancy. After many iterations/moves to an equilibrium is reached where everyone is "happy" but there is significant segregation.

The figure is taken from here.

A major conclusion is that motives at the individual level are not the same as the outcomes at the macro level. People may be very tolerant of diversity (e.g. only have a preference that 30 per cent of their neighbours be like them) but collectively this results in them living in very segregated neighbourhoods.

There are several variants of the model that Schelling presented in later papers and in an influential book Micromotives and Macrobehavior, published in 1978. He received the Nobel Prize in Economics in 2005 for work in game theory.

There is a nice simulation of the model in NetLogo. For example, you can see how if you set the individual preference for similarity at 30% one ends up with a local similarity of 70%.
In the Coursera, Model Thinking, Scott Page has a helpful lecture about the model.

This can be considered to be the first agent-based model. It is fascinating that Schelling did not use a computer but rather did his ``simulation'' manually on a checkerboard!

Physicists have considered variants of Schelling's model that can be connected to more familiar lattice models from statistical mechanics, particularly the Ising model. Examples include

Ising, Schelling and self-organising segregation 
D. Stauffer and S. Solomon

Phase diagram of a Schelling segregation model
L. Gauvin, J. Vannimenus, J.-P. Nadal
This connects to classical spin-1 models such as the Blume-Capel model.

A unified framework for Schelling's model of segregation 
Tim Rogers and Alan J McKane

Competition between collective and individual dynamics 
Sébastian Grauwin, Eric Bertin, Rémi Lemoy, and Pablo Jensen

Schelling's model is a nice example of emergence in a social system. A new entity [highly segregated neighbourhoods] emerges in the whole system that was not anticipated based on a knowledge of the properties of the components of the system.