Schelling's model of spatial segregation is a milestone in the study of emergent global phenomena based on local social interactions (Schelling 1969; Schelling 1971; Schelling 1978). Schelling's model illustrates how clearly distinctive patterns of spatial segregation (e.g. ghettos) can emerge even if individuals are only weakly segregationist.

Almost at the same time as Schelling, Sakoda (1971) proposed a very similar model based on the same principles. To analyse their models, Schelling and Sakoda used a chessboard and several coins of different colours. Since then, there have been plenty of studies that consider different versions and refinements of these models (Aydinonat 2007). In most cases, the more recent models have replaced the chessboard and the coins with computational grids and agents (Benenson and Torrens 2004).

In this appendix we analyse a computer implementation of
Schelling's model based on Edmonds and Hales' (2005). The spatial structure of
the model is a 2-dimensional grid of 20×20 cells. There are 133 green
agents and 133 red agents who are initially located on the grid at
random. Agents may be happy or unhappy. Agents are unhappy if the proportion of other agents
of its same colour in its Moore neighbourhood of radius 1 is below a certain critical threshold (which is a parameter
of the model); otherwise the agent is said to be happy. In each iteration of the model one unhappy agent is
randomly selected to move to a random empty cell in the lattice^{ [1]}. Applet 1 below is a NetLogo (Wilensky 1999) implementation of this model.

Applet 1. Applet of the model as described in this appendix. [Download]. |

This model can be represented as a time homogeneous Markov chain. For that, we can define the state of the system as a vector of dimension 400 where each component corresponds to a certain cell in the grid. Every component of this vector can take 3 different values to denote whether the cell is empty, occupied by a green agent or occupied by a red agent. With this definition the number of possible states is considerable:

Number of possible states: |

Deriving the whole transition matrix of a system of such
dimension is quite an effort. Having said that, note that most values
in the transition matrix are zero, and it is not difficult to calculate
any given transition probability. Assuming the system is in state *i*:

- If all agents in
*i*are happy then the system remains in the same state*i*(i.e.*p*= 1 and_{i,i}*p*= 0 for all_{i,j}*j ≠ i*). These are the absorbing states of the system. - If there is any unhappy agent, then
*p*= 0 and there are few states_{i,i}*j ≠ i*such that*p*> 0. These states differ from the previous one_{i,j}*i*only in that the value of one component corresponding to a cell occupied by an unhappy agent swaps with the value of a component corresponding to an empty cell. Given*i*and*j*, the probability of this transition (*p*) is equal to 1 divided by the number of unhappy agents in_{i,j}*i*(i.e. the probability of selecting the particular unhappy agent that moves) and by the number of empty cells (134) (i.e. the probability of moving exactly to the particular empty cell).

Therefore, in particular, the transition probability between
any two states that differ in more than 2 values is necessarily zero.
Also, any state *i* where all agents are happy is
absorbing, i.e. the system will stay in it with probability *p _{i,i}*
= 1.

Interestingly, with these
parameter values, there are always absorbing states regardless of the
critical threshold value^{ [2]}.
Since it is possible to reach (at least) one absorbing state from any
given state, we can assert that there isn't any closed
communicating class with more than one state in the model. Therefore,
using Proposition 2 we can partition the state space as follows:
*S*
= {*abs*_{1}} ∪ {*abs*_{2}} ∪
… ∪ {*abs _{k}*} ∪

In this model, the higher the value of the critical threshold the fewer the number of absorbing states (states which are absorbing for low values of the critical threshold may become transient for higher values of the threshold). This implies that, in general, simulations with high values of the critical threshold tend to last longer until they reach an absorbing state. Sometimes it may be necessary to wait for a very long time until convergence is achieved. This argument explains the counterintuitive results obtained by Edmonds and Hales (2005), who observed that the average proportion of neighbours of the same colour as a function of the critical threshold decreased for critical thresholds higher than 0.65. This is counterintuitive because it seems to suggest that –beyond a certain point– more segregationist preferences lead to weaker patterns of segregation.

The crucial point to realise is that Edmonds and Hales (2005)
compute the statistics running the model *for a certain number
of
time-steps*. If the simulations were run for long enough, an
absorbing
state would be reached sooner or later, and therefore the final average
proportion of neighbours of the same colour would be necessarily
greater than or equal to the critical threshold. The picture is different
when simulations are run for a limited number of iterations. For low
critical thresholds there is a high probability of reaching an
absorbing state in a short time but, on the other hand, the average
proportion of same-colour neighbours in some absorbing states may be
quite low. As the critical threshold increases, some of the previously
absorbing states with lower average proportion of same-colour
neighbours become transient but, on the other hand, the system tends to
wander around for longer before reaching an absorbing state. Due to the
highly stochastic nature of the model, the system –in its way
towards an absorbing state– may visit areas of the state
space with very low average proportions of same-colour neighbours, and
the simulation may stop when the system is in one of these states.
However it is important to emphasise that sooner or later the system
will end up in an absorbing state, so we would expect to see the
counterintuitive result disappear if the model were to be run until an
absorbing state is reached. Having said that, the length of time
required to reach an absorbing state may well exceed the time frame of
the underlying social process being modelled, so our analysis does not
diminish the value of Edmond and Hales' findings in any way.

The Schelling-Sakoda model has been generalised using other topologies and different types of (regular and irregular) neighbourhoods (Flache and Hegselmann 2001; Hegselmann and Flache 1998). These more sophisticated models can still be analysed using the Markov chain framework.

AYDINONAT N E (2007) Models, conjectures and exploration: An
analysis of Schelling's checkerboard model of residential segregation.
*Journal of Economic Methodology*, 14(4), pp. 429-454

BENENSON I and Torrens P M (2004) *Geosimulation:
automata-based modeling of urban phenomena*. Chichester, UK:
John Wiley
and Sons.

EDMONDS B and Hales D (2005) Computational Simulation as
Theoretical Experiment. *Journal of Mathematical Sociology*,
29(3), pp. 209-232.

FLACHE A and Hegselmann R (2001) Do Irregular Grids make a
Difference? Relaxing the Spatial Regularity Assumption in Cellular Models of Social Dynamics. *Journal of Artificial Societies and Social Simulation*, 4(4)6. https://www.jasss.org/4/4/6.html.

HEGSELMANN R and Flache A (1998) Understanding Complex Social
Dynamics: A Plea For Cellular Automata Based Modelling. *Journal of Artificial Societies and Social Simulation*, 1(3)1.
https://www.jasss.org/1/3/1.html.

SAKODA J M (1971) The Checkerboard Model of Social
Interaction. *Journal of Mathematical Sociology*,
1(1), pp. 119-132

SCHELLING T C (1969) Models of Segregation. *American
Economic Review*, 59(2), pp. 488-493

SCHELLING T C (1971) Dynamic Models of Segregation. *Journal
of
Mathematical Sociology*, 1(2), pp. 143-186

SCHELLING T C (1978) *Micromotives and macrobehavior*.
New York: Norton.

WILENSKY U (1999). NetLogo. Evanston, IL Center for Connected Learning and Computer-Based Modeling, Northwestern University. http://ccl.northwestern.edu/netlogo/.

^{1}
There are alternative definitions of the dynamics of the basic model.
For instance, one can define a model where in each iteration every
agent (rather than just one) is given the opportunity to move. In that
case, the order in which agents are considered in each iteration may be
predefined (as in Edmonds and Hales (2005)) or random. Each of these possible models may have different state definitions and different transition matrices, but all the conclusions provided in the analysis section of this appendix are valid for any of these alternative definitions.

^{2}
The
state of absolute segregation (i.e. two monochromatic
subpopulations separated by no man's land) is absorbing
regardless of the critical threshold.