Computer simulations of language change notes
This website collects my personal notes on Computer simulations of language change. These notes are provided to bring full transparency to my research process. Of course, since they are only notes, they do not reflect my final thoughts on a topic, and should not be interpreted as such. To read finished papers, please consult my website. Do not use these notes as a basis for your own scientific research. Start from high-quality, peer-reviewed scientific literature instead.
this chapter
further
segregation
Map of racial and ethnic divisions in the New York City metropolitan area
Possible explanations for segregation
1. organised action
2. socioeconomic filters
3. individual preferences / homophily
↳ why take individual preferences into account?
↓
↳ Schelling model
model as a drastic simplification
individuals
A funny thing about building models is that a model almost always contains models within it. A model of segregation needs a model of geographical space and models of individuals’ ethnicity and behaviour. It’s models all the way down.
group identities
other facilities
defining neighbourhoods
The grid is discrete rather than continuous (as it was in the Particle World model) because dwellings in most urban areas are discrete. Homes tend to have fixed locations, and you cannot move your home arbitrarily close to another.
A square lattice
Important properties of a square grid
1. intuitive to visualise
2. symmetrical
boundaries and boundary effects
3. no effects of preferred axis of orientation
4. easy to code
Properties a square grid ignores
1. features common to real cities influencing desirability
You might find yourself a bit uncomfortable with this oversimplification. This discomfort is healthy— hold on to it. It is important to keep in mind the simplifying assumptions one makes when modelling, because the results of our simulations are direct consequences of those assumptions. […] Nevertheless, we must always do violence to reality to make any sense of our world, and we have to start somewhere.
agent location
agent ethnicity
↳ problems
Because this is a simple baseline model, this is all OK.
agent behaviour
It is worth noting that other formal decision rules consistent with the idea that “individuals prefer to live near similar neighbors” are possible, and that using these may somewhat change the model outcomes (Bruch and Mare, 2006).
‘neighbourhood’
| Moore neighbourhood | Von Neumann Neighbourhood |
|---|---|
 |
 |
| eight closest cells | four closest cells |
| includes diagonals | cardinal directions only |
↳ larger neighbourhoods
| r = 1 | r = 2 | r = 3 | |
|---|---|---|---|
| Moore neighbourhood |
|
|
|
Von Neumann neighbourhood |
|
|
|
In bounded finite spaces, the neighbourhoods will become truncated, leaving agents near the edges of the space with fewer neighbours than those nearer the middle. Toroidal neighbourhoods fix this issue again.
How does the world change?
time step core loop
neighbourhood calculation
An agent doesn’t necessarily mind being a minority in its neighbourhood, but it wants to have at least some similar neighbours.
‘moving’
↳ equilibrium
What do we want to learn from this model?
once equilibrium
↓
1. average similarity
2. other measures TODO exploration
| simple models | complex models |
|---|---|
| can be fully described in just a few paragraphs | may require you to partition your model description ↓ important computations in the main text, specific implementation in the appendix |
Partitioning your model description is advised for readability
variables
Consider an 𝐿 × 𝐿 square grid with toroidal boundaries. At initialization, one agent is placed upon each cell with a probability p, which characterizes the population density relative to the available space (0 < 𝑝 < 1, so that each agent can relocate to an empty location). The expected population size is therefore 𝑁 = 𝑝𝐿². The population is divided into 𝐺 groups, such that each agent 𝑖 is randomly assigned a fixed group identity 𝑔𝑖 ∈ {1, 2, . . . , 𝐺}, each chosen with equal probability. For all of our analyses, we will use 𝐿 = 51 and 𝐺 = 2. The population is also defined by a similarity threshold, 𝑆, which defines the minimum proportion of an agent’s neighbors that must be similar for it to refrain from moving. Note that the use of probabilistic assignments means that, even holding 𝑝 and 𝐺 constant, there will be some variation between simulation runs in terms of exactly how large the population is and how many agents belong to each group. This stochasticity is often seen as a positive because it allows us to assess how robust the model is to minor fluctuations. However, one could also impose stricter requirements, so that, for example, the population size was always the nearest integer value of 𝑝𝐿².
At any given time, each agent is either “happy,” in which case its neighbors are sufficiently similar and it will not move, or “unhappy,” in which case it will move because its neighborhood does not contain enough similar neighbors.
At the beginning of each time step, each agent 𝑖 first determines whether or not it is happy. The agent considers the other agents in its neighborhood (defined as a Moore neighborhood with radius r =1) and determines the proportion of its neighbors that share its group identity, 𝑠𝑖. If 𝑠𝑖 < 𝑆, the agent is unhappy, otherwise it is happy. After each agent has made this assessment, each unhappy agent, in random order, moves to a random empty cell (happy agents remain in their current locations). These dynamics repeat until no agents are unhappy. And they all lived happily ever after.
possible dynamics
cascade effects
play
Schelling conclusions
model analysis
| models based on equations | models using agents |
|---|---|
| deterministic | stochastic |
| mathematically tractable | mathematically untractable |
| we precisely can compute how model parameters relate to outcomes | we cannot compute how model parameters relate to outcomes |
| less realistic | more realistic |
↓ how to report on stochastic outcomes?
↳ batch run / batch
# of runs required?
how long to run a simulation?
different parameters,
different outcomes
dynamic range of parameters
We should use parameters that are meaningful or realistic.
consider arbitrary / implicit assumptions
Implicit assumptions of the Schelling model
↳ problem with assumptions
curse of dimensionality
Playing with your model can help you realize which parameters are probably important to explore and at which range of values, as well as which parameters you can more safely ignore, or run for only a couple of values to ensure robustness.
null conditions
inferential statistics?
To what extent do individual preferences for living among similar neighbors yield segregated communities?
↳ modus operandi
here
how many runs?
qualitative results
similarity-threshold increases, we get more segregationAt lower densities, the population becomes much more segregated than it does at high densities. This is partly because at lower densities, individuals are likely to have only one or two neighbors, making the threshold harder to reach. Imagine you are in an isolated cluster of three agents. If you have only two neighbors, the only way to have at least 30% of them be the same color as yourself is for at least one of them to be that color. But if the remaining agent is a different color, then 100% of their neighbors are a different color, and so they will move.
tipping point events
depressed loners?
Being a loner should be fairly common under very low densities. If loners are always unhappy, they are more likely to end up in mixed neighborhoods, thereby decreasing overall segregation.
Notice that something funny seems to be happening for the lowest density and highest similarity threshold condition, as seen in the upper right of the rightmost plot in Figure 3.10. Average segregation dips down as a result of greater variation in outcomes. Playing around with individual runs shows that this is due to longer times needed to reach equilibrium for the lonely loners, often greater than the 100 time steps we allotted. The result is that the simulation ends before the population has reached its peak level of segregation. The spatial patterns of segregation that emerge at low densities also differ between the two conditions, as depicted in Figure 3.11.
This figure illustrates both the importance of examining the effects of seemingly minor assumptions, as well as the ways in which summary statistics can be limited in describing the true patterns present in a model system (or, indeed, in a real-world system).
high abstraction
Computer simulations of language change notes
This website collects my personal notes on Computer simulations of language change. These notes are provided to bring full transparency to my research process. Of course, since they are only notes, they do not reflect my final thoughts on a topic, and should not be interpreted as such. To read finished papers, please consult my website. Do not use these notes as a basis for your own scientific research. Start from high-quality, peer-reviewed scientific literature instead.