This page examines cellular automata and emergent behavior in the context of a NetLogo model inspired by Thomas Shelling’s theory of spatial segregation.

Part 1. What minimum level of self-preference leads to emergent patterns of segregation?

The ‘%-similar-wanted,’ or self-preference variable determines how many turtles of the same color an agent is striving to be surrounded by. For example, if the self-preference variable has a value of 50, an agent will arrange itself, to the best of its ability, so that at least half of its neighbors are the same color as itself.

Given an environment with 2,000 agents, and a self-preference value of anything greater than 0, we will see the agents readjust their position. However, this rearrangement does not produce any visible pattern until we reach a self-preference value of around 35%. The video below display the model with the stated parameters:
%-similar-wanted = 35%

At the end of the video, we can begin to discern a pattern out of the rearranged agents. We can distinguish separate masses of yellow and orange turtles, and see empty spaces bordering these masses, though they are not well defined.

Part 2. Does a specific threshold level exist that tips the system into an unstable state?

A very explicit threshold tips this system into an unsteady state when it is passed. With respect to a turtle population of 2000, this threshold may be visualized when the %-similar-wanted (or self-preference) value equals 75%. To illustrate just how rigid this threshold is, two videos with a small parameter change are posted below. In each video, both systems have a population of 2000 turtles, but the first video shows the system with a self-preference value of 75%, and the only change made in the second video is raising the self-preference value by 1.

In the latter half of the first video, we begin to see the segregation patterns take shape and eventually flesh-out. However, the second video runs for three lengths of the duration of the first, and shows no interest in settling.

Part 3. How does an increasing level of self-preference change the nature of the segregation patterns that emerge?

As defined earlier, the self-preference value denotes the percentage of similarly-colored turtles each individual is striving to be surrounded by. Since the turtles move about the space randomly, this means that the higher the self-preference value, more movement (and thus, time) will need to take place for the turtles to become situated in a ‘happy’ position.

Aside from the time it takes for the agents to coalesce into their respective masses, these masses become larger and their borders more defined. This pattern is consistent for each self-preference value below our previously determined threshold of 75. As discussed in part 2, the behavior of the system becomes unstable after this point. At a value of 76, the agents never seem to settle.

Below are three separate videos depicting each of these scenarios:

Part 4. What does the change in the unhappy-ratio at each time step tell you about the relationship between self-preference and the number of similar neighbors in one neighborhood?

The ‘unhappy-ratio’ is essentially a comparison of two pre-existing ratios in the model. That is the percentage of unhappy turtles in the population compared with the average number of similar neighbors an agent has in its neighborhood.

The graphs below explore the unhappy-ratio and percent-similar value in the context of a rising self-preference value:

%-similar-wanted = 35    %-similar-wanted = 75     %-similar-wanted = 80

Generally, if the self-preference value is set below the threshold, we will see the ratio decrease exponentially, or linearly, and the percentage of similar ratios increase in the same ways. Above the threshold, the values oscillate.

When the ratio is high, this means that the percentage of unhappy turtles is higher than the percentage of similar neighbors in the average neighborhood. This makes sense due to the fact that an agent’s happiness is increased by a larger percentage of similar neighbors. Thus, as agents begin to rearrange themselves, and become successful in finding similar neighbors, this ratio will become smaller.

Additionally, the higher self-preference value, the higher the similarity percentage needs to be in order for an agent to be ‘happy.’ This notion helps explain the  linear depression in the second graph, since the numerator in this case will remain large even as the similarity percentage increases.

Part 5. What general conclusion can be drawn from your answers to questions 1 and 3?

What the answers from parts 1 and 3 essentially tell us is that this system has chaotic properties. As we tamper with the parameters of this model, we find that we are not necessarily able to predict the outcome–the behavior of the model enduring the parameter changes cannot be described as variations of the same pattern. Sometimes the pattern changes completely, other times, it is virtually undetectable.

Since I have already posted videos illustrating these behaviors, I will simply note them here (see videos in pt. 3), and explore this behavior further with some light, graphical analysis:

%-similar-wanted = 76           %-similar-wanted = 75              %-similar-wanted = 35

The graphs above reinforce the speculations made by analyzing our videos–that is, as we increase one parameter (namely, the self-preference value) we see the system undergo a variety of changes. As the parameter increases, we see the similarity and turtle unhappiness percentages increase and decrease in a logarithmic and exponential fashion. As the value approaches the threshold, we see this pattern shift into a linear shape. We also note the explicit change in behavior after the threshold is exceeded. At this time, the system becomes unstable and the percentages oscillate continuously, denoting the inability for agents to settle in accordance to the self-preference value.