MAED5851 Mii-ec 1 Re
Student name: Xin Wei 03 Apr 2020 Instructor name: Prof. Shingyu Leung Abstract
This project creates a simulation of Schelling Model with 2 types of agents to study the issues of social segregation. The figures of different satisfactory levels reveal that the social segregation issue is more and more serious with the increase of the satisfactory level. In addition, there is a positive correlation between the number of iterations needed and the satisfactory level. However, it seems that there does not exist a strong correlation between the number of iterations and the size of the matrix with the same satisfactory level.
1. Introduction
Today, social segregation is a very common phenomenon. People with the same or similar characteristics will gather and live together. For instance, there are China Town, Mini Italy and Latin Area, etc. in some big cities such as New York. It reveals that people with similar racial and cultural backgrounds tend to live together. In addition, there are wealthy areas and poor areas (slum) in most of the cities where people are divided by their financial capabilities. There are numerous cases of this kind of social segregation issue which implies it is not an accidental or random event. Indeed, American economist Thomas Schelling came up with a model to study this issue [1]. Schelling Segegain Mdel indicae ha every indiidal efeence in he imilai leel f ne neighb ill lead he gaheing f ele in he same type and eventually cause the social segregation of the entire society. Based on Schelling Model, this project will study the effect of different parameters on final situations.
2. Schelling Segegai Mde
There are some modified versions of the Schelling Model. However, this project will use the most basic one (see [1] [2] [3]). This model simulated the area being studied as a m × n matrix. People are categorized into different types of agents with different percentages of the whole population. They are randomly assigned to a place in this matrix, i.e. living in a house. Those cells with nobody living in are empty cells. Eene similarity level of neighbors will be calculated and compared with the satisfactory level set initially. The result is the indicator of moving to another cell or not. The loop repeats until the maximum number of iterations is reached or everyone is satisfied. The detailed definitions of the abovementioned parameters and mechanism will be introduced as follows.
2.1 Percentage of Different Types of Agents and Empty Space
For instance, there are 2 types of agents (agent A and agent B) in a 5×5 matrix (25 cells in total). If the percentage of these two types of agents are 40%, then there are 10 agent A and 10 agent B. Meanwhile, the empty ratio is 1-40%-40%=20% and there are 5 empty cells in this matrix. These 20 agents are randomly assigned into one of these 25 cells and Figure 1 demonstrates one case under this condition.
Figure 1
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2.2. Neighbors
Generally, the neighbors of an agent are the remaining 8 items in the 3 × 3 matrix centered on that agent. However, the empty cell will not be count as a neighbor. Meanwhile, the agents in the cells on the edge do not have 8 neighbors. Figure 2 below shows the number of neighbors of some items.
Item (1,1) — A Item (3,3) — A Item (3,5) — B
3 neighbors: 1 A and 2 B 5 neighbors: 3 A and 2 B 4 neighbors: 2 A and 2 B Figure 2
2.3. Similarity Level and Satisfactory Level
The similarity level of each person is the ratio of the number of same-type agents and the number of neighbors. If the similarity level is greater than or equal to the satisfactory level, this person will remain unchanged. Otherwise, this unsatisfied will move to another empty cell. Figure 3 below analyzed the similarity level of those three persons and whether they are satisfied or not under the different sets of satisfactory levels.
Figure 3
2.4 Relocation
Once the item is labeled as unsatisfied, it will move to another cell. Suppose there are 7 unsatisfied items and 5 empty cells. Those 7 unsatisfied items will move out and be assigned randomly into 7+5=12 empty cells, not just the original 5 empty cells. Then, the similarity level of each item will be calculated once again. The iteration of calculation and relocation will repeat again and again until the maximum number of iterations is reached.
3. Results
The following parameters in Figure 4 are kept invariant in all tests. Therefore, the main focus is the impact of the size of the matrix and satisfactory level.
Figure 4 3.1. Figures of Different Satisfactory Level
In a 30×30 matrix, the satisfactory level is set to be 20%, 30%, 40%, 50%, 60% and 70%. In Figure 5, it can be easily found out that segregation is more and more serious with the increase of the satisfactory level. The number of areas with a unique type of agent is decreasing. There are only 6 and 3 areas in the cases of 60% and 70% respectively.
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Figure 5
3.2. The Number of Iterations Needed in Different Satisfactory Level
After getting an initial impression of the figures of different satisfactory levels, the number of iterations needed (all agents are satisfied) in each case is also worth paying attention to. In a 20×20 matrix, the satisfactory level is set to be 20%, 30%, 40%, 50%, 55%, 60%, 65%, 70%. If the satisfactory is set to be 80%, in most trials, a 100% satisfaction still cannot achieve even after 10000 iterations. In each case, 30 trials have been done to calculate the average number of iterations needed and the standard deviation is also listed.
In Figure 6, it can be found out that the overall trend is on the rise thus there is a positive relationship between the number of iterations and the satisfactory level. For the interval of 20% to 50%, the number of iterations is less than 20 and the rise is quite slow. After that, the number of iterations needed has an obvious positive correlation with the satisfactory level. The data of standard deviation show a similar trend.
Figure 6
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3.3. The Number of Iterations Needed in Different Size
n this section, the satisfactory levels are set to be 55% and 60%, while the target is the size of the matrix. The size is set to be 10×10, 15×15, 20×20, 25×25, 30×30, 35×35 and 40×40. In each case, 30 trials have been done to calculate the average number of iterations needed and the standard deviation is also listed. In the case of the size of 10×10 and the satisfactory level of 60%, it is hard to get a 100% satisfaction within 1000 times of iterations. In Figure 7, there does not exist a strong correlation between the average number of iterations and the size of the matrix.
Figure 7 3.4. Limitations and Lines for Further Research
The sample size in this project is quite small. More data under different sets of parameters will be collected and analyzed in the future. Meanwhile, in section 3.2, there is a sharp increase after the satisfaction level is greater than 50%. The mathematical principles and reasons behind are needed to be figured out.
4. Conclusions
This project firstly introduced Schelling Model, which studied the issues of social segregation. After given the definitions of some parameters, the effects of them have been discussed separately. It can be easily found out that social segregation is more and more serious with the increase of the satisfactory level. In addition, the number of iterations needed has an obvious positive correlation with the satisfactory level. However, it seems that there does not exist a strong correlation between the number of iterations and the size of the matrix if the satisfactory level is invariant.
References
[1] [2] [3]
Schelling, T. C. (1971). Dynamic models of segregation. Journal of mathematical sociology, 1(2),
143-186.
Hatna, E., & Benenson, I. (2012). The Schelling model of ethnic residential dynamics: Beyond the
integrated-segregated dichotomy of patterns. Journal of Artificial Societies and Social Simulation, 15(1), 6.
Easley, D., & Kleinberg, J. (2010). Networks, crowds, and markets (Vol. 8). Cambridge: Cambridge
university press.
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