# Examining the Schelling Model Simulation through an Estimation of Its Entropy

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Schelling Model Outline

#### 2.2. Estimating the Entropy of the Schelling Model from the Microstate and Macrostate Assignments

## 3. Results and Discussion

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Note on the use of the Terminology for the Agent Satisfaction

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**Figure 1.**These figures examine the value of the macrostates ${R}_{t}$ (Equation (7)) over the course of the Schelling model simulation. The time t is the number of iterations and the value of R is the total number of agents which have their locality homogeneity requirement satisfied at $h=4$. Here $N=100$, $|{m}_{agent1}|=45,|{m}_{agent2}|=45$ and ${m}_{empty}=10$. Subfigure (

**a**) shows an example single run of the simulation macrostate, ${R}_{t}$, values over time. The green line denotes the region where there are no more changes to the R values. Subfigure (

**b**) shows the macrostate, ${R}_{t}$, values over a set of 1000 runs with box plots (5-point statistics). The dashed line shows the theoretical maximum value that could be achieved.

**Figure 2.**The Schelling model used here has a lattice size of $N=100$ and non-empty agents of $|{m}_{agent1}|=45$ and $|{m}_{agent2}|=45$ with the empty being ${m}_{empty}=10$. The simulations were run 10K times with independent initializations. Subfigure (

**a**) shows the results from a Monte Carlo sampling scheme for the values of the macrostate value R (Equation (7)), based upon independently drawn initial configurations of the residential actors of the Schelling model. The density of the different R values can be seen in the height of the bars proportional to the observations of that value. The dashed line represents the largest value of R obtainable for this simulation setup; 90. Subfigure (

**b**) shows the final values of R for a set of independent Schelling model simulations which have been allowed the necessary time steps until their configurations remain static. The dashed line represents the maximum possible R value in this setup. (Both simulations were run with 10K runs and tested for convergence)

**Figure 3.**The Schelling model is simulated for a lattice size of $N=100$, non-empty agents of 45 members each and 10 empty cells. Over the course of a simulation the lattice view is displayed at the initialization, $t=1$ in Subfigure (

**a**), and the time point where the repositioning of the residents ceases, $t=18$ in Subfigure (

**b**). The titles report the time point that the lattice depicted was taken from, the macrostate value R (aggregate number of homogeneity satisfied residents), and the entropy value ${S}_{t}$ (Equation (8)). The Schelling model effectively manages to reduce the entropy of the macrostate R to zero by the end of the simulation.

**Figure 4.**The plots show the trace of the estimated entropy values across time (${S}_{t}$ Equation (8)) for random initializations of the Schelling model of a 10 × 10 lattice. There are 2 groups of residential occupants of cells with 45 members each and the remaining 10 are occupied by a group categorized as ‘empty’. Subfigure (

**a**) shows the entropy trace of a single randomly initialized Schelling model. The simulation has reached a configuration of the residents where there is no subsequent movement, after $t=4$. The value of the entropy drops to zero and remains at that value. Subfigure (

**b**) shows the maximum (black), mean (blue) and minimum (black) values of the entropy over 100 independent simulations. It can be seen how there is a spread over the entropy values for each time point although the model simulation successfully finds the minimum entropy value.

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**MDPI and ACS Style**

Mantzaris, A.V.; Marich, J.A.; Halfman, T.W.
Examining the Schelling Model Simulation through an Estimation of Its Entropy. *Entropy* **2018**, *20*, 623.
https://doi.org/10.3390/e20090623

**AMA Style**

Mantzaris AV, Marich JA, Halfman TW.
Examining the Schelling Model Simulation through an Estimation of Its Entropy. *Entropy*. 2018; 20(9):623.
https://doi.org/10.3390/e20090623

**Chicago/Turabian Style**

Mantzaris, Alexander V., John A. Marich, and Tristin W. Halfman.
2018. "Examining the Schelling Model Simulation through an Estimation of Its Entropy" *Entropy* 20, no. 9: 623.
https://doi.org/10.3390/e20090623