# Diffusive Resettlement: Irreversible Urban Transitions in Closed Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dynamics of Intra-Urban Migration

## 3. Results

#### 3.1. Revealing the Two-Component Structure of Intra-Urban Evolution

#### 3.2. Intra-Urban Migration as Diffusion

#### 3.3. Predicting Equilibrium Population Distribution

## 4. Robustness Evaluation

#### 4.1. Heterogeneity of the Population

#### 4.2. Solution Space of the Two-Component Model

#### 4.3. Component-Specific Relocation Matrix

#### 4.4. Sydney Case Study

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Mathematical Derivations

#### Appendix A.1. Convergence to the Same Equilibrium in (11)

**Proposition**

**A1.**

**Proof.**

#### Appendix A.2. Derivation of Equation (8)

#### Appendix A.3. Migration Model with Memory

## Appendix B. Additional Figures

#### Appendix B.1. Independence of the Equilibrium State on the Relocation Frequency

**Figure A1.**Convergence of ${X}_{k}\left(t\right)$ to the equilibrium ${\alpha}_{k}{X}_{eq}$ for ${\alpha}_{1}=0.7$, ${\alpha}_{2}=0.3$, ${\u03f5}_{1}=0.05$, and ${\u03f5}_{2}=0.5$; matrix H and initial conditions $X\left(0\right)$ are random, and the total population is 1. Equilibrium ${X}_{eq}$ is calculated as a left eigenvector of matrix H. Dotted lines 1 and 2 correspond to deviation $\parallel {X}_{k}\left(t\right)-{\alpha}_{k}{X}_{eq}\parallel $ as a function of time step t; the solid line corresponds to the deviation of the total structure $\parallel {\sum}_{k=1}^{C}{X}_{k}\left(t\right)-{X}_{eq}\parallel $.

#### Appendix B.2. Predicting the Number of Movers with the 2011 Data Set

**Figure A2.**Number of movers in the five-year migration data: actual (${\sum}_{i\ne j}{T}_{ij}^{\left(5\right)}\left(2011\right)$) vs. predicted by ${P}^{5}$ (${\sum}_{i\ne j}{\hat{T}}_{ij}^{\left(5\right)}\left(2011\right)$), with each dot representing one suburb. Red dots correspond to the one-component model, and the green dots correspond to the two-component model. The blue solid line has the slope of 1, showing the ideal prediction. The corresponding calibration errors are shown in Table A1.

#### Appendix B.3. Actual Population Density Map of the Australian Capital Cities

**Figure A3.**Actual population density map of the Australian capital cities (2016 Census). Scale bars in the lower-left corners indicate distances equivalent to 20 km.

#### Appendix B.4. Long-Term Prediction Comparison: 2011 vs. 2016

**Figure A4.**The equilibrium population of city suburbs predicted with the 2016 data set is plotted against the 2011 prediction. The red solid line has a slope of 1, showing the ideal consistency.

## Appendix C. Additional Tables

#### Appendix C.1. Relative Error of Relocation Prediction

GCA | 2011 | 2016 | ||
---|---|---|---|---|

1-Component | 2-Component | 1-Component | 2-Component | |

Sydney | 40% | 12% | 37% | 13% |

Melbourne | 45% | 14% | 42% | 11% |

Brisbane | 42% | 10% | 43% | 10% |

Adelaide | 50% | 9% | 46% | 7% |

Perth | 46% | 9% | 43% | 10% |

Hobart | 54% | 10% | 48% | 9% |

Darwin | 59% | 15% | 64% | 16% |

Canberra | 51% | 18% | 40% | 18% |

#### Appendix C.2. Five-Year Predictions of the Single-Component Model

**Table A2.**Share of people who do not change their place of residence within a five-year period (actual vs. predicted) based on the 2011–2016 migration data.

GCA | Actual | Approach 1 | Approach 2 | Approach 3 |
---|---|---|---|---|

Sydney | 0.733 | 0.630 | 0.624 | 0.630 |

Melbourne | 0.734 | 0.618 | 0.610 | 0.613 |

Brisbane | 0.702 | 0.569 | 0.560 | 0.545 |

Adelaide | 0.752 | 0.635 | 0.632 | 0.642 |

Perth | 0.709 | 0.584 | 0.579 | 0.588 |

Hobart | 0.783 | 0.678 | 0.674 | 0.669 |

Darwin | 0.713 | 0.527 | 0.518 | 0.511 |

Canberra | 0.720 | 0.615 | 0.605 | 0.585 |

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**Figure 1.**Number of movers in five-year migration data: actual (${\sum}_{i\ne j}{T}_{ij;\phantom{\rule{0.166667em}{0ex}}5Y}\left(2016\right)$) vs. predicted (${\sum}_{i\ne j}{\hat{T}}_{ij;\phantom{\rule{0.166667em}{0ex}}5Y}\left(2016\right)$), with each dot representing one suburb. Red dots correspond to the one-component model, and the green ones correspond to the two-component model. The blue solid line has a slope of 1, showing the ideal prediction. The corresponding calibration errors are shown in Table A1.

**Figure 2.**Exponential convergence of ${U}_{k}\left(t\right)$ for each of the population groups, $k=1,2$: (

**A**) $log\parallel {U}_{k}\left(t\right)\parallel $ is plotted against time step t; (

**B**) $\parallel {Q}_{k}\left(t\right)\parallel $ is plotted against $\parallel {U}_{k}\left(t\right)\parallel $ (thick dotted curves) and the tangential lines with the slope $1-{\lambda}_{k}$ (solid straight lines), where ${\lambda}_{k}$ is the second eigenvalue of the group relocation matrix. For illustration purposes, both ${U}_{k}\left(t\right)$ and ${Q}_{k}\left(t\right)$ are normalised by the total number of residents, ${\alpha}_{k}\overline{x}$, in the corresponding group.

**Figure 3.**Long-run population structure prediction based on eigenvectors of the migration matrix, which were obtained from the 2016 Census data. The scale bars in lower-left corners indicate distances equivalent to 20 km.

**Figure 4.**The share of people who do not change their place of residence within period t plotted against the length of this period. Green dots correspond to the actual values for Sydney (${s}_{0Y}=1$; ${s}_{1Y}=0.91$; ${s}_{5Y}=0.73$). Solid curves correspond to the model where people do not relocate within $\tau $ years after their last relocation ($\tau $ ranges from 0 to 5; see Equation (A10)). The dotted curve corresponds to the two-component model ($\alpha =0.9$; see Equation (6)). All models are calibrated to the Sydney relocation data. All solid curves pass through the actual one-year relocation rate ${s}_{1Y}=0.91$, but go well below the corresponding five-year value, ${s}_{5Y}=0.73$.

**Figure 5.**Number of solutions to (8) depending on $\alpha $ and ${s}_{5Y}$ for (

**A**) ${s}_{1Y}=0.2$, (

**B**) ${s}_{1Y}=0.3$, (

**C**) ${s}_{1Y}=0.4$, (

**D**) ${s}_{1Y}=0.5$, (

**E**) ${s}_{1Y}=0.6$, (

**F**) ${s}_{1Y}=0.7$, (

**G**) ${s}_{1Y}=0.8$, (

**H**) ${s}_{1Y}=0.9$, (

**I**) ${s}_{1Y}=0.98$, and (

**J**) ${s}_{1Y}=0.99$. Yellow areas correspond to two distinct solutions, green areas represent one solution, and dark purple stands for no solution.

**Figure 6.**The share of people who do not change their place of residence within period t plotted against the length of this period. The dotted curve corresponds to the naive single-component model (calibrated to the one-year value, ${s}_{1Y}=0.91$). Solid lines describe a family of the two-component model predictions matching actual one-year and five-year values (Sydney values, ${s}_{1Y}=0.91$ and ${s}_{5Y}=0.73$, are taken as an example) for different levels of $\alpha $. All solid curves pass through three common points (green): ${s}_{0Y}=1$; ${s}_{1Y}=0.91$; ${s}_{5Y}=0.73$. The dotted curve passes through the first two green points, and its five-year prediction is marked in red.

**Figure 7.**Stationary population structure for the case where components 1 and 2 have migration flows with opposite directions: (

**A**) $\alpha =0.1$; (

**B**) $\alpha =0.5$; (

**C**) $\alpha =0.9$. The component-wise population structure is shown in the left column. The total population structure is shown in the right column. For all values of $\alpha $, the approximations ${\hat{X}}_{eq}$ obtained from the observable matrix $\hat{H}$ (red solid line) are almost indistinguishable from the ground-truth equilibria ${X}_{eq}$ (green bars).

**Figure 8.**Stationary population structure for the case where the first group members always relocate to the central districts, while the second group members migrate to the peripheral ones: (

**A**) $\alpha =0.1$; (

**B**) $\alpha =0.5$; (C)$\alpha =0.9$. The component-wise population structure is shown in the left column. The total population structure is shown in the right column. For all values of $\alpha $, the approximations ${\hat{X}}_{eq}$ obtained from the observable matrix $\hat{H}$ (red solid line) are almost indistinguishable from the ground-truth equilibria ${X}_{eq}$ (green bars).

**Figure 9.**Equilibrium population density in Sydney for the case of heterogeneous relocation matrices ${H}_{k}$. (

**A**) First group’s equilibrium structure ${X}_{1,eq}$; (

**B**) second group’s equilibrium structure ${X}_{2,eq}$; (

**C**) total equilibrium structure ${X}_{eq}$; (

**D**) approximation ${\hat{X}}_{eq}$ obtained from the overall relocation matrix H. The scale bar in the lower-left corner indicates a distance equivalent to 20 km.

**Table 1.**Share of people who do not change their place of residence (actual vs. predicted). The values are based on 2016 Census data.

GCA | 1Y Actual | 5Y Actual | 5Y Predicted 1-Component | 5Y Predicted 2-Component |
---|---|---|---|---|

Sydney | 0.909 | 0.733 | 0.630 | 0.735 |

Melbourne | 0.905 | 0.734 | 0.618 | 0.736 |

Brisbane | 0.889 | 0.702 | 0.569 | 0.704 |

Adelaide | 0.911 | 0.752 | 0.635 | 0.754 |

Perth | 0.895 | 0.71 | 0.584 | 0.712 |

Hobart | 0.923 | 0.783 | 0.678 | 0.788 |

Darwin | 0.874 | 0.713 | 0.527 | 0.717 |

Canberra | 0.904 | 0.720 | 0.615 | 0.722 |

**Table 2.**Spreading index calculated for both the current and predicted long-run structure of the Australian cities.

GCA | Current | Predicted |
---|---|---|

Sydney | 0.29 | 0.74 |

Melbourne | 0.26 | 0.43 |

Brisbane | 0.32 | 0.37 |

Adelaide | 0.54 | 0.53 |

Perth | 0.44 | 0.58 |

Hobart | 0.38 | 0.25 |

Darwin | 0.75 | 0.79 |

Canberra | 1.06 | 0.92 |

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Slavko, B.; Prokopenko, M.; Glavatskiy, K.S. Diffusive Resettlement: Irreversible Urban Transitions in Closed Systems. *Entropy* **2021**, *23*, 66.
https://doi.org/10.3390/e23010066

**AMA Style**

Slavko B, Prokopenko M, Glavatskiy KS. Diffusive Resettlement: Irreversible Urban Transitions in Closed Systems. *Entropy*. 2021; 23(1):66.
https://doi.org/10.3390/e23010066

**Chicago/Turabian Style**

Slavko, Bohdan, Mikhail Prokopenko, and Kirill S. Glavatskiy. 2021. "Diffusive Resettlement: Irreversible Urban Transitions in Closed Systems" *Entropy* 23, no. 1: 66.
https://doi.org/10.3390/e23010066