推荐阅读：
City seeds: Geography and the origins of the European city system. M. Bosker & E. Buringh. JUE. 2017
Human behavior and the Principle of Least Effort: An Introduction to Human Ecology
Zipf’s Law:
\[logS=logm-qlogR\]城市的位序规模，Yule-Simon Model: 词频、论文数、人口数、收入等的分布都是幂律分布
计：前\(k\)个词中出现次数为\(i\)的词共有\(f(i,k)\)个。做如下假设：
Gibrat’s law
growth rates of individual cities are independently and identically distributed (iid) random variables：城市（人口）的增长速度与城市规模无关
Mori et al., Common power laws for cities and spatial fractal structures, 2020, PNAS
引入交互项后的城市增长：Verbavatz V, Barthelemy M. The growth equation of cities[J]. Nature, 2020, 587(7834): 397-401.
\[\frac{\partial{S_i}}{\partial{t}}=\eta_iS_i+\sum_{j\epsilon N(i)}{(J_{j\rightarrow i}-J_{i\rightarrow j})}\]where the quantity \(\eta_i\) is a random variable accounting for the ‘out-of-system’ growth of city ; the data show that \(\eta_i\) is Gaussian-distributed.The flow \(J_{j\rightarrow i}\) is the number of individuals moving from city \(i\) to city \(j\) during aperiod of time \(dt\). If there is an exact balance of migration flows (\(J_{j\rightarrow i}=J_{i\rightarrow j}\)), the equation becomes equivalent to Gibrat’s model which predicts a log-normal distribution of populations.
使用重力模型评估\(J_{j\rightarrow i}\):
\[J_{j\rightarrow i}=I_0\frac{S_i^{\mu}S_j^{\nu}}{d_{ij}^{\sigma}}\]distance free:
\[J_{j\rightarrow i}=I_0S_i^{\mu}S_j^{\nu}\]distance-free后模型可可解析但丢失了空间性。
引入其他干扰项\(\chi_{ij}\)，则：\(J_{j\rightarrow i}=I_0S_i^{\mu}S_j^{\nu}\chi_{ij}\)，令人均迁入比\(I_{ji}=J_{j\rightarrow i}/S_i\)，则有：
\[\frac{I_{ij}}{I_{ji}}=(\frac{S_i}{S_j})^{1-\mu+\nu}\frac{\chi_{ji}}{\chi_{ij}}\]通过实证得到\(\frac{\chi_{ji}}{\chi_{ij}}\approx 1\)，进而通过实证得到\(\mu\approx\nu\)，得到最终的模型形式：
\[I_{ij}=\frac{I_0S_j^{\nu}S_i^{\nu}\chi_{ij}}{S_j}=I_0S_j^{\nu-1}S_i^{\nu}\chi_{ij}\]则：
\(J_{i\rightarrow j}=I_0S_j^{\nu}S_i^{\nu}\chi_{ij}\) \(J_{j\rightarrow i}-J_{i\rightarrow j}=I_0S_i^{\nu}S_j^{\nu}(\chi_{ji}-\chi_{ij})\)
\[\sum_{j\epsilon N(i)}{J_{j\rightarrow i}-J_{i\rightarrow j}}=I_0S_i^{\nu}\sum_{j\epsilon N(i)}{S_j^{\nu}(\chi_{ji}-\chi_{ij})}=I_0S_i^{\nu}\sum_{j\epsilon N(i)}{\chi_{ij}}\]此外：
\[\zeta_i=\frac{1}{|N(i)|^{1/{\alpha}}}\sum_{j\epsilon N(i)}{\chi_{ij}}\]其中\(\zeta_i\)是Lévy分布的特征函数值
最终得到：
\[\frac{\partial{S_i}}{\partial{t}}=\eta_iS_i+DS_i^{\beta}\zeta_i\]其中：\(D\propto{I_0},\beta=\nu+\gamma/\alpha,\eta_i\)是有平均增长率\(r\)和标准差\(\sigma\)的高斯噪声
central place theory: Central places in Southern Germany
中心地理论的相关研究：
使用超均匀分布对人类居住点的空间分布进行分析：Hyperuniform organization in human settlements
中心地理论与幂律分布：人口密度和邻域内设施密度的数量关系
优化项：
\[T=\frac{S}{v}+\frac{h}{P}\]其中\(T\)是行政单元的平均到达社会服务时间，\(S\)是平均到中心的距离，\(v\)是公交的速度，\(h\)是time of maintaining the center，\(P\)是人口总量。如果以\(A\)表示面积，\(D\)表示人口密度，\(g\)表示领域的形状密度，则有：
\(T=\frac{g\sqrt{A}}{v}+\frac{h}{AD}\) \(\frac{dT}{dA}=\frac{g}{2v\sqrt{A}}+\frac{h}{A^2D}\)
令导数=0，有：
\(A=(\frac{2vh}{gD})^{2/3}\) \(\log{A}=\frac{2}{3}\log{\frac{2vh}{g}}-\frac{2}{3}\log{D}\) \(\log{A}=K-\frac{2}{3}\log{D}\)
可以简单地推出“人口密度越高，区划面积越小”的结论，但显然\(D\)和\(A\)是有关联的，如果将\(D=\frac{P}{A}\)代回，则有：
\[A=\frac{4v^2h^2}{P^2g^2}\]与一般认知似乎不符…