Dynamic models of gentrification
绅士化的动态模型

We model the urban environment as a 7×7 grid, with each cell representing a city neighbourhood (Figure 1a). We populate this grid with $2^{12}$ agents, categorised into three socioeconomic groups: low-income ($L$), middle-income ($M$), and high-income ($H$). At the beginning of a simulation, each agent is assigned a fixed income $w$, sampled from real-world data, and all agents are divided into three groups according to their assigned incomes: $L$ accounting for 38% of the population; $M$ accounting for 57% of the population; and $H$ corresponding to the remaining 5% of the population of the model city. The method of income assignment is based on data from the 2022 USA Social Security Administration report[39] (see Methods for further details) and allows income variation within each agent class. Figure 1b shows one realisation of income assignment to the agents resulting in a heavy-tailed agent income distribution (more details in Methods).
我们将城市环境建模为7 × 7网格，每个单元格代表一个城市社区 (图1a)。我们用 $2^{12}$ 代理人填充这个网格，分为三个社会经济群体: 低收入 ( $L$ ) 、中等收入 ( $M$ ) 和高收入 ( $H$ )。在模拟开始时，每个代理被分配一个固定收入 $w$ ，从现实世界的数据中采样，所有代理根据其分配的收入分为三组: $L$ 占人口的38%； $M$ 占人口的57%; 和 $H$ 对应于模型城市人口的剩余5%。收入分配方法基于2022美国社会保障管理局报告 [39] 的数据 (更多细节请参见方法)，并允许每个代理人类别内的收入变化。 图1b显示了对代理商的收入分配的一种实现，从而导致了沉重的代理商收入分配 (方法中的更多细节)。

The spatial distribution of the agents follows a socioeconomic radial gradient: $H$ agents predominantly occupy the city centre, $M$ agents populate the inner areas, and $L$ agents are concentrated in the periphery (Figure 1a). This mono-centric structure reflects the presence of a dominant central business district, found in many cities or metropolitan areas of varying sizes[40, 41]. Each cell (neighbourhood) $j$ has a fixed maximum capacity $K$, which limits its occupancy, that is, how many agents can stay there. When $n(j)$, the number of agents in the cell $j$, reaches maximum capacity $K$, agents are allocated to the nearest available locations. Our model operates based on two key parameters: the parameter $p_{H}$ that corresponds to the probability that a $H$ agent relocates from its current cell to a new one where the median agent-income is increasing; the parameter $\epsilon$ that is the width of the time window over which $H$ agents evaluate cell growth trends. Our model simulates a 7x7 grid city comprising 49 neighbourhoods, a scale consistent with moderately large urban areas. To ensure the robustness of our findings, we extended our analysis to an even larger urban environment (9$\times$9 grid, 81 neighbourhoods), obtaining similar results (see Supplementary Information 4).
代理人的空间分布遵循社会经济径向梯度: $H$ 代理人主要占据市中心， $M$ 代理人居住在内部区域， $L$ 代理人集中在外围 (图1a)。这种单一中心的结构反映了一个占主导地位的中央商务区的存在，在许多城市或不同规模的大都市地区发现 [40，41]。每个小区 (邻域) $j$ 都有一个固定的最大容量 $K$ ，这限制了它的占用率，即有多少座席可以留在那里。当 $n(j)$ (单元格 $j$ 中的座席数) 达到最大容量 $K$ 时，座席被分配到最近的可用位置。 我们的模型基于两个关键参数运行: 参数 $p_{H}$ ，它对应于 $H$ 代理从其当前单元格重新定位到中值代理收入增加的新单元格的概率；参数 $\epsilon$ 是 $H$ 代理评估细胞生长趋势的时间窗口的宽度。我们的模型模拟了一个由49个社区组成的7x7网格城市，其规模与中等大的城市地区一致。为了确保研究结果的稳健性，我们将分析扩展到更大的城市环境 (9 $\times$ 9个网格，81个社区)，获得了类似的结果 (请参阅补充信息4)。

Our model implements income-dependent relocation rules for three agent classes across a set of cells, where each cell $i$ has a maximum capacity $K$. Let $n(i)$ denote the number of agents in cell $i$ at any given time. Figure LABEL:fig:fig_0a-f illustrates the rules behind agents’ behaviours.
我们的模型在一组单元格中为三个代理类实现了与收入相关的重定位规则，其中每个单元格 $i$ 具有最大容量 $K$ 。令 $n(i)$ 表示在任何给定时间单元格 $i$ 中的代理数量。图标签: 图: fig_0a-f说明了代理人行为背后的规则。

A $L$ agent is likely to move away from a cell if they are significantly poorer than other agents in the cell. The agent moves, with higher probability, to a cell where the income gap with current residents is smaller. The probability of a $L$ agent leaving cell $i$ at time $t$, shown in Figure LABEL:fig:fig_0a, is given by:
如果 $L$ 代理明显比细胞中的其他代理差，则它们可能会远离细胞。代理人以更高的概率移动到与当前居民收入差距较小的单元格。 $L$ 代理在时间 $t$ 离开单元格 $i$ 的概率，如图所示: fig_0a，由下式给出:

where $\gamma^{L}_{i}(t)\in[0,1]$ represents the agent’s relative income percentile within cell $i$’s income distribution at time $t$. Upon deciding to move, $L$ agents select from the set $\mathcal{J}$ of available cells, defined as cells where $n(j)<K$. As illustrated in Figure LABEL:fig:fig_0b, the probability of selecting cell $j$ is:
其中 $\gamma^{L}_{i}(t)\in[0,1]$ 表示代理在单元格 $i$ 的时间 $t$ 的收入分布中的相对收入百分位数。在决定移动时， $L$ 代理从可用单元的集合 $\mathcal{J}$ 中选择，定义为 $n(j)<K$ 的单元。如图所示: fig_0b，选择单元格 $j$ 的概率为:

where $q^{L}_{j}(t)$ is the cell’s attractiveness score based on the agent’s prospective income percentile in cell $j$.
其中 $q^{L}_{j}(t)$ 是单元格的吸引力分数，基于单元格 $j$ 中代理的预期收入百分位数。

A $M$ agent is likely to move away when its income significantly deviates from the median income in the current cell. The agent will move to a cell where the gap with the median income is lower. Their relocation probability, visualized in Figure LABEL:fig:fig_0c, follows::
当 $M$ 代理的收入明显偏离当前单元格中的收入中位数时，代理很可能会离开。代理人将移动到与中位数收入差距较低的单元格。它们的重新定位概率，在图标签中可视化: fig:fig_0c，如下:

where $\gamma^{M}_{i}(t)\in[0,1]$ is the $M$ agent’s relative income percentile in cell $i$. Figure LABEL:fig:fig_0d shows how their destination selection probability is determined by:
其中 $\gamma^{M}_{i}(t)\in[0,1]$ 是单元格 $i$ 中 $M$ 座席的相对收入百分位数。fig_0d显示了他们的目的地选择概率是如何确定的:

where $q^{M}_{j}(t)$ scores cells based on proximity to their median income.
其中 $q^{M}_{j}(t)$ 根据与收入中位数的接近程度对单元格进行评分。

For both $L$ and $M$ agents, we choose nonlinear functional forms (square root for $L$, quadratic for $M$) to ensure that small differences in percentiles lead to larger differences in probabilities when agents are far from their preferred positions. The denominators in $\rho^{L}_{j}(t)$ and $\rho^{M}_{j}(t)$ serve as normalisation factors, ensuring proper probability distributions.
对于 $L$ 和 $M$ 代理，我们都选择非线性函数形式 ( $L$ 的平方根，对于 $M$ 而言是二次的)，以确保当座席远离其首选位置时，百分位数的微小差异会导致更大的概率差异。 $\rho^{L}_{j}(t)$ 和 $\rho^{M}_{j}(t)$ 中的分母作为归一化因子，确保适当的概率分布。

A $H$ agent moves with a fixed probability $p_{H}\in[0,1]$ that reflects the frequency with which profit-orientated investments are made in the city. Let $\tilde{w}_{j}(t)$ denote the median income in cell $j$ at time $t$. $H$ agents select from set $\mathcal{H}$ of cells satisfying both $n(j)<K$ and $\phi_{j}^{\epsilon}(t)>\phi_{i}^{\epsilon}(t)$, where $\phi_{j}^{\epsilon}(t)$ is the average growth rate of median income over the past $\epsilon$ time steps, as reported in (Figure LABEL:fig:fig_0e-f):
$H$ 代理以固定概率 $p_{H}\in[0,1]$ 移动，该概率反映了在城市进行以利润为导向的投资的频率。令 $\tilde{w}_{j}(t)$ 表示单元格 $j$ 在时间 $t$ 中的中值收入。 $H$ 代理从满足 $n(j)<K$ 和 $\phi_{j}^{\epsilon}(t)>\phi_{i}^{\epsilon}(t)$ 的单元格集 $\mathcal{H}$ 中选择，其中 $\phi_{j}^{\epsilon}(t)$ 是过去 $\epsilon$ 时间步长中位数收入的平均增长率，如 (图标签: fig:fig_0e-f) 所示:

Their destination probability is:
他们的目的地概率是:

The simulation terminates at time $T$ when all $L$ agents can only find cells that would place them in the lowest income percentile, or after 300 time steps. We denote the termination time as T. All simulations were implemented using the Python library mesa[42]. The complete source code for model implementation and experimental procedures is available at: https://github.com/mauruscz/Gentrification
模拟在时间 $T$ 终止，此时所有 $L$ 代理只能找到将其置于最低收入百分位数的单元格，或在300个时间步长之后。我们将终止时间表示为T。所有模拟都是使用Python库mesa[ 42] 实现的。模型实现和实验过程的完整源代码可在: https://github.com/mauruscz/Gentrification

Gentrification is often defined as a period in which a neighbourhood (a cell in the grid and a node in the temporal network), previously populated by a majority of lower-income citizens, undergoes a gradual replacement of its population with middle- and higher-income citizens[10, 2]. To capture this process, we introduce a measure of gentrification for each node in the network, denoted as $\mathcal{G}_{count}^{i}(t,\Delta)$, based on the number of agents from each socioeconomic class present in node $i$ at two time points: the current time $t$ and an earlier time $t-\Delta$. For each time point, we consider the counts of $H$, $M$ and $L$ agents. At time $t$, these counts are represented by $n^{i}_{H}(t)$, $n^{i}_{M}(t)$, and $n^{i}_{L}(t)$, respectively. Similarly, at time $t-\Delta$, the counts are denoted by $n^{i}_{H}(t-\Delta)$, $n^{i}_{M}(t-\Delta)$, and $n^{i}_{L}(t-\Delta)$. The definition of $\mathcal{G}_{count}^{i}(t,\Delta)$ is as follows:
绅士化通常被定义为这样一个时期，在这个时期中，以前由大多数低收入公民居住的社区 (网格中的一个单元和时间网络中的一个节点) 逐渐被中等收入和高收入公民所取代 [10，2]。为了捕捉这个过程，我们为网络中的每个节点引入了一种绅士化度量，表示为 $\mathcal{G}_{count}^{i}(t,\Delta)$ ，基于节点 $i$ 中存在的每个社会经济类别的代理数量在两个时间点: 当前时间 $t$ 和较早时间 $t-\Delta$ 。对于每个时间点，我们考虑 $H$ ， $M$ 和 $L$ 代理的计数。在时间 $t$ ，这些计数分别由 $n^{i}_{H}(t)$ 、 $n^{i}_{M}(t)$ 和 $n^{i}_{L}(t)$ 表示。类似地，在时间 $t-\Delta$ ，计数由 $n^{i}_{H}(t-\Delta)$ 、 $n^{i}_{M}(t-\Delta)$ 和 $n^{i}_{L}(t-\Delta)$ 表示。 $\mathcal{G}_{count}^{i}(t,\Delta)$ 的定义如下:

$\mathcal{G}_{count}^{i}(t,\Delta)$ represents the average fraction of $H$ and $M$ agents in node $i$ over the time window $[t-\Delta,t]$. We establish a significance threshold $n_{H,M}^{*}$, defined as the expected node-wise fraction of $H$ and $M$ agents under uniform random distribution across the grid. Values of $\mathcal{G}_{count}^{i}(t,\Delta)$ exceeding $n_{H,M}^{*}$ indicate an over-representation of $H$ agents. Gentrification is identified when $\mathcal{G}_{count}^{i}(t,\Delta)$ transitions from below to above this threshold. This metric functions as a node property, independent of the network edge dynamics.
$\mathcal{G}_{count}^{i}(t,\Delta)$ 表示节点 $i$ 中的 $H$ 和 $M$ 代理在时间窗口 $[t-\Delta,t]$ 上的平均分数。我们建立了一个显著性阈值 $n_{H,M}^{*}$ ，定义为在网格上均匀随机分布下 $H$ 和 $M$ 代理的预期节点分数。 $\mathcal{G}_{count}^{i}(t,\Delta)$ 超过 $n_{H,M}^{*}$ 的值表示 $H$ 代理的过度表示。当 $\mathcal{G}_{count}^{i}(t,\Delta)$ 从低于该阈值转换到高于该阈值时，识别出绅士化。该度量用作节点属性，独立于网络边缘动态。

To identify gentrification, we establish a critical threshold $n_{M,H}^{*}=0.62$, representing the city-wide proportion of $M$ and $H$ agents. To precisely capture gentrification events, we introduce the binary indicator $\mathcal{G}_{bin}$:
为了识别绅士化，我们建立了一个临界阈值 $n_{M,H}^{*}=0.62$ ，代表 $M$ 和 $H$ 代理在全市范围内的比例。为了精确捕获高档化事件，我们引入了二进制指标 $\mathcal{G}_{bin}$ :

We define $t_{shift}$, the onset of a gentrification event, as the moment when $\mathcal{G}_{bin}$ shifts from 0 to 1, indicating that a neighborhood has crossed the critical $M,H$ population threshold. More mathematical details about $t_{shift}$ can be found in Methods.
我们将 $t_{shift}$ ，即绅士化事件的开始，定义为 $\mathcal{G}_{bin}$ 从0变为1的时刻，表明一个邻域已经越过临界 $M,H$ 人口阈值。关于 $t_{shift}$ 的更多数学细节可以在Methods中找到。

The count-based measure, while providing an intuitive quantification of gentrification, has notable limitations: it requires a significance threshold and only detects completed transitions, disregarding inter-neighbourhood dynamics. To overcome these constraints, we define a gentrification measure based on the temporal relocation network, $\mathcal{G}_{net}^{i}(t,\Delta)$ that captures relocation patterns between neighborhoods. For any node $i$ in the network, this measure considers the net-outflow of $L$ agents, $\varphi^{\text{out}}_{i}(t)$, and the net-inflow of $M$ and $H$ agents, $\varphi^{\text{in}}_{i}(t)$:
基于计数的测量虽然提供了对高档化的直观量化，但具有明显的局限性: 它需要一个显着性阈值，并且仅检测已完成的过渡，而忽略了邻域间的动态。为了克服这些限制，我们基于时间重定位网络 $\mathcal{G}_{net}^{i}(t,\Delta)$ 定义了一种绅士化度量，该度量捕获了邻域之间的重定位模式。对于网络中的任何节点 $i$ ，此措施都考虑 $L$ 代理， $\varphi^{\text{out}}_{i}(t)$ ， $M$ 和 $H$ 代理的净流入， $\varphi^{\text{in}}_{i}(t)$ :

$\varphi^{\text{out}}_{i}(\tau)$ is high when the outflow of $L$ agents from node $i$ corresponds to a high fraction of the overall outflow of agents from $i$, i.e., the denominator in Equation (11); $\varphi^{\text{in}}_{i}(\tau)$ is high when the inflow of $M$ and $H$ agents to node $i$ corresponds to a high fraction of the overall inflow of agents towards $i$, i.e., the denominator in Equation (12). $\mathcal{G}_{net}^{i}(t,\Delta)$ is therefore defined as the geometric mean of the averages of $\varphi^{\text{out}}_{i}(t)$ and $\varphi^{\text{in}}_{i}(t)$ over the last steps $\Delta$:
当来自节点 $i$ 的 $L$ 代理的流出对应于来自 $i$ 的代理的总流出的高比例时， $\varphi^{\text{out}}_{i}(\tau)$ 高，即等式 (11) 中的分母； $\varphi^{\text{in}}_{i}(\tau)$ 当 $M$ 和 $H$ 代理向节点 $i$ 的流入对应于代理向 $i$ 的总流入的大部分时， $\varphi^{\text{in}}_{i}(\tau)$ 高，即，式 (12) 中的分母。因此， $\mathcal{G}_{net}^{i}(t,\Delta)$ 被定义为 $\varphi^{\text{out}}_{i}(t)$ 和 $\varphi^{\text{in}}_{i}(t)$ 在最后一步的平均值的几何平均值 $\Delta$ :

In Equation (13), we consider only positive values of $\varphi^{\text{out}}_{i}(t)$ and $\varphi^{\text{in}}_{i}(t)$. This approach ensures that high values of $\mathcal{G}_{net}^{i}(t,\Delta)$ occur only when both the outflow of $L$ agents and the inflow of $M$ and $H$ agents are substantial during the same period. Including negative values would erroneously indicate gentrification in areas where $L$ agents replace $M$ and $H$ agents, which actually signals neighbourhood impoverishment.
在等式 (13) 中，我们仅考虑 $\varphi^{\text{out}}_{i}(t)$ 和 $\varphi^{\text{in}}_{i}(t)$ 的正值。这种方法确保只有当 $L$ 代理的流出和 $M$ 和 $H$ 代理的流入在同一时期大量时， $\mathcal{G}_{net}^{i}(t,\Delta)$ 才会出现高值。包括负值将错误地指示 $L$ 代理取代 $M$ 和 $H$ 代理的地区的绅士化，这实际上标志着邻里贫困。

We define $t_{peak}$ as the time of a gentrification event, corresponding to a local maximum in $\mathcal{G}_{net}^{i}(t,\Delta)$ or the onset of a plateau after rapid growth (see Methods for details).
我们将 $t_{peak}$ 定义为绅士化事件的时间，对应于 $\mathcal{G}_{net}^{i}(t,\Delta)$ 中的局部最大值或快速生长后平台的开始 (有关详细信息，请参见方法)。

To quantify gentrification at the city scale, we introduce two cumulative metrics: $\mathcal{G}_{bin}^{city}$, the percentage of nodes experiencing transitions in $\mathcal{G}_{bin}$, and $\mathcal{G}_{net}^{city}$, the percentage of nodes showing peaks in $\mathcal{G}_{net}$ (see Methods for details).
为了量化城市规模的绅士化，我们引入了两个累积指标: $\mathcal{G}_{bin}^{city}$ ，在 $\mathcal{G}_{bin}$ 和 $\mathcal{G}_{net}^{city}$ 中经历转换的节点的百分比，在 $\mathcal{G}_{net}$ 中显示峰的节点的百分比 (有关详细信息，请参见方法)。