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arXiv:2410.18004v1 [physics.soc-ph] 23 Oct 2024
arXiv:2410.18004v1 [physics.soc-ph] 2024年10月23日

Dynamic models of gentrification
绅士化的动态模型

Giovanni Mauro1,2,3,4, Nicola Pedreschi5, Renaud Lambiotte5, Luca Pappalardo1,2
Abstract 摘要

The phenomenon of gentrification of an urban area is characterized by the displacement of lower-income residents due to rising living costs and an influx of wealthier individuals. This study presents an agent-based model that simulates urban gentrification through the relocation of three income groups – low, middle, and high – driven by living costs. The model incorporates economic and sociological theories to generate realistic neighborhood transition patterns. We introduce a temporal network-based measure to track the outflow of low-income residents and the inflow of middle- and high-income residents over time. Our experiments reveal that high-income residents trigger gentrification and that our network-based measure consistently detects gentrification patterns earlier than traditional count-based methods, potentially serving as an early detection tool in real-world scenarios. Moreover, the analysis also highlights how city density promotes gentrification. This framework offers valuable insights for understanding gentrification dynamics and informing urban planning and policy decisions.
城市地区的高档化现象的特点是,由于生活成本上升和较富裕的个人涌入,低收入居民流离失所。这项研究提出了一个基于代理的模型,该模型通过生活成本驱动的三个收入群体的搬迁来模拟城市高档化-低,中,高。该模型结合了经济和社会学理论,以生成现实的邻里过渡模式。我们引入了一种基于时间网络的措施来跟踪低收入居民的流出以及随着时间的推移中高收入居民的流入。我们的实验表明,高收入居民会触发高档化,并且我们基于网络的措施比传统的基于计数的方法更早地检测出高档化模式,这可能是现实世界场景中的早期检测工具。此外,分析还强调了城市密度如何促进绅士化。 该框架为理解高档化动态以及为城市规划和政策决策提供了宝贵的见解。

1Institute of Information Science and Technologies (ISTI), National Research Council (CNR), Pisa, Italy
1 意大利比萨国家研究委员会 (CNR) 信息科学与技术研究所 (ISTI)

2Scuola Normale Superiore, Pisa, Italy
2 Scuola Normale Superiore,比萨,意大利

3Department of Computer Science, University of Pisa, Pisa, Italy
3 比萨大学计算机科学系,比萨,意大利

4IMT School for Advanced Studies, Lucca, Italy
4 IMT高级研究学院,卢卡,意大利

5Mathematical Institute, University of Oxford, United Kingdom
5 英国牛津大学数学研究所

1 Introduction 1简介

Cities are dynamic systems[1, 2, 3] in which the interactions between numerous agents determine the emergence of non-trivial patterns at different scales, such as traffic congestion[4], epidemic spreading[5, 6], and socioeconomic segregation[7, 8, 9]. Gentrification, first defined by Ruth Glass in 1964[10], describes the transformation of neighborhoods from working-class to affluent areas, often displacing original residents, and potentially undermining urban diversity and affordability[2]. Slater[11] divides gentrification research into two main strands: production-side and consumption-side theories. Both reject the view that gentrification is a benign return to urban centers[12, 13], with production-side theories linking it to economic factors such as the rent gap[14, 15]and housing quality decline[16]. Conversely, consumption-side theories emphasise the growing appeal of city centers[17], where proximity to urban amenities fuels demand. Ley[18] argues that artists, drawn by the cultural and social vitality of these areas, act as early catalysts for gentrification, eventually attracting wealthier residents and driving up property values[19].
城市是动态系统 [1,2,3],其中众多主体之间的相互作用决定了不同规模的非平凡模式的出现,例如交通拥堵 [4],流行病传播 [5,6] 和社会经济隔离 [7,8,9]。中产阶级化首先由露丝·格拉斯1964年定义 [10],描述了社区从工人阶级向富裕地区的转变,往往取代原居民,并可能破坏城市多样性和负担能力 [2]。Slater[ 11] 将绅士化研究分为两个主要方面: 生产方面和消费方面的理论。两者都拒绝认为高档化是对城市中心的良性回归 [12,13],生产方面的理论将其与租金差距 [14,15] 和住房质量下降 [16] 等经济因素联系起来。相反,消费侧理论强调了城市中心日益增长的吸引力 [17],靠近城市便利设施的地方推动了需求。 莱伊 [18] 认为,艺术家被这些地区的文化和社会活力所吸引,是中产阶级化的早期催化剂,最终吸引了更富裕的居民并推高了房地产价值 [19]。

Computational studies have blurred the lines between these perspectives, focusing on housing market dynamics, particularly fluctuations in rent and housing prices, as these reflect the real-world data used to validate their models. O’Sullivan[20] proposes a model incorporating housing markets, social networks, history, and policies, illustrating how gentrification operates in cycles influenced by these factors. Redfern[21] introduces the “investment gap,” emphasizing the difference between non-modernized homes and their potential if modernized, with domestic technologies driving gentrification. Other computational models simulate household movements[22], considering vacancies, accessibility, socioeconomic status, and urban policies[23, 24]. Alternatively, machine learning approaches have been implemented in an attempt to predict gentrification events[25, 26], in some limited cases taking into account proxies for human mobility in urban areas[27]. A recent work by Shaw et al. [28] showed how even a simple dynamical system model of gentrification, focused on neighborhood attractiveness and artist populations, can generate complex temporal patterns including synchronized oscillations and transient chaos.
计算研究模糊了这些观点之间的界限,重点关注住房市场动态,特别是租金和房价的波动,因为这些反映了用于验证其模型的现实数据。O'Sullivan[ 20] 提出了一个包含住房市场,社会网络,历史和政策的模型,说明了绅士化如何在受这些因素影响的周期中运作。Redfern[ 21] 引入了 “投资差距”,强调了非现代化住宅与现代化潜力之间的差异,国内技术推动了中产阶级化。其他计算模型模拟家庭流动 [22],考虑空缺,可达性,社会经济地位和城市政策 [23,24]。或者,机器学习方法已被实施以试图预测高档化事件 [25,26],在一些有限的情况下,考虑到城市地区人类移动性的代理 [27]。Shaw等人最近的工作。 [28] 展示了即使是一个简单的高档化动力系统模型,专注于邻里吸引力和艺术家群体,也可以产生复杂的时间模式,包括同步振荡和瞬态混沌。

In this study, we present an agent-based model of gentrification, inspired by the work of Schelling on urban segregation[7]. Rather than focusing on replicating housing market dynamics, we base our analysis of gentrification on the relocation flows of citizens in a stylised urban grid. Our model is founded on the key assumption that gentrification is driven by socioeconomic inequality and additionally by differing relocation strategies across income levels. Based on insights from the literature [29, 30, 31], we assume that agents relocate according to their socioeconomic conditions[32]: low-income agents move when priced out of a neighbourhood, medium-income agents gravitate toward areas with similar economic conditions and quality of life, while high-income agents are attracted to areas undergoing economic growth where they can maximise investment returns. These contrasting behaviours, along with a heavy-tailed income distribution, are the only drivers of neighbourhood transformations. While our agent-based model builds on simple rules, it generates complex and emergent dynamics, requiring a rigorous complex systems approach to quantify and interpret the multifaceted aspects of gentrification.
在这项研究中,我们提出了一个基于代理的绅士化模型,灵感来自谢林对城市隔离的工作 [7]。我们没有将重点放在复制住房市场动态上,而是将对高档化的分析基于程式化城市网格中公民的搬迁流。我们的模型基于以下关键假设: 高档化是由社会经济不平等以及不同收入水平的搬迁策略驱动的。根据文献 [29,30,31] 的见解,我们假设代理人根据他们的社会经济条件 [32] 重新安置: 低收入代理人在定价时离开社区,中等收入代理人倾向于经济条件和生活质量相似的地区,而高收入代理商被吸引到经济增长的地区,在那里他们可以最大限度地提高投资回报。 这些截然不同的行为,以及沉重的收入分配,是邻里转型的唯一驱动因素。虽然我们基于代理的模型建立在简单的规则上,但它会产生复杂而紧急的动态,需要严格的复杂系统方法来量化和解释高档化的多方面。

Within our modeling framework, we show that gentrification emerges only when high-income residents have some mobility, even if minimal, highlighting how their movement patterns catalyse the process. We treat relocation flows of agents in our city as time-varying edges in a temporal network[33, 34], leveraging established tools from network science and human mobility research[35, 36, 37, 38, 3]. We introduce two novel measures to quantify gentrification within our theoretical framework. The first measure translates the conventional definition of gentrification into a metric based on the over-representation of middle- and high-income agents in a given area. The second measure captures the dynamics of gentrification by tracking the inflow of these agents alongside the simultaneous outflow of low-income residents. Our findings demonstrate that this dynamic measure can consistently detect gentrification earlier than traditional count-based approaches, making it a potential early-warning indicator for policymakers aiming to mitigate its impacts. Additionally, our framework can simulate the effects of various urban planning strategies and city characteristics on gentrification. Notably, we observe a direct correlation between urban density and the frequency of gentrification events. Overall, our model and measures offer a comprehensive perspective on gentrification, shedding new light on this complex urban phenomenon.
在我们的建模框架中,我们表明,高档化只有在高收入居民有一定的流动性时才会出现,即使流动性很小,突出了他们的运动模式如何催化这一过程。我们利用网络科学和人类移动性研究 [35,36,37,38,3] 的既定工具,将我们城市中的代理的重新安置流视为时间网络中的时变边缘 [33,34]。我们在我们的理论框架内引入了两种量化绅士化的新方法。第一项措施将传统的高档化定义转换为基于给定地区中高收入代理商的过度代表的度量。第二项措施通过跟踪这些代理商的流入以及低收入居民的同时流出来捕获高档化的动态。 我们的研究结果表明,这种动态措施可以比传统的基于计数的方法更早地发现高档化,使其成为旨在减轻其影响的政策制定者的潜在预警指标。此外,我们的框架可以模拟各种城市规划策略和城市特征对高档化的影响。值得注意的是,我们观察到城市密度与高档化事件发生频率之间存在直接相关性。总体而言,我们的模型和措施为高档化提供了全面的视角,为这一复杂的城市现象提供了新的视角。

2 Agent-based model of gentrification
2基于代理的绅士化模型

Refer to caption
Figure 1: Income and spatial distribution: a) Heatmap reporting the median income of each cell at the beginning of a simulation. The darker the colour, the higher the median income of the cell. b) Log-log plot of the Initial distribution of the income, sampled from the 2022 USA Social Security Administration report[39], of the agents in a simulation with N = 212superscript2122^{12}2 start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPT agents.
图1: 收入和空间分布: a) 热图在模拟开始时报告每个单元格的收入中位数。颜色越深,单元格的收入中位数越高。b) 从2022的美国社会保障管理局报告 [39] 中采样的收入初始分配的对数-对数图,在N = 212superscript2122^{12}2 start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPT 代理的模拟中。

Modelling the city and its citizens.
城市和市民的模型。

We model the urban environment as a 7×7 grid, with each cell representing a city neighbourhood (Figure 1a). We populate this grid with 212superscript2122^{12}2 start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPT agents, categorised into three socioeconomic groups: low-income (L𝐿Litalic_L), middle-income (M𝑀Mitalic_M), and high-income (H𝐻Hitalic_H). At the beginning of a simulation, each agent is assigned a fixed income w𝑤witalic_w, sampled from real-world data, and all agents are divided into three groups according to their assigned incomes: L𝐿Litalic_L accounting for 38% of the population; M𝑀Mitalic_M accounting for 57% of the population; and H𝐻Hitalic_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 212superscript2122^{12}2 start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPT 代理人填充这个网格,分为三个社会经济群体: 低收入 ( LLitalic_L ) 、中等收入 ( MMitalic_M ) 和高收入 ( HHitalic_H )。在模拟开始时,每个代理被分配一个固定收入 wwitalic_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: LLitalic_L 占人口的38%; MMitalic_M 占人口的57%; 和 HHitalic_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: H𝐻Hitalic_H agents predominantly occupy the city centre, M𝑀Mitalic_M agents populate the inner areas, and L𝐿Litalic_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) j𝑗jitalic_j has a fixed maximum capacity K𝐾Kitalic_K, which limits its occupancy, that is, how many agents can stay there. When n(j)𝑛𝑗n(j)italic_n ( italic_j ), the number of agents in the cell j𝑗jitalic_j, reaches maximum capacity K𝐾Kitalic_K, agents are allocated to the nearest available locations. Our model operates based on two key parameters: the parameter pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT that corresponds to the probability that a H𝐻Hitalic_H agent relocates from its current cell to a new one where the median agent-income is increasing; the parameter ϵitalic-ϵ\epsilonitalic_ϵ that is the width of the time window over which H𝐻Hitalic_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: HHitalic_H 代理人主要占据市中心, MMitalic_M 代理人居住在内部区域, LLitalic_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) jjitalic_j 都有一个固定的最大容量 KKitalic_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)n(j)italic_n ( italic_j ) (单元格 jjitalic_j 中的座席数) 达到最大容量 KKitalic_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 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ,它对应于 HHitalic_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 ϵ\epsilonitalic_ϵHHitalic_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)。

[Uncaptioned image]

Agent relocation dynamics.
代理搬迁动态。

Our model implements income-dependent relocation rules for three agent classes across a set of cells, where each cell i𝑖iitalic_i has a maximum capacity K𝐾Kitalic_K. Let n(i)𝑛𝑖n(i)italic_n ( italic_i ) denote the number of agents in cell i𝑖iitalic_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 iiitalic_i 具有最大容量 KKitalic_K 。令 n(i)n(i)italic_n ( italic_i ) 表示在任何给定时间单元格 iiitalic_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 L𝐿Litalic_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 L𝐿Litalic_L agent leaving cell i𝑖iitalic_i at time t𝑡titalic_t, shown in Figure LABEL:fig:fig_0a, is given by:
如果 LLitalic_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。 LLitalic_L 代理在时间 ttitalic_t 离开单元格 iiitalic_i 的概率,如图所示: fig_0a,由下式给出:

piL(t)=1γiL(t)12,subscriptsuperscript𝑝𝐿𝑖𝑡1subscriptsuperscript𝛾𝐿𝑖superscript𝑡12p^{L}_{i}(t)=1-\gamma^{L}_{i}(t)^{\frac{1}{2}},italic_p start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) = 1 - italic_γ start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT , (1)

where γiL(t)[0,1]subscriptsuperscript𝛾𝐿𝑖𝑡01\gamma^{L}_{i}(t)\in[0,1]italic_γ start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ∈ [ 0 , 1 ] represents the agent’s relative income percentile within cell i𝑖iitalic_i’s income distribution at time t𝑡titalic_t. Upon deciding to move, L𝐿Litalic_L agents select from the set 𝒥𝒥\mathcal{J}caligraphic_J of available cells, defined as cells where n(j)<K𝑛𝑗𝐾n(j)<Kitalic_n ( italic_j ) < italic_K. As illustrated in Figure LABEL:fig:fig_0b, the probability of selecting cell j𝑗jitalic_j is:
其中 γiL(t)[0,1]subscriptsuperscript01\gamma^{L}_{i}(t)\in[0,1]italic_γ start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ∈ [ 0 , 1 ] 表示代理在单元格 iiitalic_i 的时间 ttitalic_t 的收入分布中的相对收入百分位数。在决定移动时, LLitalic_L 代理从可用单元的集合 𝒥\mathcal{J}caligraphic_J 中选择,定义为 n(j)<Kn(j)<Kitalic_n ( italic_j ) < italic_K 的单元。如图所示: fig_0b,选择单元格 jjitalic_j 的概率为:

ρjL(t)=qjL(t)n𝒥qnL(t),qjL(t)=γjL(t)12,formulae-sequencesubscriptsuperscript𝜌𝐿𝑗𝑡subscriptsuperscript𝑞𝐿𝑗𝑡subscript𝑛𝒥subscriptsuperscript𝑞𝐿𝑛𝑡subscriptsuperscript𝑞𝐿𝑗𝑡subscriptsuperscript𝛾𝐿𝑗superscript𝑡12\rho^{L}_{j}(t)=\frac{q^{L}_{j}(t)}{\sum_{n\in\mathcal{J}}q^{L}_{n}(t)},\quad q% ^{L}_{j}(t)=\gamma^{L}_{j}(t)^{\frac{1}{2}},italic_ρ start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) = divide start_ARG italic_q start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_J end_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) end_ARG , italic_q start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) = italic_γ start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT , (2)

where qjL(t)subscriptsuperscript𝑞𝐿𝑗𝑡q^{L}_{j}(t)italic_q start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) is the cell’s attractiveness score based on the agent’s prospective income percentile in cell j𝑗jitalic_j.
其中 qjL(t)subscriptsuperscriptq^{L}_{j}(t)italic_q start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) 是单元格的吸引力分数,基于单元格 jjitalic_j 中代理的预期收入百分位数。

A M𝑀Mitalic_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::
MMitalic_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:

piM(t)=4(γiM(t)0.5)2,subscriptsuperscript𝑝𝑀𝑖𝑡4superscriptsubscriptsuperscript𝛾𝑀𝑖𝑡0.52p^{M}_{i}(t)=4(\gamma^{M}_{i}(t)-0.5)^{2},italic_p start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) = 4 ( italic_γ start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) - 0.5 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , (3)

where γiM(t)[0,1]subscriptsuperscript𝛾𝑀𝑖𝑡01\gamma^{M}_{i}(t)\in[0,1]italic_γ start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ∈ [ 0 , 1 ] is the M𝑀Mitalic_M agent’s relative income percentile in cell i𝑖iitalic_i. Figure LABEL:fig:fig_0d shows how their destination selection probability is determined by:
其中 γiM(t)[0,1]subscriptsuperscript01\gamma^{M}_{i}(t)\in[0,1]italic_γ start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ∈ [ 0 , 1 ] 是单元格 iiitalic_iMMitalic_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:

ρjM(t)=qjM(t)n𝒥qnM(t),qjM(t)=14(γjM(t)0.5)2,formulae-sequencesubscriptsuperscript𝜌𝑀𝑗𝑡subscriptsuperscript𝑞𝑀𝑗𝑡subscript𝑛𝒥subscriptsuperscript𝑞𝑀𝑛𝑡subscriptsuperscript𝑞𝑀𝑗𝑡14superscriptsubscriptsuperscript𝛾𝑀𝑗𝑡0.52\rho^{M}_{j}(t)=\frac{q^{M}_{j}(t)}{\sum_{n\in\mathcal{J}}q^{M}_{n}(t)},\quad q% ^{M}_{j}(t)=1-4(\gamma^{M}_{j}(t)-0.5)^{2},italic_ρ start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) = divide start_ARG italic_q start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_J end_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) end_ARG , italic_q start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) = 1 - 4 ( italic_γ start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) - 0.5 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , (4)

where qjM(t)subscriptsuperscript𝑞𝑀𝑗𝑡q^{M}_{j}(t)italic_q start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) scores cells based on proximity to their median income.
其中 qjM(t)subscriptsuperscriptq^{M}_{j}(t)italic_q start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) 根据与收入中位数的接近程度对单元格进行评分。

For both L𝐿Litalic_L and M𝑀Mitalic_M agents, we choose nonlinear functional forms (square root for L𝐿Litalic_L, quadratic for M𝑀Mitalic_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 ρjL(t)subscriptsuperscript𝜌𝐿𝑗𝑡\rho^{L}_{j}(t)italic_ρ start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) and ρjM(t)subscriptsuperscript𝜌𝑀𝑗𝑡\rho^{M}_{j}(t)italic_ρ start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) serve as normalisation factors, ensuring proper probability distributions.
对于 LLitalic_LMMitalic_M 代理,我们都选择非线性函数形式 ( LLitalic_L 的平方根,对于 MMitalic_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。 ρjL(t)subscriptsuperscript\rho^{L}_{j}(t)italic_ρ start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t )ρjM(t)subscriptsuperscript\rho^{M}_{j}(t)italic_ρ start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) 中的分母作为归一化因子,确保适当的概率分布。

A H𝐻Hitalic_H agent moves with a fixed probability pH[0,1]subscript𝑝𝐻01p_{H}\in[0,1]italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ∈ [ 0 , 1 ] that reflects the frequency with which profit-orientated investments are made in the city. Let w~j(t)subscript~𝑤𝑗𝑡\tilde{w}_{j}(t)over~ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) denote the median income in cell j𝑗jitalic_j at time t𝑡titalic_t. H𝐻Hitalic_H agents select from set \mathcal{H}caligraphic_H of cells satisfying both n(j)<K𝑛𝑗𝐾n(j)<Kitalic_n ( italic_j ) < italic_K and ϕjϵ(t)>ϕiϵ(t)superscriptsubscriptitalic-ϕ𝑗italic-ϵ𝑡superscriptsubscriptitalic-ϕ𝑖italic-ϵ𝑡\phi_{j}^{\epsilon}(t)>\phi_{i}^{\epsilon}(t)italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT ( italic_t ) > italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT ( italic_t ), where ϕjϵ(t)superscriptsubscriptitalic-ϕ𝑗italic-ϵ𝑡\phi_{j}^{\epsilon}(t)italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT ( italic_t ) is the average growth rate of median income over the past ϵitalic-ϵ\epsilonitalic_ϵ time steps, as reported in (Figure LABEL:fig:fig_0e-f):
HHitalic_H 代理以固定概率 pH[0,1]subscript01p_{H}\in[0,1]italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ∈ [ 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 w~j(t)subscript\tilde{w}_{j}(t)over~ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) 表示单元格 jjitalic_j 在时间 ttitalic_t 中的中值收入。 HHitalic_H 代理从满足 n(j)<Kn(j)<Kitalic_n ( italic_j ) < italic_Kϕjϵ(t)>ϕiϵ(t)superscriptsubscriptsuperscriptsubscript\phi_{j}^{\epsilon}(t)>\phi_{i}^{\epsilon}(t)italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT ( italic_t ) > italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT ( italic_t ) 的单元格集 \mathcal{H}caligraphic_H 中选择,其中 ϕjϵ(t)superscriptsubscript\phi_{j}^{\epsilon}(t)italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT ( italic_t ) 是过去 ϵ\epsilonitalic_ϵ 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:

ϕjϵ(t)=1ϵτ=tϵt[w~j(τ)w~j(τ1)].superscriptsubscriptitalic-ϕ𝑗italic-ϵ𝑡1italic-ϵsuperscriptsubscript𝜏𝑡italic-ϵ𝑡delimited-[]subscript~𝑤𝑗𝜏subscript~𝑤𝑗𝜏1\phi_{j}^{\epsilon}(t)=\frac{1}{\epsilon}\sum_{\tau=t-\epsilon}^{t}[\tilde{w}_% {j}(\tau)-\tilde{w}_{j}(\tau-1)].italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT ( italic_t ) = divide start_ARG 1 end_ARG start_ARG italic_ϵ end_ARG ∑ start_POSTSUBSCRIPT italic_τ = italic_t - italic_ϵ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT [ over~ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_τ ) - over~ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_τ - 1 ) ] . (5)

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

qjH(t)=ϕjϵ(t)nϕnϵ(t).subscriptsuperscript𝑞𝐻𝑗𝑡superscriptsubscriptitalic-ϕ𝑗italic-ϵ𝑡subscript𝑛superscriptsubscriptitalic-ϕ𝑛italic-ϵ𝑡q^{H}_{j}(t)=\frac{\phi_{j}^{\epsilon}(t)}{\sum_{n\in\mathcal{H}}\phi_{n}^{% \epsilon}(t)}.italic_q start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) = divide start_ARG italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT ( italic_t ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_H end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT ( italic_t ) end_ARG . (6)

The simulation terminates at time T𝑇Titalic_T when all L𝐿Litalic_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 TTitalic_T 终止,此时所有 LLitalic_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

3 Measures of gentrification
3高档化措施

Our gentrification model generates dynamic flows of agents of the three types among grid cells (see Figure LABEL:fig:fig_0g). We model these time-varying flows as a dynamic network where nodes represent grid cells and edges correspond to movements of agents of the three types. The network consists of three layers, each representing flows of L𝐿Litalic_L, M𝑀Mitalic_M, or H𝐻Hitalic_H agents. This representation results in a temporal network[43, 44, 33],which is multi-layer (one per each edge type)[45] and weighted[46], defined as follows:
我们的绅士化模型在网格单元之间生成三种类型的代理的动态流 (见图标签: fig_0g)。我们将这些时变流建模为动态网络,其中节点表示网格单元,而边缘对应于三种类型的主体的运动。网络由三个层组成,每个层代表 LLitalic_LMMitalic_MHHitalic_H 代理的流。这种表示产生了一个时间网络 [43,44,33],它是多层的 (每个边缘类型一个)[ 45] 和加权的 [46],定义如下:

Gα(t)={Ai,jα(t)|i,jV;α{L,M,H};t[0,T]}superscriptG𝛼𝑡conditional-setsuperscriptsubscript𝐴𝑖𝑗𝛼𝑡formulae-sequence𝑖𝑗𝑉formulae-sequence𝛼𝐿𝑀𝐻𝑡0𝑇\text{G}^{\alpha}(t)=\Big{\{}A_{i,j}^{\alpha}(t)|i,j\in V;\alpha\in\{L,M,H\};t% \in[0,T]\Big{\}}G start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_t ) = { italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_t ) | italic_i , italic_j ∈ italic_V ; italic_α ∈ { italic_L , italic_M , italic_H } ; italic_t ∈ [ 0 , italic_T ] } (7)

where V𝑉Vitalic_V denotes the set of nodes (grid cells) in the network, α𝛼\alphaitalic_α represents the layer index corresponding to L𝐿Litalic_L, M𝑀Mitalic_M or H𝐻Hitalic_H agents, and Aα(t)superscript𝐴𝛼𝑡A^{\alpha}(t)italic_A start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_t ) is the adjacency matrix corresponding to the network in layer α𝛼\alphaitalic_α at time t𝑡titalic_t. The matrix elements of 𝐀α(t)superscript𝐀𝛼𝑡\mathbf{A}^{\alpha}(t)bold_A start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_t ) correspond to weighted, directed edges connecting node pairs in layer α𝛼\alphaitalic_α, representing the relocation flows of agents of type α𝛼\alphaitalic_α between contiguous time steps of the agent-based model. For each layer α[L,M,H]𝛼𝐿𝑀𝐻\alpha\in[L,M,H]italic_α ∈ [ italic_L , italic_M , italic_H ] at time t𝑡titalic_t, we define three quantities: nαi(t)subscriptsuperscript𝑛𝑖𝛼𝑡n^{i}_{\alpha}(t)italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_t ), the number of agents in node i𝑖iitalic_i, and si,αin(t)subscriptsuperscript𝑠in𝑖𝛼𝑡s^{\text{in}}_{i,\alpha}(t)italic_s start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_α end_POSTSUBSCRIPT ( italic_t ) and si,αout(t)subscriptsuperscript𝑠out𝑖𝛼𝑡s^{\text{out}}_{i,\alpha}(t)italic_s start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_α end_POSTSUBSCRIPT ( italic_t ), the in- and out-strength of node i𝑖iitalic_i, respectively. The latter two quantify the flow of agents in [L,M,H]𝐿𝑀𝐻[L,M,H][ italic_L , italic_M , italic_H ] moving to or from node i𝑖iitalic_i between t1𝑡1t-1italic_t - 1 and t𝑡titalic_t:
其中 VVitalic_V 表示网络中的节点 (网格单元) 集合, α\alphaitalic_α 表示与 LLitalic_LMMitalic_MHHitalic_H 代理对应的层索引,并且 Aα(t)superscriptA^{\alpha}(t)italic_A start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_t ) 是在 ttitalic_t 时刻 α\alphaitalic_α 层中的网络对应的邻接矩阵。 𝐀α(t)superscript\mathbf{A}^{\alpha}(t)bold_A start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_t ) 的矩阵元素对应于连接层 α\alphaitalic_α 中的节点对的加权有向边,表示基于代理的模型的连续时间步长之间 α\alphaitalic_α 类型的代理的重定位流。对于时间 ttitalic_t 的每个层 α[L,M,H]\alpha\in[L,M,H]italic_α ∈ [ italic_L , italic_M , italic_H ] ,我们定义三个量: nαi(t)subscriptsuperscriptn^{i}_{\alpha}(t)italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_t ) ,节点中的代理数 iiitalic_isi,αin(t)subscriptsuperscripts^{\text{in}}_{i,\alpha}(t)italic_s start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_α end_POSTSUBSCRIPT ( italic_t )si,αout(t)subscriptsuperscripts^{\text{out}}_{i,\alpha}(t)italic_s start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_α end_POSTSUBSCRIPT ( italic_t ) ,分别是节点 iiitalic_i 的输入和输出强度。后两者量化了 [L,M,H][L,M,H][ italic_L , italic_M , italic_H ]t11t-1italic_t - 1ttitalic_t 之间往返于节点 iiitalic_i 的代理流:

si,αin(t)=iAi,jα(t),si,αout(t)=jAi,jα(t),α=L,M,H.formulae-sequencesubscriptsuperscript𝑠in𝑖𝛼𝑡subscript𝑖subscriptsuperscript𝐴𝛼𝑖𝑗𝑡formulae-sequencesubscriptsuperscript𝑠out𝑖𝛼𝑡subscript𝑗subscriptsuperscript𝐴𝛼𝑖𝑗𝑡𝛼𝐿𝑀𝐻\begin{split}s^{\text{in}}_{i,\alpha}(t)&=\sum_{i}A^{\alpha}_{i,j}(t),\\ s^{\text{out}}_{i,\alpha}(t)&=\sum_{j}A^{\alpha}_{i,j}(t),\\ \alpha&=L,M,H.\end{split}start_ROW start_CELL italic_s start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_α end_POSTSUBSCRIPT ( italic_t ) end_CELL start_CELL = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_t ) , end_CELL end_ROW start_ROW start_CELL italic_s start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_α end_POSTSUBSCRIPT ( italic_t ) end_CELL start_CELL = ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_t ) , end_CELL end_ROW start_ROW start_CELL italic_α end_CELL start_CELL = italic_L , italic_M , italic_H . end_CELL end_ROW (8)

where Ai,jα(t)subscriptsuperscript𝐴𝛼𝑖𝑗𝑡A^{\alpha}_{i,j}(t)italic_A start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_t ) represents the (i,j)𝑖𝑗(i,j)( italic_i , italic_j ) element of the weighted adjacency matrix for layer α𝛼\alphaitalic_α at time t𝑡titalic_t.
其中 Ai,jα(t)subscriptsuperscriptA^{\alpha}_{i,j}(t)italic_A start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( italic_t ) 表示层 α\alphaitalic_α 在时间 ttitalic_t 的加权邻接矩阵的 (i,j)(i,j)( italic_i , italic_j ) 元素。

Count-based measure. 基于计数的度量。

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 𝒢counti(t,Δ)superscriptsubscript𝒢𝑐𝑜𝑢𝑛𝑡𝑖𝑡Δ\mathcal{G}_{count}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ), based on the number of agents from each socioeconomic class present in node i𝑖iitalic_i at two time points: the current time t𝑡titalic_t and an earlier time tΔ𝑡Δt-\Deltaitalic_t - roman_Δ. For each time point, we consider the counts of H𝐻Hitalic_H, M𝑀Mitalic_M and L𝐿Litalic_L agents. At time t𝑡titalic_t, these counts are represented by nHi(t)subscriptsuperscript𝑛𝑖𝐻𝑡n^{i}_{H}(t)italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_t ), nMi(t)subscriptsuperscript𝑛𝑖𝑀𝑡n^{i}_{M}(t)italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ( italic_t ), and nLi(t)subscriptsuperscript𝑛𝑖𝐿𝑡n^{i}_{L}(t)italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_t ), respectively. Similarly, at time tΔ𝑡Δt-\Deltaitalic_t - roman_Δ, the counts are denoted by nHi(tΔ)subscriptsuperscript𝑛𝑖𝐻𝑡Δn^{i}_{H}(t-\Delta)italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_t - roman_Δ ), nMi(tΔ)subscriptsuperscript𝑛𝑖𝑀𝑡Δn^{i}_{M}(t-\Delta)italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ( italic_t - roman_Δ ), and nLi(tΔ)subscriptsuperscript𝑛𝑖𝐿𝑡Δn^{i}_{L}(t-\Delta)italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_t - roman_Δ ). The definition of 𝒢counti(t,Δ)superscriptsubscript𝒢𝑐𝑜𝑢𝑛𝑡𝑖𝑡Δ\mathcal{G}_{count}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) 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 𝒢counti(t,Δ)superscriptsubscript\mathcal{G}_{count}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) ,基于节点 iiitalic_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 ttitalic_t 和较早时间 tΔt-\Deltaitalic_t - roman_Δ 。对于每个时间点,我们考虑 HHitalic_HMMitalic_MLLitalic_L 代理的计数。在时间 ttitalic_t ,这些计数分别由 nHi(t)subscriptsuperscriptn^{i}_{H}(t)italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_t )nMi(t)subscriptsuperscriptn^{i}_{M}(t)italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ( italic_t )nLi(t)subscriptsuperscriptn^{i}_{L}(t)italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_t ) 表示。类似地,在时间 tΔt-\Deltaitalic_t - roman_Δ ,计数由 nHi(tΔ)subscriptsuperscriptn^{i}_{H}(t-\Delta)italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_t - roman_Δ )nMi(tΔ)subscriptsuperscriptn^{i}_{M}(t-\Delta)italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ( italic_t - roman_Δ )nLi(tΔ)subscriptsuperscriptn^{i}_{L}(t-\Delta)italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_t - roman_Δ ) 表示。 𝒢counti(t,Δ)superscriptsubscript\mathcal{G}_{count}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) 的定义如下:

𝒢counti(t,Δ)=1Δτ=tΔtnHi(τ)+nMi(τ)nHi(τ)+nMi(τ)+nLi(τ)superscriptsubscript𝒢𝑐𝑜𝑢𝑛𝑡𝑖𝑡Δ1Δsuperscriptsubscript𝜏𝑡Δ𝑡subscriptsuperscript𝑛𝑖𝐻𝜏subscriptsuperscript𝑛𝑖𝑀𝜏subscriptsuperscript𝑛𝑖𝐻𝜏subscriptsuperscript𝑛𝑖𝑀𝜏subscriptsuperscript𝑛𝑖𝐿𝜏\mathcal{G}_{count}^{i}(t,\Delta)=\frac{1}{\Delta}\sum_{\tau=t-\Delta}^{t}% \frac{n^{i}_{H}(\tau)+n^{i}_{M}(\tau)}{n^{i}_{H}(\tau)+n^{i}_{M}(\tau)+n^{i}_{% L}(\tau)}caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) = divide start_ARG 1 end_ARG start_ARG roman_Δ end_ARG ∑ start_POSTSUBSCRIPT italic_τ = italic_t - roman_Δ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT divide start_ARG italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_τ ) + italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ( italic_τ ) end_ARG start_ARG italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_τ ) + italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ( italic_τ ) + italic_n start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_τ ) end_ARG (9)

𝒢counti(t,Δ)superscriptsubscript𝒢𝑐𝑜𝑢𝑛𝑡𝑖𝑡Δ\mathcal{G}_{count}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) represents the average fraction of H𝐻Hitalic_H and M𝑀Mitalic_M agents in node i𝑖iitalic_i over the time window [tΔ,t]𝑡Δ𝑡[t-\Delta,t][ italic_t - roman_Δ , italic_t ]. We establish a significance threshold nH,Msuperscriptsubscript𝑛𝐻𝑀n_{H,M}^{*}italic_n start_POSTSUBSCRIPT italic_H , italic_M end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT, defined as the expected node-wise fraction of H𝐻Hitalic_H and M𝑀Mitalic_M agents under uniform random distribution across the grid. Values of 𝒢counti(t,Δ)superscriptsubscript𝒢𝑐𝑜𝑢𝑛𝑡𝑖𝑡Δ\mathcal{G}_{count}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) exceeding nH,Msuperscriptsubscript𝑛𝐻𝑀n_{H,M}^{*}italic_n start_POSTSUBSCRIPT italic_H , italic_M end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT indicate an over-representation of H𝐻Hitalic_H agents. Gentrification is identified when 𝒢counti(t,Δ)superscriptsubscript𝒢𝑐𝑜𝑢𝑛𝑡𝑖𝑡Δ\mathcal{G}_{count}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) transitions from below to above this threshold. This metric functions as a node property, independent of the network edge dynamics.
𝒢counti(t,Δ)superscriptsubscript\mathcal{G}_{count}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) 表示节点 iiitalic_i 中的 HHitalic_HMMitalic_M 代理在时间窗口 [tΔ,t][t-\Delta,t][ italic_t - roman_Δ , italic_t ] 上的平均分数。我们建立了一个显著性阈值 nH,Msuperscriptsubscriptn_{H,M}^{*}italic_n start_POSTSUBSCRIPT italic_H , italic_M end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ,定义为在网格上均匀随机分布下 HHitalic_HMMitalic_M 代理的预期节点分数。 𝒢counti(t,Δ)superscriptsubscript\mathcal{G}_{count}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) 超过 nH,Msuperscriptsubscriptn_{H,M}^{*}italic_n start_POSTSUBSCRIPT italic_H , italic_M end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT 的值表示 HHitalic_H 代理的过度表示。当 𝒢counti(t,Δ)superscriptsubscript\mathcal{G}_{count}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) 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 nM,H=0.62superscriptsubscript𝑛𝑀𝐻0.62n_{M,H}^{*}=0.62italic_n start_POSTSUBSCRIPT italic_M , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = 0.62, representing the city-wide proportion of M𝑀Mitalic_M and H𝐻Hitalic_H agents. To precisely capture gentrification events, we introduce the binary indicator 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT:
为了识别绅士化,我们建立了一个临界阈值 nM,H=0.62superscriptsubscript0.62n_{M,H}^{*}=0.62italic_n start_POSTSUBSCRIPT italic_M , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = 0.62 ,代表 MMitalic_MHHitalic_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 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT :

𝒢bin(t,Δ)={1,if 𝒢count(t,Δ)>nH,M.0,otherwise.subscript𝒢𝑏𝑖𝑛𝑡Δcases1if 𝒢count(t,Δ)>nH,M0otherwise\mathcal{G}_{bin}(t,\Delta)=\begin{cases}1,&\text{if $\mathcal{G}_{count}(t,% \Delta)>n_{H,M}^{*}$}.\\ 0,&\text{otherwise}.\end{cases}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT ( italic_t , roman_Δ ) = { start_ROW start_CELL 1 , end_CELL start_CELL if caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT ( italic_t , roman_Δ ) > italic_n start_POSTSUBSCRIPT italic_H , italic_M end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT . end_CELL end_ROW start_ROW start_CELL 0 , end_CELL start_CELL otherwise . end_CELL end_ROW (10)

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

Network-based measure. 基于网络的度量。

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, 𝒢neti(t,Δ)superscriptsubscript𝒢𝑛𝑒𝑡𝑖𝑡Δ\mathcal{G}_{net}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) that captures relocation patterns between neighborhoods. For any node i𝑖iitalic_i in the network, this measure considers the net-outflow of L𝐿Litalic_L agents, φiout(t)subscriptsuperscript𝜑out𝑖𝑡\varphi^{\text{out}}_{i}(t)italic_φ start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ), and the net-inflow of M𝑀Mitalic_M and H𝐻Hitalic_H agents, φiin(t)subscriptsuperscript𝜑in𝑖𝑡\varphi^{\text{in}}_{i}(t)italic_φ start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_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 𝒢neti(t,Δ)superscriptsubscript\mathcal{G}_{net}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) 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 iiitalic_i ,此措施都考虑 LLitalic_L 代理, φiout(t)subscriptsuperscript\varphi^{\text{out}}_{i}(t)italic_φ start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t )MMitalic_MHHitalic_H 代理的净流入, φiin(t)subscriptsuperscript\varphi^{\text{in}}_{i}(t)italic_φ start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) :

φiout(t)si,Lout(t)si,Lin(t)si,Lout(t)+si,Mout(t)+si,Hout(t)subscriptsuperscript𝜑out𝑖𝑡subscriptsuperscript𝑠out𝑖𝐿𝑡subscriptsuperscript𝑠in𝑖𝐿𝑡subscriptsuperscript𝑠out𝑖𝐿𝑡subscriptsuperscript𝑠out𝑖𝑀𝑡subscriptsuperscript𝑠out𝑖𝐻𝑡\varphi^{\text{out}}_{i}(t)\equiv\frac{s^{\text{out}}_{i,L}(t)-s^{\text{in}}_{% i,L}(t)}{s^{\text{out}}_{i,L}(t)+s^{\text{out}}_{i,M}(t)+s^{\text{out}}_{i,H}(% t)}italic_φ start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ≡ divide start_ARG italic_s start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_L end_POSTSUBSCRIPT ( italic_t ) - italic_s start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_L end_POSTSUBSCRIPT ( italic_t ) end_ARG start_ARG italic_s start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_L end_POSTSUBSCRIPT ( italic_t ) + italic_s start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_M end_POSTSUBSCRIPT ( italic_t ) + italic_s start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_H end_POSTSUBSCRIPT ( italic_t ) end_ARG (11)
φiin(t)si,M,Hin(t)si,M,Hout(t)si,Lin+si,Min(t)+si,Hin(t)subscriptsuperscript𝜑in𝑖𝑡subscriptsuperscript𝑠in𝑖𝑀𝐻𝑡subscriptsuperscript𝑠out𝑖𝑀𝐻𝑡subscriptsuperscript𝑠in𝑖𝐿subscriptsuperscript𝑠in𝑖𝑀𝑡subscriptsuperscript𝑠in𝑖𝐻𝑡\varphi^{\text{in}}_{i}(t)\equiv\frac{s^{\text{in}}_{i,M,H}(t)-s^{\text{out}}_% {i,M,H}(t)}{s^{\text{in}}_{i,L}+s^{\text{in}}_{i,M}(t)+s^{\text{in}}_{i,H}(t)}italic_φ start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ≡ divide start_ARG italic_s start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_M , italic_H end_POSTSUBSCRIPT ( italic_t ) - italic_s start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_M , italic_H end_POSTSUBSCRIPT ( italic_t ) end_ARG start_ARG italic_s start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_L end_POSTSUBSCRIPT + italic_s start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_M end_POSTSUBSCRIPT ( italic_t ) + italic_s start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_H end_POSTSUBSCRIPT ( italic_t ) end_ARG (12)

where si,M,Hin(t)=si,Min(t)+si,Hin(t)subscriptsuperscript𝑠in𝑖𝑀𝐻𝑡subscriptsuperscript𝑠in𝑖𝑀𝑡subscriptsuperscript𝑠in𝑖𝐻𝑡s^{\text{in}}_{i,M,H}(t)=s^{\text{in}}_{i,M}(t)+s^{\text{in}}_{i,H}(t)italic_s start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_M , italic_H end_POSTSUBSCRIPT ( italic_t ) = italic_s start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_M end_POSTSUBSCRIPT ( italic_t ) + italic_s start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_H end_POSTSUBSCRIPT ( italic_t ) and si,M,Hout(t)=si,Mout(t)si,Hout(t)subscriptsuperscript𝑠out𝑖𝑀𝐻𝑡subscriptsuperscript𝑠out𝑖𝑀𝑡subscriptsuperscript𝑠out𝑖𝐻𝑡s^{\text{out}}_{i,M,H}(t)=s^{\text{out}}_{i,M}(t)-s^{\text{out}}_{i,H}(t)italic_s start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_M , italic_H end_POSTSUBSCRIPT ( italic_t ) = italic_s start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_M end_POSTSUBSCRIPT ( italic_t ) - italic_s start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_H end_POSTSUBSCRIPT ( italic_t ).
其中 si,M,Hin(t)=si,Min(t)+si,Hin(t)subscriptsuperscriptsubscriptsuperscriptsubscriptsuperscripts^{\text{in}}_{i,M,H}(t)=s^{\text{in}}_{i,M}(t)+s^{\text{in}}_{i,H}(t)italic_s start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_M , italic_H end_POSTSUBSCRIPT ( italic_t ) = italic_s start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_M end_POSTSUBSCRIPT ( italic_t ) + italic_s start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_H end_POSTSUBSCRIPT ( italic_t )si,M,Hout(t)=si,Mout(t)si,Hout(t)subscriptsuperscriptsubscriptsuperscriptsubscriptsuperscripts^{\text{out}}_{i,M,H}(t)=s^{\text{out}}_{i,M}(t)-s^{\text{out}}_{i,H}(t)italic_s start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_M , italic_H end_POSTSUBSCRIPT ( italic_t ) = italic_s start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_M end_POSTSUBSCRIPT ( italic_t ) - italic_s start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_H end_POSTSUBSCRIPT ( italic_t )

φiout(τ)subscriptsuperscript𝜑out𝑖𝜏\varphi^{\text{out}}_{i}(\tau)italic_φ start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_τ ) is high when the outflow of L𝐿Litalic_L agents from node i𝑖iitalic_i corresponds to a high fraction of the overall outflow of agents from i𝑖iitalic_i, i.e., the denominator in Equation (11); φiin(τ)subscriptsuperscript𝜑in𝑖𝜏\varphi^{\text{in}}_{i}(\tau)italic_φ start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_τ ) is high when the inflow of M𝑀Mitalic_M and H𝐻Hitalic_H agents to node i𝑖iitalic_i corresponds to a high fraction of the overall inflow of agents towards i𝑖iitalic_i, i.e., the denominator in Equation (12). 𝒢neti(t,Δ)superscriptsubscript𝒢𝑛𝑒𝑡𝑖𝑡Δ\mathcal{G}_{net}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) is therefore defined as the geometric mean of the averages of φiout(t)subscriptsuperscript𝜑out𝑖𝑡\varphi^{\text{out}}_{i}(t)italic_φ start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) and φiin(t)subscriptsuperscript𝜑in𝑖𝑡\varphi^{\text{in}}_{i}(t)italic_φ start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) over the last steps ΔΔ\Deltaroman_Δ:
当来自节点 iiitalic_iLLitalic_L 代理的流出对应于来自 iiitalic_i 的代理的总流出的高比例时, φiout(τ)subscriptsuperscript\varphi^{\text{out}}_{i}(\tau)italic_φ start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_τ ) 高,即等式 (11) 中的分母; φiin(τ)subscriptsuperscript\varphi^{\text{in}}_{i}(\tau)italic_φ start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_τ )MMitalic_MHHitalic_H 代理向节点 iiitalic_i 的流入对应于代理向 iiitalic_i 的总流入的大部分时, φiin(τ)subscriptsuperscript\varphi^{\text{in}}_{i}(\tau)italic_φ start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_τ ) 高,即,式 (12) 中的分母。因此, 𝒢neti(t,Δ)superscriptsubscript\mathcal{G}_{net}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) 被定义为 φiout(t)subscriptsuperscript\varphi^{\text{out}}_{i}(t)italic_φ start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t )φiin(t)subscriptsuperscript\varphi^{\text{in}}_{i}(t)italic_φ start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) 在最后一步的平均值的几何平均值 Δ\Deltaroman_Δ :

𝒢neti(t,Δ)(1Δτ=tΔtφiout(τ))(1Δτ=tΔtφiin(τ)).superscriptsubscript𝒢𝑛𝑒𝑡𝑖𝑡Δ1Δsuperscriptsubscript𝜏𝑡Δ𝑡subscriptsuperscript𝜑out𝑖𝜏1Δsuperscriptsubscript𝜏𝑡Δ𝑡subscriptsuperscript𝜑in𝑖𝜏\mathcal{G}_{net}^{i}(t,\Delta)\equiv\sqrt{\Bigg{(}\frac{1}{\Delta}\sum_{\tau=% t-\Delta}^{t}\varphi^{\text{out}}_{i}(\tau)\Bigg{)}\cdot\Bigg{(}\frac{1}{% \Delta}\sum_{\tau=t-\Delta}^{t}\varphi^{\text{in}}_{i}(\tau)\Bigg{)}}.caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) ≡ square-root start_ARG ( divide start_ARG 1 end_ARG start_ARG roman_Δ end_ARG ∑ start_POSTSUBSCRIPT italic_τ = italic_t - roman_Δ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_φ start_POSTSUPERSCRIPT out end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_τ ) ) ⋅ ( divide start_ARG 1 end_ARG start_ARG roman_Δ end_ARG ∑ start_POSTSUBSCRIPT italic_τ = italic_t - roman_Δ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_φ start_POSTSUPERSCRIPT in end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_τ ) ) end_ARG . (13)

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

We define tpeaksubscript𝑡𝑝𝑒𝑎𝑘t_{peak}italic_t start_POSTSUBSCRIPT italic_p italic_e italic_a italic_k end_POSTSUBSCRIPT as the time of a gentrification event, corresponding to a local maximum in 𝒢neti(t,Δ)superscriptsubscript𝒢𝑛𝑒𝑡𝑖𝑡Δ\mathcal{G}_{net}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) or the onset of a plateau after rapid growth (see Methods for details).
我们将 tpeaksubscriptt_{peak}italic_t start_POSTSUBSCRIPT italic_p italic_e italic_a italic_k end_POSTSUBSCRIPT 定义为绅士化事件的时间,对应于 𝒢neti(t,Δ)superscriptsubscript\mathcal{G}_{net}^{i}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) 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: 𝒢bincitysuperscriptsubscript𝒢𝑏𝑖𝑛𝑐𝑖𝑡𝑦\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT, the percentage of nodes experiencing transitions in 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT, and 𝒢netcitysuperscriptsubscript𝒢𝑛𝑒𝑡𝑐𝑖𝑡𝑦\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT, the percentage of nodes showing peaks in 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT (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: 𝒢bincitysuperscriptsubscript\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT ,在 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT𝒢netcitysuperscriptsubscript\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT 中经历转换的节点的百分比,在 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 中显示峰的节点的百分比 (有关详细信息,请参见方法)。

4 Results 4结果

4.1 High income agents drive gentrification.
4.1高收入代理商推动了高档化。

We examine the impact of H𝐻Hitalic_H agent mobility on gentrification dynamics. Simulations were conducted with varying H𝐻Hitalic_H agent movement probabilities (pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT), while maintaining a fixed evaluation window of ϵ=20italic-ϵ20\epsilon=20italic_ϵ = 20 time steps for node growth rates.
我们研究了 HHitalic_H 代理移动性对绅士化动力学的影响。使用变化的 HHitalic_H 代理移动概率 ( pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ) 进行模拟,同时保持节点增长率的 ϵ=2020\epsilon=20italic_ϵ = 20 时间步长的固定评估窗口。

Figure 2 illustrates the influence of pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT on spatial gentrification patterns. At pH=0subscript𝑝𝐻0p_{H}=0italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0, both 𝒢bincitysuperscriptsubscript𝒢𝑏𝑖𝑛𝑐𝑖𝑡𝑦\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT and 𝒢netcitysuperscriptsubscript𝒢𝑛𝑒𝑡𝑐𝑖𝑡𝑦\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT yield 0%, indicating complete absence of gentrification when H𝐻Hitalic_H agents are static. Introducing minimal H𝐻Hitalic_H agent mobility (pH=0.01subscript𝑝𝐻0.01p_{H}=0.01italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0.01) triggers gentrification in 20-40% of nodes, according to both metrics. This abrupt transition highlights the critical role of H𝐻Hitalic_H agent mobility in initiating the gentrification process. As pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT increases, we observe a monotonic rise in gentrification levels, with 𝒢bincitysuperscriptsubscript𝒢𝑏𝑖𝑛𝑐𝑖𝑡𝑦\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT and 𝒢netcitysuperscriptsubscript𝒢𝑛𝑒𝑡𝑐𝑖𝑡𝑦\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT showing similar trends but slightly different magnitudes.
图2说明了 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 对空间绅士化模式的影响。在 pH=0subscript0p_{H}=0italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0 时, 𝒢bincitysuperscriptsubscript\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT𝒢netcitysuperscriptsubscript\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT 均产生0%,表明当 HHitalic_H 试剂为静态时完全没有绅士化。根据两个指标,引入最小的 HHitalic_H 代理移动性 ( pH=0.01subscript0.01p_{H}=0.01italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0.01 ) 会触发20-40% 个节点的绅士化。这种突然的转变凸显了 HHitalic_H 代理移动性在启动绅士化过程中的关键作用。随着 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 的增加,我们观察到绅士化水平的单调上升, 𝒢bincitysuperscriptsubscript\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT𝒢netcitysuperscriptsubscript\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT 显示出相似的趋势,但幅度略有不同。

Refer to caption
Figure 2: Role of relocation rate of high-income agents 𝐩𝐇subscript𝐩𝐇\mathbf{p_{H}}bold_p start_POSTSUBSCRIPT bold_H end_POSTSUBSCRIPT: gentrification levels and times across 150 simulation of the model with a fixed ϵ=20italic-ϵ20\epsilon=20italic_ϵ = 20, N=212𝑁superscript212N=2^{12}italic_N = 2 start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPT and varying value of pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT according to 𝒢bincitysuperscriptsubscript𝒢𝑏𝑖𝑛𝑐𝑖𝑡𝑦\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT (a) and to the network based measure 𝒢netcitysuperscriptsubscript𝒢𝑛𝑒𝑡𝑐𝑖𝑡𝑦\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT (b).
图2: 高收入代理商的搬迁率的作用 𝐩𝐇subscript\mathbf{p_{H}}bold_p start_POSTSUBSCRIPT bold_H end_POSTSUBSCRIPT : 具有固定 ϵ=2020\epsilon=20italic_ϵ = 20 的模型的150模拟中的绅士化水平和时间, N=212superscript212N=2^{12}italic_N = 2 start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPTpHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 的变化值根据 𝒢bincitysuperscriptsubscript\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT (a) 和基于网络的测量 𝒢netcitysuperscriptsubscript\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT (b)。

4.2 Network-based measure anticipates count-based measure.
4.2基于网络的度量预期基于计数的度量。

While 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT provides an intuitive measure of gentrification by capturing demographic transitions from L𝐿Litalic_L to M/H𝑀𝐻M/Hitalic_M / italic_H agent majorities, 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT enables earlier detection by identifying patterns of coordinated movement through temporal networks, revealing gentrification dynamics before visible demographic shifts occur. Figure 3a displays 𝒢net(t,Δ)subscript𝒢𝑛𝑒𝑡𝑡Δ\mathcal{G}_{net}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT ( italic_t , roman_Δ ) and 𝒢bin(t,Δ)subscript𝒢𝑏𝑖𝑛𝑡Δ\mathcal{G}_{bin}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT ( italic_t , roman_Δ ) curves for a node in a representative simulation. In this example, the node initially experiences impoverishment, as 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT transitions from 1 to 0 at approximately t=95𝑡95t=95italic_t = 95, indicating a shift from over- to under-representation of M𝑀Mitalic_M and H𝐻Hitalic_H residents. However, 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT shows a rapid increase at t=122𝑡122t=122italic_t = 122, followed by a plateau at t=123𝑡123t=123italic_t = 123, approximately 10 steps before the abrupt transition from 0 to 1 in 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT at t=132𝑡132t=132italic_t = 132.
虽然 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 通过捕获从 LLitalic_LM/HM/Hitalic_M / italic_H 代理人多数的人口统计转变提供了一种直观的高档化措施,但 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 通过识别时间网络中协调运动的模式来实现早期检测,在可见的人口变化发生之前揭示绅士化动态。图3a显示了代表性仿真中节点的 𝒢net(t,Δ)subscript\mathcal{G}_{net}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT ( italic_t , roman_Δ )𝒢bin(t,Δ)subscript\mathcal{G}_{bin}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT ( italic_t , roman_Δ ) 曲线。在此示例中,节点最初经历贫化,因为 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 在大约 t=9595t=95italic_t = 95 处从1过渡到0,表示 MMitalic_MHHitalic_H 居民的代表性从高到低的转变。然而, 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPTt=122122t=122italic_t = 122 处显示快速增加,随后在 t=123123t=123italic_t = 123 处显示平稳,在 t=132132t=132italic_t = 132𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 中从0到1的突然转变之前大约10步。

Figure 3b illustrates relocation dynamics during and after the 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT peak for the same node (neighborhood) analyzed in Figure 3a. Red arrows indicate L𝐿Litalic_L agent outflows, and blue arrows show M+H𝑀𝐻M+Hitalic_M + italic_H inflows. At the peak (tpeaksubscript𝑡𝑝𝑒𝑎𝑘t_{peak}italic_t start_POSTSUBSCRIPT italic_p italic_e italic_a italic_k end_POSTSUBSCRIPT), both occur simultaneously, with more L𝐿Litalic_L agents leaving. After the peak, M+H𝑀𝐻M+Hitalic_M + italic_H inflows increase while L𝐿Litalic_L outflows decrease. Unlike the gross flows in the visualization, 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT captures net flows, offering a more nuanced understanding of these changes.
图3b示出了在图3a中分析的同一节点 (邻域) 的 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 峰期间和之后的重定位动态。红色箭头表示 LLitalic_L 代理流出,蓝色箭头表示 M+HM+Hitalic_M + italic_H 流入。在峰值 ( tpeaksubscriptt_{peak}italic_t start_POSTSUBSCRIPT italic_p italic_e italic_a italic_k end_POSTSUBSCRIPT ),两者同时发生,更多的 LLitalic_L 代理离开。在峰值之后, M+HM+Hitalic_M + italic_H 流入增加,而 LLitalic_L 流出减少。与可视化中的总流量不同, 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 捕获净流量,提供对这些变化的更细致的理解。

To verify the consistency with which 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT peaks precede 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT transitions across our simulations, we conduct a lagged cross-correlation analysis between 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT and 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT time series. Specifically, we first compute the cross-correlation for each node in the city grid, then average these cross-correlations across all nodes. This process is repeated for 150 independent model runs, and the results are again averaged to obtain the final cross-correlation profile, reported in Figure 3c. We compute cross-correlations for lags τ[15,+15]𝜏1515\tau\in[-15,+15]italic_τ ∈ [ - 15 , + 15 ] (see Methods for details), with significance tested against a null distribution. The highest correlation at τ=9𝜏9\tau=-9italic_τ = - 9 shows that 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT peaks systematically anticipate 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT transitions by approximately 9 time steps. We further validate these findings by developing a null model where agent movements are completely stochastic. The anticipatory behaviour of 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT relative to 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT transitions disappears in the randomised version of the model (see Supplementary Information 2 for detailed analysis). This result support our main findings, demonstrating that 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT reliably anticipates 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT only in scenarios where agents’ movements are influenced by their income distribution of in the city neighbourhoods.
为了验证在我们的模拟中 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 峰先于 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 跃迁的一致性,我们在 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 时间序列之间进行了滞后互相关分析。具体来说,我们首先计算城市网格中每个节点的互相关,然后对所有节点的这些互相关进行平均。对于150独立的模型运行重复该过程,并且再次对结果进行平均以获得最终的互相关曲线,如图3c所示。我们计算lags τ[15,+15]1515\tau\in[-15,+15]italic_τ ∈ [ - 15 , + 15 ] 的互相关 (详见方法),并针对空分布进行显著性测试。 τ=99\tau=-9italic_τ = - 9 处的最高相关性表明 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 峰系统地预测 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 约9个时间步长的过渡。我们通过开发一个空模型来进一步验证这些发现,在该模型中,代理的移动是完全随机的。 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 相对于 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 过渡的预期行为在模型的随机版本中消失了 (有关详细分析,请参见补充信息2)。这一结果支持了我们的主要发现,表明 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 仅在代理人的行动受到城市社区收入分布影响的情况下才可靠地预测 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT

Refer to caption
Figure 3: Evolution of a gentrification event for a representative node (pH=0.1subscript𝑝𝐻0.1p_{H}=0.1italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0.1, ϵ=20italic-ϵ20\epsilon=20italic_ϵ = 20, Δ=15Δ15\Delta=15roman_Δ = 15) and N=212𝑁superscript212N=2^{12}italic_N = 2 start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPT. a): Comparison of 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT (in gold) and 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT (in black) measures over time highlighting two key moments : during and after the peak in 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT, i.e., tpeaksubscript𝑡𝑝𝑒𝑎𝑘t_{peak}italic_t start_POSTSUBSCRIPT italic_p italic_e italic_a italic_k end_POSTSUBSCRIPT and tpostsubscript𝑡𝑝𝑜𝑠𝑡t_{post}italic_t start_POSTSUBSCRIPT italic_p italic_o italic_s italic_t end_POSTSUBSCRIPT, respectively. b): Relocation in- and out-flows during (top), and after (bottom) the peak of 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT. Red arrows correspond to L𝐿Litalic_L agent outflows, blue arrows correspond to M𝑀Mitalic_M and H𝐻Hitalic_H agent inflows within ΔΔ\Deltaroman_Δ. (top) Peak: Simultaneous L𝐿Litalic_L outflow and M+H𝑀𝐻M+Hitalic_M + italic_H inflow. (bottom) Post-peak: Diminishing L𝐿Litalic_L outflow, intensifying M𝑀Mitalic_M and H𝐻Hitalic_H inflow. Flows represent gross out- and in- strengths. c): Average cross-correlation Rdelimited-⟨⟩𝑅\langle R\rangle⟨ italic_R ⟩ between 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT and 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT across all nodes and 150 model runs, as a function of lag τ𝜏\tauitalic_τ. Grey area shows 95% confidence interval; dotted line indicates null distribution mean.
图3: 代表性节点 ( pH=0.1subscript0.1p_{H}=0.1italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0.1ϵ=2020\epsilon=20italic_ϵ = 20Δ=1515\Delta=15roman_Δ = 15 ) 和 N=212superscript212N=2^{12}italic_N = 2 start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPT 的绅士化事件的演变。a): 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT (金色) 和 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT (黑色) 度量随时间的比较突出了两个关键时刻: 在 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 中的峰值期间和之后,即, tpeaksubscriptt_{peak}italic_t start_POSTSUBSCRIPT italic_p italic_e italic_a italic_k end_POSTSUBSCRIPTtpostsubscriptt_{post}italic_t start_POSTSUBSCRIPT italic_p italic_o italic_s italic_t end_POSTSUBSCRIPT 。b): 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 峰值期间 (顶部) 和之后 (底部) 的流入和流出的重新定位。红色箭头对应 LLitalic_L 代理流出,蓝色箭头对应 MMitalic_MHHitalic_H Δ\Deltaroman_Δ 流出和 M+HM+Hitalic_M + italic_H 流入。(下) 峰值后: LLitalic_L 流出减少, MMitalic_MHHitalic_H 流入加剧。流量代表总外流和内流。c): 所有节点和模型运行150 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 之间的平均互相关 Rdelimited-⟨⟩\langle R\rangle⟨ italic_R ⟩ ,作为滞后 τ\tauitalic_τ 的函数。 灰色区域表示95% 置信区间; 虚线表示零分布均值。

We analyse the relationship between 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT and socioeconomic neighborhood dynamics in Figure 4. Nodes exhibiting 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT peaks are colour-coded, while those where 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT remains at zero are depicted in grey (Figure 4a). Median richness time series reveal a starting trimodal distribution: high-richness central nodes, intermediate-richness nodes, and low-richness nodes (Figure 4b). Several nodes transition from low to intermediate richness, coinciding with 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT peaks. A notable exception (pink curve) displays a sharp richness increase followed by a steep decline, ultimately transitioning from low to middle income. 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT correctly detects gentrification only after the moment this curve transitions from low to middle income, disregarding the earlier fluctuations. The overall correspondence between the colour-coded curves in both figures validates the capacity of 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT to identify gentrification solely based on relocation patterns.
我们在图4中分析了 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 与社会经济邻域动态之间的关系。显示 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 峰的节点用颜色编码,而 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 保持为零的节点用灰色表示 (图4a)。中位数丰富度时间序列揭示了一个起始的三峰分布: 高丰富度中心节点,中等丰富度节点和低丰富度节点 (图4b)。几个节点从低丰富度过渡到中等丰富度,与 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 峰重合。一个值得注意的例外 (粉红色曲线) 显示财富急剧增加,随后急剧下降,最终从低收入过渡到中等收入。 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 仅在此曲线从低收入过渡到中等收入的那一刻之后,才正确地检测到高档化,而忽略了较早的波动。两幅图中的颜色编码曲线之间的总体对应关系验证了 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 仅基于重定位模式识别高档化的能力。

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Figure 4: 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT peaks capture median richness transitions in neighbourhoods: a) Time series of 𝒢net(t,Δ)subscript𝒢𝑛𝑒𝑡𝑡Δ\mathcal{G}_{net}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT ( italic_t , roman_Δ ), with fixed values of the time window width Δ=15Δ15\Delta=15roman_Δ = 15, of only the nodes that undergo sharp peaks of 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT. b) Time series of the median richness (median of agents’ income) of all nodes in the network: the colored curves correspond to the same nodes whose time series of 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT is presented in a). The black vertical line in both plots indicates the initial time step when H𝐻Hitalic_H agents become eligible to relocate, occurring ϵitalic-ϵ\epsilonitalic_ϵ steps after the simulation’s start.
图4: 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 峰捕获邻域中的中值丰富度过渡: a) 𝒢net(t,Δ)subscript\mathcal{G}_{net}(t,\Delta)caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT ( italic_t , roman_Δ ) 的时间序列,具有时间窗口宽度 Δ=1515\Delta=15roman_Δ = 15 的固定值,只有经历 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 尖峰的节点。b) 网络中所有节点的中位数丰富度 (代理人收入的中位数) 的时间序列: 着色曲线对应于其时间序列 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 在a) 中呈现的相同节点。两个图中的黑色垂直线表示 HHitalic_H 代理有资格重新定位的初始时间步长,在模拟开始后发生 ϵ\epsilonitalic_ϵ 步骤。

4.3 Gentrification follows city density.
4.3高档化遵循城市密度。

In Figures 5a-c we show the relationship between urban density and gentrification levels across different values of the model parameter pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT. We conduct 150 simulations for each configuration, with fixed values of the parameters ϵ=20italic-ϵ20\epsilon=20italic_ϵ = 20 and Δ=10Δ10\Delta=10roman_Δ = 10. To model increasing urban density, we varied the number of agents (N𝑁Nitalic_N) present in the grid, while keeping the grid dimension constant at 7x7 along with the capacity of individual nodes K𝐾Kitalic_K.
在图5a-c中,我们显示了模型参数 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 的不同值下城市密度与高档化水平之间的关系。我们对每种配置进行150模拟,参数 ϵ=2020\epsilon=20italic_ϵ = 20Δ=1010\Delta=10roman_Δ = 10 的固定值。为了对不断增加的城市密度进行建模,我们改变了网格中存在的代理数量 ( NNitalic_N ),同时保持网格维度恒定为7x7以及各个节点的容量 KKitalic_K

City-wise gentrification levels, as captured by 𝒢netcitysuperscriptsubscript𝒢𝑛𝑒𝑡𝑐𝑖𝑡𝑦\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT (Figure 5a) and 𝒢bincitysuperscriptsubscript𝒢𝑏𝑖𝑛𝑐𝑖𝑡𝑦\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT (Figure 5b), are characterised by a clear trend: as city population density increases, so does the propensity for gentrification, in terms of number of gentrification events as observed by the two measures (see Methods for details), averaged over the different runs of the model. Furthermore, this effect is amplified by the H𝐻Hitalic_H agents relocation rate pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT, as the curves in the two figures increase monotonically with N𝑁Nitalic_N, with the exception of 0 or very low values of pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT and low N𝑁Nitalic_N for 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT, as shown in Figure 5a. In Figure 5c, we show the relationship between city density and the average convergence time Tdelimited-⟨⟩𝑇\langle T\rangle⟨ italic_T ⟩, the mean number of steps required to reach the termination condition over the 150 simulations, where L𝐿Litalic_L agents can no longer move. The average convergence time Tdelimited-⟨⟩𝑇\langle T\rangle⟨ italic_T ⟩ increases with both N𝑁Nitalic_N and pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT, except when pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0. In this case, where H𝐻Hitalic_H agents are present but stationary and the city is extremely dense, the model does not reach the termination condition within the imposed 300-step limit when N213𝑁superscript213N\geq 2^{13}italic_N ≥ 2 start_POSTSUPERSCRIPT 13 end_POSTSUPERSCRIPT.
𝒢netcitysuperscriptsubscript\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT (图5a) 和 𝒢bincitysuperscriptsubscript\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT (图5b) 所描述的城市高档化水平具有明显的趋势: 随着城市人口密度的增加,高档化的倾向,根据两种测量方法观察到的高档化事件的数量 (有关详细信息,请参见方法),在模型的不同运行中取平均值。此外,这种影响被 HHitalic_H 代理重定位率 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 放大,因为两个图中的曲线随 NNitalic_N 单调增加,除了 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPTpHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 和低 NNitalic_N 的0或非常低的值,如图5a所示。在图5c中,我们显示了城市密度与平均收敛时间 Tdelimited-⟨⟩\langle T\rangle⟨ italic_T ⟩ 之间的关系,这是在150模拟中达到终止条件所需的平均步数,其中 LLitalic_L 代理不再移动。 平均收敛时间 Tdelimited-⟨⟩\langle T\rangle⟨ italic_T ⟩NNitalic_NpHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 而增加,但 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0时除外。在这种情况下,在 HHitalic_H 代理存在但固定且城市极其密集的情况下,当 N213superscript213N\geq 2^{13}italic_N ≥ 2 start_POSTSUPERSCRIPT 13 end_POSTSUPERSCRIPT 时,模型在所施加的300步限制内未达到终止条件。

Overall these results suggest how higher urban densities lead to the emergence of more gentrification waves throughout the evolution of a city.
总体而言,这些结果表明,较高的城市密度如何导致整个城市演变过程中出现更多的高档化浪潮。

Refer to caption
Figure 5: City density drives gentrification dynamics: Average results across 150 simulations for varying agent populations (x-axis, logarithmic scale) and high-income agent relocation rates pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT (colors). a,b) City-wise gentrification levels measured by 𝒢netcitysuperscriptsubscript𝒢𝑛𝑒𝑡𝑐𝑖𝑡𝑦\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT (a) and 𝒢bincitysuperscriptsubscript𝒢𝑏𝑖𝑛𝑐𝑖𝑡𝑦\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT (b). c) Number of average simulation time steps. Higher urban densities facilitate gentrification and increase convergences time.
图5: 城市密度驱动绅士化动态: 不同代理人人口 (x轴,对数标度) 和高收入代理人重新安置率 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT (颜色) 的150模拟的平均结果。a,b) 以 𝒢netcitysuperscriptsubscript\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT (a) 和 𝒢bincitysuperscriptsubscript\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT (b) 衡量的城市绅士化水平。c) 平均仿真时间步长数。较高的城市密度促进了中产阶级化,并增加了收敛时间。

5 Discussion 5讨论

Our study introduces an agent-based model of gentrification that categorises inhabitants of a city into three income groups – low, medium and high – and simulates agent movements within a grid-based urban environment driven by socioeconomic factors. The model effectively captures the essence of gentrification dynamics, consisting in the displacement of lower income inhabitants of a neighbourhood of the city caused by a simultaneous inflow of wealthier citizens [10]. This characteristic of gentrification is evidenced by the results of our simulations and quantitatively described by our proposed network-based measure.
我们的研究引入了一种基于agent的高档化模型,该模型将城市居民分为低,中,高三个收入群体,并模拟了由社会经济因素驱动的基于网格的城市环境中的agent运动。该模型有效地捕捉了高档化动态的本质,包括由较富裕的公民同时流入引起的城市附近低收入居民的流离失所 [10]。我们的模拟结果证明了这种绅士化的特征,并通过我们提出的基于网络的措施进行了定量描述。

We find that even a small proportion of high-income agents (5% of the population) significantly affect the dynamics of gentrification. When high-income agents do not move, even if still present in the model city, no gentrification is detected, and the model does not converge within the imposed limit of 300 steps. However, introducing even a low probability for the movement of high-income agents (pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 1%) leads to model convergence in approximately 150 steps, while 40% of the city neighbourhoods experience at least one gentrification wave.
我们发现,即使一小部分高收入代理商 (占人口的5%) 也会显着影响高档化的动态。当高收入代理商不动时,即使仍然存在于模型城市中,也不会检测到绅士化,并且模型不会在300步骤的限制范围内收敛。但是,即使高收入代理商的移动概率很低 ( pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 1%),也会导致模型在大约150个步骤中收敛,而40% 城市社区至少经历了一次高档化浪潮。

Our measures, 𝒢countsubscript𝒢𝑐𝑜𝑢𝑛𝑡\mathcal{G}_{count}caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT and 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT, represent two distinct approaches to quantify gentrification. The count-based measure 𝒢countsubscript𝒢𝑐𝑜𝑢𝑛𝑡\mathcal{G}_{count}caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT evaluates the concentration of agents of the three types in each neighborhood over a time window ΔΔ\Deltaroman_Δ. A significance threshold (nH,Msuperscriptsubscript𝑛𝐻𝑀n_{H,M}^{*}italic_n start_POSTSUBSCRIPT italic_H , italic_M end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT) is needed to detect when middle- and high-income agents are over-represented and, therefore, define the binarised version of the count-based measure, 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT. Such measure thus captures neighborhood transitions from under- to over-representation of middle- and high-income agents, indicating the completion of gentrification. The network-based measure 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT tracks the net inflow of middle- and high-income agents and the simultaneous outflow of lower-income agents over ΔΔ\Deltaroman_Δ. This measure is thus rooted in temporal network analysis, where the existence of structures [47, 48, 49] in the networks under study is related to the simultaneity of the interactions (edges) between pairs or groups of nodes. Furthermore, this network-based approach avoids more or less arbitrary thresholds and aligns with available commuting flow data [35, 36, 37, 38], which could serve as a proxy given the absence of residential relocation records. 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT helps identifying early signs of gentrification by tracking peaks or plateaus in its trajectory, correlating higher-income resident influx with lower-income displacement. The cross-correlation computed between the count-based an the network-based measures highlights how peaks in 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT are consistently and significantly observed in advance with respect to 𝒢countsubscript𝒢𝑐𝑜𝑢𝑛𝑡\mathcal{G}_{count}caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT. The analysis of the randomised version of our model, in Supplementary Figure S2, shows that while gentrification events caused by random agent-relocations are still detected, no significant cross-correlation exists between our two measures, emphasising the importance of the agents’ decision-making rules in our model for predicting transitions. These results demonstrate that our minimal model captures the essential features needed to reproduce gentrification: a heavy-tailed income distribution, few income classes, and distinct relocation strategies – notably the profit-driven behavior of high-income agents. Moreover, our network-based measure 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT enables earlier detection of gentrification compared to count-based metrics, potentially aiding policymakers in preventing low-income displacement.
我们的措施, 𝒢countsubscript\mathcal{G}_{count}caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT ,代表了量化绅士化的两种不同方法。基于计数的度量 𝒢countsubscript\mathcal{G}_{count}caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT 评估在时间窗口 Δ\Deltaroman_Δ 内每个邻域中三种类型的代理的浓度。需要一个显著性阈值 ( nH,Msuperscriptsubscriptn_{H,M}^{*}italic_n start_POSTSUBSCRIPT italic_H , italic_M end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) 来检测中高收入代理何时被过度代表,因此定义基于计数的度量的二进制版本 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 。因此,这种措施可以捕捉到中高收入代理商从代表不足到代表过高的邻里过渡,表明高档化已经完成。基于网络的措施 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 跟踪中高收入代理商的净流入以及低收入代理商在 Δ\Deltaroman_Δ 上的同时流出。因此,该措施植根于时间网络分析,其中所研究的网络中结构 [47,48,49] 的存在与节点对或节点组之间的相互作用 (边) 的同时性有关。 此外,这种基于网络的方法避免了或多或少的任意阈值,并与可用的通勤流量数据 [35,36,37,38] 对齐,在没有住宅搬迁记录的情况下,通勤流量数据可以用作代理。 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 通过跟踪其轨迹中的峰值或高原,将高收入居民涌入与低收入流离失所相关联,有助于识别高档化的早期迹象。基于计数和基于网络的测量之间计算的互相关突出了 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 中的峰值如何相对于 𝒢countsubscript\mathcal{G}_{count}caligraphic_G start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t end_POSTSUBSCRIPT 被一致且显著地提前观察到。在补充图S2中,对我们模型的随机版本的分析表明,尽管仍然检测到由随机代理重定位引起的绅士化事件,但在我们的两种测量方法之间不存在显著的交叉相关性,强调了代理决策规则在我们的模型中对于预测转变的重要性。 这些结果表明,我们的最小模型捕获了重现高档化所需的基本特征: 沉重的收入分配,很少的收入类别以及独特的搬迁策略-尤其是高收入代理商的利润驱动行为。此外,与基于计数的指标相比,我们基于网络的措施 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 可以更早地检测高档化,这可能有助于政策制定者防止低收入流离失所。

While our model provides valuable insights into gentrification dynamics, it has limitations that future research could address. The constant population size and static income-group assignments could be expanded to incorporate population growth and inter-city migration flows, potentially using a network-based approach to disentangle endogenous and exogenous causes of gentrification [50]. Multiple property ownership per agent could be introduced to model wealth concentration and short-term rental effects. The grid-based urban representation could be enhanced with more complex geographical features, although our results hold for both 7×\times×7 and 9×\times×9 grids (Supplementary Information 5-7). Moreover, simulating policy interventions [51] could provide insights for urban planners, particularly regarding density restrictions given our findings on city density and gentrification probability. Finally, future research In conclusion, our agent-based model and novel quantitative measures offer a powerful framework for understanding and predicting gentrification processes. This approach not only advances our theoretical understanding of gentrification but also provides quantitative what-if tools for early detection and potential mitigation of its effects in real-world urban environments.
虽然我们的模型为绅士化动态提供了有价值的见解,但它有未来研究可以解决的局限性。恒定的人口规模和静态的收入群体分配可以扩展,以纳入人口增长和城市间的移民流动,可能使用基于网络的方法来解开中产阶级化的内生和外生原因 [50]。可以引入每个代理商的多个财产所有权来模拟财富集中和短期租金效应。基于网格的城市表示可以通过更复杂的地理特征来增强,尽管我们的结果适用于7 ×\times× 7和9 ×\times× 9个网格 (补充信息5-7)。此外,模拟政策干预 [51] 可以为城市规划者提供见解,特别是考虑到我们对城市密度和高档化概率的发现,密度限制。 最后,未来的研究结论是,我们基于代理的模型和新颖的定量措施为理解和预测绅士化过程提供了强大的框架。这种方法不仅提高了我们对绅士化的理论理解,而且还提供了定量的假设工具,用于早期检测和潜在缓解其在现实世界城市环境中的影响。

Methods 方法

Agents’ income. 代理商的收入。

At the beginning of a simulation, each agent is assigned a fixed income w𝑤witalic_w, based on data from the 2022 USA Social Security Administration report [39]. The assignment process uses the income brackets and population percentages provided in this report. Each agent is assigned to an income bracket with probability proportional to the US population within that bracket, and then the agent’s specific income w𝑤witalic_w is randomly selected from within their assigned bracket. Agents are categorized into three groups based on their assigned incomes: L𝐿Litalic_L (low-income) agents with incomes up to $29,999.99, encompassing the 2022 poverty line for a family of four ($27,750); H𝐻Hitalic_H (high-income) agents representing the top 5% of earners; and M𝑀Mitalic_M (middle-income) agents comprising all remaining individuals.
在模拟开始时,根据2022美国社会保障管理局报告 [39] 的数据,为每个代理人分配固定收入 wwitalic_w 。分配过程使用本报告中提供的收入范围和人口百分比。每个代理商都被分配到一个收入范围内,其概率与该范围内的美国人口成正比,然后从其分配的范围内随机选择代理商的特定收入 wwitalic_w 。代理人根据其分配的收入分为三类: LLitalic_L (低收入) 代理人,收入不超过29,999.99美元,包括一个四口之家的2022贫困线 (27,750美元); HHitalic_H (高收入) 代理人代表收入最高的5%; MMitalic_M (中等收入) 代理人包括所有剩余的个人。

Gentrification: peaks, shifts and aggregate measure
绅士化: 峰值、位移和聚合测量

A cell i𝑖iitalic_i undergoes gentrification events according to 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT at all times t𝑡titalic_t where there is a binary shift from 0 to 1 in the time series throughout the simulation:
单元格 iiitalic_i 始终根据 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 经历绅士化事件 ttitalic_t ,其中在整个模拟过程中,时间序列中存在从0到1的二进制移位:

Tshifti={t:𝒢bini(t,Δ)=1𝒢bini(t1,Δ)=0}superscriptsubscript𝑇𝑠𝑖𝑓𝑡𝑖conditional-set𝑡superscriptsubscript𝒢𝑏𝑖𝑛𝑖𝑡Δ1superscriptsubscript𝒢𝑏𝑖𝑛𝑖𝑡1Δ0T_{shift}^{i}=\{t:\mathcal{G}_{bin}^{i}(t,\Delta)=1\wedge\mathcal{G}_{bin}^{i}% (t-1,\Delta)=0\}italic_T start_POSTSUBSCRIPT italic_s italic_h italic_i italic_f italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = { italic_t : caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) = 1 ∧ caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t - 1 , roman_Δ ) = 0 } (14)

A cell i𝑖iitalic_i undergoes gentrification events according to 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT at all times t𝑡titalic_t where there is a peak in its time series throughout the simulation. A peak is defined as either a local maximum or the start of a plateau after a growing phase:
一个单元 iiitalic_i 根据 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 在所有时间 ttitalic_t 经历绅士化事件,其中在整个模拟过程中在其时间序列中存在峰值。峰被定义为局部最大值或生长阶段后平台期的起点:

Tpeaki={t:[(𝒢neti(t,Δ)>𝒢neti(t1,Δ)𝒢neti(t,Δ)>𝒢neti(t+1,Δ))(𝒢neti(t,Δ)>𝒢neti(t1,Δ)𝒢neti(t,Δ)=𝒢neti(t+1,Δ))]}superscriptsubscript𝑇𝑝𝑒𝑎𝑘𝑖conditional-set𝑡delimited-[]superscriptsubscript𝒢𝑛𝑒𝑡𝑖𝑡Δsuperscriptsubscript𝒢𝑛𝑒𝑡𝑖𝑡1Δsuperscriptsubscript𝒢𝑛𝑒𝑡𝑖𝑡Δsuperscriptsubscript𝒢𝑛𝑒𝑡𝑖𝑡1Δsuperscriptsubscript𝒢𝑛𝑒𝑡𝑖𝑡Δsuperscriptsubscript𝒢𝑛𝑒𝑡𝑖𝑡1Δsuperscriptsubscript𝒢𝑛𝑒𝑡𝑖𝑡Δsuperscriptsubscript𝒢𝑛𝑒𝑡𝑖𝑡1ΔT_{peak}^{i}=\left\{t:\left[\begin{array}[]{c}(\mathcal{G}_{net}^{i}(t,\Delta)% >\mathcal{G}_{net}^{i}(t-1,\Delta)\wedge\mathcal{G}_{net}^{i}(t,\Delta)>% \mathcal{G}_{net}^{i}(t+1,\Delta))\\ \vee\\ (\mathcal{G}_{net}^{i}(t,\Delta)>\mathcal{G}_{net}^{i}(t-1,\Delta)\wedge% \mathcal{G}_{net}^{i}(t,\Delta)=\mathcal{G}_{net}^{i}(t+1,\Delta))\end{array}% \right]\right\}italic_T start_POSTSUBSCRIPT italic_p italic_e italic_a italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = { italic_t : [ start_ARRAY start_ROW start_CELL ( caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) > caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t - 1 , roman_Δ ) ∧ caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) > caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t + 1 , roman_Δ ) ) end_CELL end_ROW start_ROW start_CELL ∨ end_CELL end_ROW start_ROW start_CELL ( caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) > caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t - 1 , roman_Δ ) ∧ caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t , roman_Δ ) = caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t + 1 , roman_Δ ) ) end_CELL end_ROW end_ARRAY ] } (15)

We define the gentrification level of the city as the percentage of cells of the city that experience at least one gentrification event according to each measure:
我们将城市的高档化水平定义为根据每种措施经历至少一次高档化事件的城市单元格的百分比:

𝒢bincity=i=1Ncells𝟏(|Tshifti|>0)Ncells×100,𝒢netcity=i=1Ncells𝟏(|Tpeaki|>0)Ncells×100\mathcal{G}_{bin}^{city}=\frac{\sum_{i=1}^{N_{\text{cells}}}\mathbf{1}(|T_{% shift}^{i}|>0)}{N_{\text{cells}}}\times 100\quad,\quad\mathcal{G}_{net}^{city}% =\frac{\sum_{i=1}^{N_{\text{cells}}}\mathbf{1}(|T_{peak}^{i}|>0)}{N_{\text{% cells}}}\times 100caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT cells end_POSTSUBSCRIPT end_POSTSUPERSCRIPT bold_1 ( | italic_T start_POSTSUBSCRIPT italic_s italic_h italic_i italic_f italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT | > 0 ) end_ARG start_ARG italic_N start_POSTSUBSCRIPT cells end_POSTSUBSCRIPT end_ARG × 100 , caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT cells end_POSTSUBSCRIPT end_POSTSUPERSCRIPT bold_1 ( | italic_T start_POSTSUBSCRIPT italic_p italic_e italic_a italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT | > 0 ) end_ARG start_ARG italic_N start_POSTSUBSCRIPT cells end_POSTSUBSCRIPT end_ARG × 100 (16)

where 𝟏()1\mathbf{1}(\cdot)bold_1 ( ⋅ ) is the indicator function.
其中 𝟏()1\mathbf{1}(\cdot)bold_1 ( ⋅ ) 是指标函数。

Lagged cross-correlation 滞后互相关

We compute the mean cross-correlation Rdelimited-⟨⟩𝑅\langle R\rangle⟨ italic_R ⟩ by calculating the lagged cross-correlation between pairs of time series of 𝒢neti(t,Δ)subscriptsuperscript𝒢𝑖𝑛𝑒𝑡𝑡Δ\mathcal{G}^{i}_{net}(t,\Delta)caligraphic_G start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT ( italic_t , roman_Δ ) and 𝒢bini(t,Δ)subscriptsuperscript𝒢𝑖𝑏𝑖𝑛𝑡Δ\mathcal{G}^{i}_{bin}(t,\Delta)caligraphic_G start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT ( italic_t , roman_Δ ) corresponding to each cell on the grid-view of a model city, for several values of the lag τ𝜏\tauitalic_τ, and then averaging over all cells. To compute the cross-correlation Risuperscript𝑅𝑖R^{i}italic_R start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT between the two time series for a cell i𝑖iitalic_i, we transform the two time series into two binary vectors 𝒢~netisuperscriptsubscript~𝒢𝑛𝑒𝑡𝑖\widetilde{\mathcal{G}}_{net}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT and 𝒢~binisuperscriptsubscript~𝒢𝑏𝑖𝑛𝑖\widetilde{\mathcal{G}}_{bin}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT of length T𝑇Titalic_T, where the tksubscript𝑡𝑘t_{k}italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT-th entry is 1111 if the original corresponding time series has a peak or 0-1 transition at time tk[0,T]subscript𝑡𝑘0𝑇t_{k}\in[0,T]italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ [ 0 , italic_T ], respectively:
我们通过计算与模型城市的网格视图上的每个单元相对应的 𝒢neti(t,Δ)subscriptsuperscript\mathcal{G}^{i}_{net}(t,\Delta)caligraphic_G start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT ( italic_t , roman_Δ )𝒢bini(t,Δ)subscriptsuperscript\mathcal{G}^{i}_{bin}(t,\Delta)caligraphic_G start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT ( italic_t , roman_Δ ) 的时间序列对之间的滞后互相关来计算平均互相关 Rdelimited-⟨⟩\langle R\rangle⟨ italic_R ⟩ ,对于lag τ\tauitalic_τ 的几个值,然后对所有单元进行平均。要计算单元格 iiitalic_i 的两个时间序列之间的互相关 RisuperscriptR^{i}italic_R start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ,我们将两个时间序列转换为两个长度为 TTitalic_T 的二进制向量 𝒢~netisuperscriptsubscript\widetilde{\mathcal{G}}_{net}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT𝒢~binisuperscriptsubscript\widetilde{\mathcal{G}}_{bin}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ,其中 tksubscriptt_{k}italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT -th条目是 1111 如果原始对应的时间序列分别在时间 tk[0,T]subscript0t_{k}\in[0,T]italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ [ 0 , italic_T ] 处具有峰值或0-1过渡:

(𝒢~neti)tk={1,if tkTpeaki,0,otherwise.subscriptsuperscriptsubscript~𝒢𝑛𝑒𝑡𝑖subscript𝑡𝑘cases1if tkTpeaki0otherwise(\widetilde{\mathcal{G}}_{net}^{i})_{t_{k}}=\begin{cases}1,&\text{if $t_{k}\in T% _{peak}^{i}$},\\ 0,&\text{otherwise}.\end{cases}( over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT = { start_ROW start_CELL 1 , end_CELL start_CELL if italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ italic_T start_POSTSUBSCRIPT italic_p italic_e italic_a italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL 0 , end_CELL start_CELL otherwise . end_CELL end_ROW (17)
(𝒢~bini)tk={1,if tkTshifti,0,otherwise.subscriptsuperscriptsubscript~𝒢𝑏𝑖𝑛𝑖subscript𝑡𝑘cases1if tkTshifti0otherwise(\widetilde{\mathcal{G}}_{bin}^{i})_{t_{k}}=\begin{cases}1,&\text{if $t_{k}\in T% _{shift}^{i}$},\\ 0,&\text{otherwise}.\end{cases}( over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT = { start_ROW start_CELL 1 , end_CELL start_CELL if italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ italic_T start_POSTSUBSCRIPT italic_s italic_h italic_i italic_f italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL 0 , end_CELL start_CELL otherwise . end_CELL end_ROW (18)

For each value of the lag τ[15,+15]𝜏1515\tau\in[-15,+15]italic_τ ∈ [ - 15 , + 15 ], we compute the lagged cross-correlation between the two vectors and obtain a value of Ri(τ)superscript𝑅𝑖𝜏R^{i}(\tau)italic_R start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_τ ) for each cell i𝑖iitalic_i on the grid. We then calculate Rdelimited-⟨⟩𝑅\langle R\rangle⟨ italic_R ⟩ by averaging, for each value of the lag τ𝜏\tauitalic_τ, the values of Ri(τ)superscript𝑅𝑖𝜏R^{i}(\tau)italic_R start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_τ ) over all cells i[0,Ncells]𝑖0subscript𝑁cellsi\in[0,N_{\text{cells}}]italic_i ∈ [ 0 , italic_N start_POSTSUBSCRIPT cells end_POSTSUBSCRIPT ]:
对于滞后 τ[15,+15]1515\tau\in[-15,+15]italic_τ ∈ [ - 15 , + 15 ] 的每个值,我们计算两个向量之间的滞后互相关,并为网格上的每个单元格 iiitalic_i 获得 Ri(τ)superscriptR^{i}(\tau)italic_R start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_τ ) 的值。然后,我们通过对滞后 τ\tauitalic_τ 的每个值对所有单元格 i[0,Ncells]0subscripti\in[0,N_{\text{cells}}]italic_i ∈ [ 0 , italic_N start_POSTSUBSCRIPT cells end_POSTSUBSCRIPT ] 上的 Ri(τ)superscriptR^{i}(\tau)italic_R start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_τ ) 的值求平均值来计算 Rdelimited-⟨⟩\langle R\rangle⟨ italic_R ⟩ :

R(τ)=i[0,Ncells]Ri(τ)Ncells.delimited-⟨⟩𝑅𝜏subscript𝑖0subscript𝑁cellssuperscript𝑅𝑖𝜏subscript𝑁cells\langle R(\tau)\rangle=\frac{\sum_{i\in[0,N_{\text{cells}}]}R^{i}(\tau)}{N_{% \text{cells}}}.⟨ italic_R ( italic_τ ) ⟩ = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i ∈ [ 0 , italic_N start_POSTSUBSCRIPT cells end_POSTSUBSCRIPT ] end_POSTSUBSCRIPT italic_R start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_τ ) end_ARG start_ARG italic_N start_POSTSUBSCRIPT cells end_POSTSUBSCRIPT end_ARG . (19)

To establish a baseline for comparison, we generate a null distribution by computing, for each cell i𝑖iitalic_i, the cross-correlation R𝑅Ritalic_R between 50505050 pairs of randomly reshuffled versions of the two vectors 𝒢~netisuperscriptsubscript~𝒢𝑛𝑒𝑡𝑖\widetilde{\mathcal{G}}_{net}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT and 𝒢~binisuperscriptsubscript~𝒢𝑏𝑖𝑛𝑖\widetilde{\mathcal{G}}_{bin}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT.
为了建立比较的基线,我们通过计算生成一个空分布,对于每个单元格 iiitalic_i ,两个向量 𝒢~netisuperscriptsubscript\widetilde{\mathcal{G}}_{net}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT𝒢~binisuperscriptsubscript\widetilde{\mathcal{G}}_{bin}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT 的随机重组版本对 50505050 之间的互相关 RRitalic_R

Code availability statement
代码可用性语句

The code for implement our model and reproduce our analyses can be found at https://github.com/mauruscz/Gentrification.
实现我们的模型和重现我们的分析的代码可以在 https://github.com/mauruscz/Gentrification. 找到

Acknowledgements 致谢

We thank Daniele Fadda for its precious support with the visualizations. We thank Dino Pedreschi, Timothy LaRock, Rohit Sahasrabuddhe, Andrea Beretta, Giuliano Cornacchia, Margherita Lalli, Daniele Gambetta, Emanuele Ferragina and Salvatore Citraro for their useful suggestions.
我们感谢Daniele Fadda对可视化的宝贵支持。我们感谢Dino Pedreschi,Timothy LaRock,Rohit Sahasrabuddhe,Andrea Beretta,Giuliano Cornacchia,Margherita Lalli,Daniele Gambetta,Emanuele Ferragina和Salvatore Citraro的有益建议。

Funding 资金

This work has been partially supported by: EU project H2020 SoBigData++ G.A. 871042; PNRR (Piano Nazionale di Ripresa e Resilienza) in the context of the research program 20224CZ5X4 PE6 PRIN 2022 “URBAI – Urban Artificial Intelligence” (CUP B53D23012770006), funded by the European Commission under the Next Generation EU programme; and by PNRR - M4C2 - Investimento 1.3, Partenariato Esteso PE00000013 - “FAIR – Future Artificial Intelligence Research” – Spoke 1 “Human-centered AI”, funded by the European Commission under the NextGeneration EU programme; project “SoBigData.it - Strengthening the Italian RI for Social Mining and Big Data Analytics”, prot. IR0000013, avviso n. 3264 on 28/12/2021.
这项工作得到了以下项目的部分支持: 欧盟项目H2020 SoBigData g.a.871042; PNRR (钢琴Nazionale di ripresilienza e Resilienza) 在研究计划20224CZ5X4 PE6 PRIN 2022 “城市人工智能” (CUP b53d2301270006) 的背景下,由欧盟委员会在下一代欧盟计划下资助; 以及由PNRR-M4C2-投资1.3,partenariato Esteso PE00000013-“公平-未来人工智能研究”-发言1 “以人为中心的人工智能”,由欧盟委员会在下一代欧盟计划下资助; 项目 “SoBigData.it-加强意大利RI用于社会挖掘和大数据分析”,prot。IR0000013, avviso n.3264在28/12/2021上。

NP and RL received funding from EPSRC Grant Ref EP/V013068/1.
NP和RL从EPSRC拨款编号EP/V013068/1获得资金。

Author contributions 作者贡献

GM: study conceptualisation, model implementation, experiment design, execution of experiments, code implementation, interpretation of results, writing, plots and images, study management. NP: study conceptualisation, model implementation, experiment design, interpretation of results, writing, plots design. LP: study conceptualisation, experiment design, interpretation of results, writing, study direction. RL: study conceptualisation, experiment design, interpretation of results, writing, study direction. All authors read and approved the final manuscript.
GM: 研究概念化,模型实现,实验设计,实验执行,代码实现,结果解释,写作,情节和图像,研究管理。NP: 研究概念化,模型实施,实验设计,结果解释,写作,情节设计。LP: 研究概念化,实验设计,结果解释,写作,研究方向。RL: 研究概念化,实验设计,结果解释,写作,研究方向。所有作者均阅读并批准了最终手稿。

Competing interests 利益冲突

The authors declare no competing interests.
作者声明没有竞争利益。

Corresponding authors 通讯作者

Giovanni Mauro - giovanni.mauro@sns.it

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Supplementary Information
补充资料

Supplementary Information 1: Binary Vector Transformation for Cross-Correlation Analysis
补充信息1: 用于互相关分析的二进制矢量变换

In our analysis, we compute the cross-correlation Risuperscript𝑅𝑖R^{i}italic_R start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT between two time series for a cell i𝑖iitalic_i. To facilitate this process, we transform both time series into binary “barcodes” that represent significant changes or peaks in the original signals. This section details the process of converting the original time series into their corresponding binary vectors, as illustrated in Figure S1.
在我们的分析中,我们计算单元格 iiitalic_i 的两个时间序列之间的互相关 RisuperscriptR^{i}italic_R start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT 。为了促进这一过程,我们将两个时间序列转换为表示原始信号中的显著变化或峰值的二进制 “条形码”。本节详细介绍了将原始时间序列转换为其相应的二进制向量的过程,如图s1所示。

The upper part of Figure S1 displays two example time series, denoted as 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT and 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT. We identify key events in these time series: peaks in the continuous signal 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT and transitions in the binary signal 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT. These events are then encoded as binary vectors, 𝒢~netisuperscriptsubscript~𝒢𝑛𝑒𝑡𝑖\widetilde{\mathcal{G}}_{net}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT and 𝒢~binisuperscriptsubscript~𝒢𝑏𝑖𝑛𝑖\widetilde{\mathcal{G}}_{bin}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT, representing the time points at which the respective events occur. Specifically:
图S1的上部显示了两个示例时间序列,分别表示为 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 。我们识别这些时间序列中的关键事件: 连续信号 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 中的峰值和二进制信号 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 中的转变。然后将这些事件编码为二进制向量 𝒢~netisuperscriptsubscript\widetilde{\mathcal{G}}_{net}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT𝒢~binisuperscriptsubscript\widetilde{\mathcal{G}}_{bin}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ,表示相应事件发生的时间点。具体而言:

  • For 𝒢~netisuperscriptsubscript~𝒢𝑛𝑒𝑡𝑖\widetilde{\mathcal{G}}_{net}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT: A peak detection algorithm is applied to the time series 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT. If a peak is detected at a specific time tksubscript𝑡𝑘t_{k}italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, we assign (𝒢~neti)tk=1subscriptsuperscriptsubscript~𝒢𝑛𝑒𝑡𝑖subscript𝑡𝑘1(\widetilde{\mathcal{G}}_{net}^{i})_{t_{k}}=1( over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 1; otherwise, (𝒢~neti)tk=0subscriptsuperscriptsubscript~𝒢𝑛𝑒𝑡𝑖subscript𝑡𝑘0(\widetilde{\mathcal{G}}_{net}^{i})_{t_{k}}=0( over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0. Thus, the binary vector 𝒢~netisuperscriptsubscript~𝒢𝑛𝑒𝑡𝑖\widetilde{\mathcal{G}}_{net}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT captures the occurrence of peaks in 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT over the time period T𝑇Titalic_T.


    • 对于 𝒢~netisuperscriptsubscript\widetilde{\mathcal{G}}_{net}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT : 将峰值检测算法应用于时间序列 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 。如果在特定时间 tksubscriptt_{k}italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 检测到峰值,则分配 (𝒢~neti)tk=1subscriptsuperscriptsubscriptsubscript1(\widetilde{\mathcal{G}}_{net}^{i})_{t_{k}}=1( over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 1 ; 否则,分配 (𝒢~neti)tk=0subscriptsuperscriptsubscriptsubscript0(\widetilde{\mathcal{G}}_{net}^{i})_{t_{k}}=0( over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0 。因此,二进制向量 𝒢~netisuperscriptsubscript\widetilde{\mathcal{G}}_{net}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPTTTitalic_T 时间段内捕获 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT 中的峰值的出现。
  • For 𝒢~binisuperscriptsubscript~𝒢𝑏𝑖𝑛𝑖\widetilde{\mathcal{G}}_{bin}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT: Similarly, we track the 0-1 transitions in the binary signal 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT. If a transition from 0 to 1 occurs at time tksubscript𝑡𝑘t_{k}italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, we assign (𝒢~bini)tk=1subscriptsuperscriptsubscript~𝒢𝑏𝑖𝑛𝑖subscript𝑡𝑘1(\widetilde{\mathcal{G}}_{bin}^{i})_{t_{k}}=1( over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 1; otherwise, (𝒢~bini)tk=0subscriptsuperscriptsubscript~𝒢𝑏𝑖𝑛𝑖subscript𝑡𝑘0(\widetilde{\mathcal{G}}_{bin}^{i})_{t_{k}}=0( over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0. This forms the binary vector 𝒢~binisuperscriptsubscript~𝒢𝑏𝑖𝑛𝑖\widetilde{\mathcal{G}}_{bin}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT, encoding significant shifts in 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT.


    • 对于 𝒢~binisuperscriptsubscript\widetilde{\mathcal{G}}_{bin}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT : 类似地,我们跟踪二进制信号 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 中的0-1跃迁。如果从0到1的转换发生在时间 tksubscriptt_{k}italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ,我们分配 (𝒢~bini)tk=1subscriptsuperscriptsubscriptsubscript1(\widetilde{\mathcal{G}}_{bin}^{i})_{t_{k}}=1( over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 1 ; 否则, (𝒢~bini)tk=0subscriptsuperscriptsubscriptsubscript0(\widetilde{\mathcal{G}}_{bin}^{i})_{t_{k}}=0( over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0 。这形成二进制向量 𝒢~binisuperscriptsubscript\widetilde{\mathcal{G}}_{bin}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ,编码 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 中的显著移位。

The lower part of Figure S1 illustrates the resulting binary vectors, 𝒢~netisuperscriptsubscript~𝒢𝑛𝑒𝑡𝑖\widetilde{\mathcal{G}}_{net}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT and 𝒢~binisuperscriptsubscript~𝒢𝑏𝑖𝑛𝑖\widetilde{\mathcal{G}}_{bin}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT, which are derived from the original time series. These binary representations, also referred to as ”barcodes,” simplify the identification of coinciding events (peaks and transitions) between the two time series, making it easier to compute the cross-correlation Risuperscript𝑅𝑖R^{i}italic_R start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT.
图S1的下部说明了从原始时间序列导出的结果二进制向量 𝒢~netisuperscriptsubscript\widetilde{\mathcal{G}}_{net}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT𝒢~binisuperscriptsubscript\widetilde{\mathcal{G}}_{bin}^{i}over~ start_ARG caligraphic_G end_ARG start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT 。这些二进制表示,也称为 “条形码”,简化了两个时间序列之间的重合事件 (峰值和过渡) 的识别,使其更容易计算互相关 RisuperscriptR^{i}italic_R start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT

It is important to note that in our analysis, the ”left” time series (i.e., the series corresponding to negative lags) represents 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT, while the ”right” time series (i.e., the series corresponding to positive lags) represents 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT.
值得注意的是,在我们的分析中,“左” 时间序列 (即,对应于负滞后的序列) 表示 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT ,而 “右” 时间序列 (即,对应于正滞后的级数) 表示 𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT

Refer to caption
Figure S1: Example of barcode
图S1: 条形码示例

Supplementary Information 2: Comparison with null model
补充信息2: 与null模型的比较

We devised a null model out of our main model. In this version, agents behave as pure random walkers:
我们从主模型中设计了一个空模型。在这个版本中,代理的行为就像纯粹的随机漫步者:

  • At each time step, every agent decides whether to move with a uniformly random probability (e.g., 50% chance of moving).


    • 在每个时间步,每个代理决定是否以均匀随机概率 (例如,50% 移动的机会) 移动。
  • If an agent decides to move, it relocates to a cell with space chosen with a uniformly random probability throughout the grid.


    • 如果智能体决定移动,则其重新定位到具有在整个网格内以均匀随机概率选择的空间的单元。

The null model completely removes both the evaluation criteria and targeted relocation, providing a baseline for comparison with purely random motion. We applied the same statistical analysis used in the main text to assess the statistical relevance of 𝒢net𝒢𝑛𝑒𝑡\mathcal{G}{net}caligraphic_G italic_n italic_e italic_t anticipating 𝒢bin𝒢𝑏𝑖𝑛\mathcal{G}{bin}caligraphic_G italic_b italic_i italic_n for this baseline model. The results are presented in Figure S2. Two key observations emerge:
空模型完全消除了评估标准和目标重定位,为与纯随机运动进行比较提供了基线。我们应用了正文中使用的相同统计分析来评估 𝒢net\mathcal{G}{net}caligraphic_G italic_n italic_e italic_t 预期 𝒢bin\mathcal{G}{bin}caligraphic_G italic_b italic_i italic_n 对于该基线模型的统计相关性。结果显示在图s2中。出现了两个关键的观察结果:

  1. 1.

    The null model does not produce statistically significant results, indicating an absence of observable anticipatory patterns.


    1.空模型不产生统计上显著的结果,表明不存在可观察的预期模式。
  2. 2.

    The peak of the cross-correlation, although not significant, occurs at τ=1𝜏1\tau=-1italic_τ = - 1. This is an expected outcome, as the 𝒢net𝒢𝑛𝑒𝑡\mathcal{G}{net}caligraphic_G italic_n italic_e italic_t measure is based on flow data, while 𝒢bin𝒢𝑏𝑖𝑛\mathcal{G}{bin}caligraphic_G italic_b italic_i italic_n is derived from count data. Specifically, the flow of agents moving at one time step (𝒢net𝒢𝑛𝑒𝑡\mathcal{G}{net}caligraphic_G italic_n italic_e italic_t) is reflected in the count of agents (𝒢bin𝒢𝑏𝑖𝑛\mathcal{G}{bin}caligraphic_G italic_b italic_i italic_n) in the subsequent time step, naturally creating a lag of -1.


    2.互相关的峰值虽然不显著,但出现在 τ=11\tau=-1italic_τ = - 1 处。这是预期的结果,因为 𝒢net\mathcal{G}{net}caligraphic_G italic_n italic_e italic_t 度量基于流数据,而 𝒢bin\mathcal{G}{bin}caligraphic_G italic_b italic_i italic_n 从计数数据导出。具体而言,在一个时间步长 ( 𝒢net\mathcal{G}{net}caligraphic_G italic_n italic_e italic_t ) 移动的代理流反映在随后的时间步长中的代理计数 ( 𝒢bin\mathcal{G}{bin}caligraphic_G italic_b italic_i italic_n ) 中,自然会产生-1的滞后。

The absence of statistically significant anticipatory patterns in the null model reinforces the validity of our main model’s findings. It suggests that the observed anticipatory behaviour is not a random artefact but rather an emergent property arising from the interplay of our model’s evaluation criteria and targeted relocation strategies. This comparison provides strong evidence for the robustness of our main model and the significance of its results in capturing complex socio-economic dynamics.
零模型中没有统计上显着的预期模式,这增强了我们主要模型发现的有效性。这表明观察到的预期行为不是随机的伪像,而是由我们模型的评估标准和目标重定位策略的相互作用而产生的紧急属性。这种比较为我们的主要模型的稳健性及其结果在捕获复杂的社会经济动态方面的重要性提供了有力的证据。

Refer to caption
Figure S2: Cross-correlation analysis of 𝒢net𝒢𝑛𝑒𝑡\mathcal{G}{net}caligraphic_G italic_n italic_e italic_t and 𝒢bin𝒢𝑏𝑖𝑛\mathcal{G}{bin}caligraphic_G italic_b italic_i italic_n for the null model. The x-axis represents the lag τ𝜏\tauitalic_τ, while the y-axis shows the cross-correlation coefficient. The shaded area indicates the 95% confidence interval. Note the lack of statistically significant anticipatory patterns in the null model, with peak at τ=1𝜏1\tau=-1italic_τ = - 1 due to the nature of the measured variables.
图S2: 空模型的 𝒢net\mathcal{G}{net}caligraphic_G italic_n italic_e italic_t𝒢bin\mathcal{G}{bin}caligraphic_G italic_b italic_i italic_n 的互相关分析。X轴表示滞后 τ\tauitalic_τ ,而y轴表示互相关系数。阴影区域表示95% 置信区间。请注意,由于测量变量的性质,在零模型中缺乏统计上显着的预期模式,在 τ=11\tau=-1italic_τ = - 1 处具有峰值。

Supplementary Information 3: Early warning consistency across parameters and measures
补充信息3: 参数和措施之间的预警一致性

In the main text, we presented the results of the cross-correlation analysis for a single value of pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT, the probability of high-income agents relocating. Here, we demonstrate that the observed anticipatory patterns are robust across different values of pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT and alternative analytical measures.
在正文中,我们介绍了 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 的单个值的互相关分析结果,即高收入代理商搬迁的可能性。在这里,我们证明了观察到的预期模式在 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 的不同值和替代分析措施中是稳健的。

Figure S3 illustrates the consistency of the cross-correlation analysis results for various pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT values. The anticipatory relationship between 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT and 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT remains stable across different probabilities of high-income agent relocation, indicating that this phenomenon is not sensitive to specific parameter choices within our model.
图S3说明了各种 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 值的互相关分析结果的一致性。 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 之间的预期关系在高收入代理人搬迁的不同概率上保持稳定,表明这种现象对我们模型内的特定参数选择不敏感。

To further validate our findings, we conducted an additional analysis using Mutual Information (MI) instead of cross-correlation, as shown in Figure S4. MI provides a more general measure of statistical dependence, capturing both linear and non-linear relationships between variables. The results from this analysis corroborate our cross-correlation findings, exhibiting similar anticipatory patterns. It’s worth noting that while the overall trends are consistent, the absolute values of MI are approximately one order of magnitude smaller than those of the cross-correlation analysis.
为了进一步验证我们的发现,我们使用互信息 (MI) 而不是互相关进行了额外的分析,如图s4所示。MI提供了统计相关性的更一般的度量,捕获变量之间的线性和非线性关系。此分析的结果证实了我们的互相关发现,显示出类似的预期模式。值得注意的是,虽然整体趋势是一致的,但MI的绝对值比互相关分析的绝对值小大约一个数量级。

The consistency of results across different pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT values and analytical methods reinforces the robustness of our findings, suggesting that the anticipatory patterns observed in our model are a fundamental feature of the gentrification dynamics we’ve simulated, rather than an artifact of specific parameter choices or analytical approaches.
不同 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 值和分析方法的结果的一致性加强了我们研究结果的稳健性,这表明在我们的模型中观察到的预期模式是我们模拟的绅士化动力学的基本特征,而不是特定参数选择或分析方法的产物。

Refer to caption
Figure S3: Cross-correlation analysis between 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT and 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT for various pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT values. Each panel represents a different pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT. The x-axis shows the lag τ𝜏\tauitalic_τ, and the y-axis represents the cross-correlation coefficient. Shaded areas indicate 95% confidence intervals. The consistent peak at negative lags across all pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT values demonstrates the robustness of the anticipatory relationship between 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT and 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT, regardless of the probability of high-income agent relocation.
图S3: 对于各种 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 值, 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 之间的互相关分析。每个面板代表不同的 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 。X轴表示滞后 τ\tauitalic_τ ,y轴表示互相关系数。阴影区域表示95% 的置信区间。所有 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 值的负滞后处的一致峰值证明了 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 之间的预期关系的稳健性,而不考虑高收入代理商搬迁的可能性。
Refer to caption
Figure S4: Mutual Information (MI) analysis between 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT and 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT for various pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT values. Each panel corresponds to a different pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT. The x-axis represents the lag τ𝜏\tauitalic_τ, while the y-axis shows the MI value. Shaded areas indicate 95% confidence intervals. The patterns observed are consistent with the cross-correlation analysis, exhibiting peaks at negative lags. Note that MI values are approximately one order of magnitude smaller than cross-correlation coefficients, which is expected due to the different nature of these measures. The consistency across pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT values further supports the robustness of the anticipatory relationship between 𝒢netsubscript𝒢𝑛𝑒𝑡\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT and 𝒢binsubscript𝒢𝑏𝑖𝑛\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT.
图S4: 各种 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 值的 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 之间的互信息 (MI) 分析。每个面板对应于不同的 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 。X轴表示滞后 τ\tauitalic_τ ,而y轴表示MI值。阴影区域表示95% 的置信区间。观察到的模式与互相关分析一致,表现出负滞后的峰值。注意,MI值大约比互相关系数小一个数量级,这是由于这些度量的不同性质而预期的。跨 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 值的一致性进一步支持 𝒢netsubscript\mathcal{G}_{net}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT𝒢binsubscript\mathcal{G}_{bin}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT 之间的预期关系的鲁棒性。

Supplementary Information 4: Results are Consistent for a Larger (9x9) City
补充信息4: 结果与较大 (9x9) 城市一致

To assess the scalability and robustness of our findings, we extended our analysis to a larger urban environment, specifically a 9x9 grid city. This expansion allows us to verify whether the patterns observed in our original model persist in a more complex urban setting with a greater number of neighborhoods and potential interactions.
为了评估研究结果的可扩展性和稳健性,我们将分析扩展到更大的城市环境,特别是9x9网格城市。这种扩展使我们能够验证在我们的原始模型中观察到的模式是否在具有更多社区和潜在相互作用的更复杂的城市环境中持续存在。

Figure S5 presents the spatial arrangement and income distribution for this larger city model.
图S5显示了这个较大的城市模型的空间安排和收入分配。

Refer to caption
Figure S5: Income distribution and spatial arrangement in a 9x9 city model. Left panel: Heatmap depicting the median income of each cell at the initial state of the simulation for a 9x9 grid city. Darker colors indicate higher median incomes. The hierarchical structure and spatial clustering of income levels are preserved in this larger model. Right panel: log-log plot of the initial distribution of agent incomes in the 9x9 city simulation, sampled from the 2022 USA Social Security Administration report. The distribution maintains the same shape as in the main model. N = 212superscript2122^{12}2 start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPT agents.
图S5: 9x9城市模型中的收入分布和空间安排。左图: 热图,描绘了9x9网格城市模拟初始状态下每个单元的收入中位数。较深的颜色表示收入中位数较高。在这个更大的模型中,收入水平的层次结构和空间聚类得以保留。右图: 9x9城市模拟中代理人收入初始分布的对数-对数图,抽样于2022美国社会保障局的报告。分布保持与主模型中相同的形状。N = 212superscript2122^{12}2 start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPT 代理。

Figure S6 presents the results of simulations conducted on the 9x9 city grid, mirroring the analysis shown in Figure 2 of the main text. Remarkably, the gentrification patterns observed in this larger model closely resemble those of the original smaller model, providing strong evidence for the scalability and consistency of our findings.
图S6显示了在9x9城市网格上进行的模拟结果,反映了正文图2中所示的分析。值得注意的是,在这个较大的模型中观察到的绅士化模式与原始较小模型的绅士化模式非常相似,为我们发现的可扩展性和一致性提供了有力的证据。

In the 9x9 city model, we observe that the critical role of H𝐻Hitalic_H agent mobility in initiating gentrification is preserved. At pH=0subscript𝑝𝐻0p_{H}=0italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0, both 𝒢bincitysuperscriptsubscript𝒢𝑏𝑖𝑛𝑐𝑖𝑡𝑦\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT and 𝒢netcitysuperscriptsubscript𝒢𝑛𝑒𝑡𝑐𝑖𝑡𝑦\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT show 0% gentrification, indicating a complete absence of the phenomenon when high-income agents are static. The abrupt transition to gentrification with minimal H𝐻Hitalic_H agent mobility (pH=0.01subscript𝑝𝐻0.01p_{H}=0.01italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0.01) is also evident in the larger model, with both metrics showing gentrification in a significant proportion of cells. As pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT increases, we see a similar monotonic rise in gentrification levels, with 𝒢bincitysuperscriptsubscript𝒢𝑏𝑖𝑛𝑐𝑖𝑡𝑦\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT and 𝒢netcitysuperscriptsubscript𝒢𝑛𝑒𝑡𝑐𝑖𝑡𝑦\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT exhibiting comparable trends but slightly different magnitudes, consistent with the original model.
在9x9城市模型中,我们观察到 HHitalic_H 代理移动性在启动绅士化中的关键作用得以保留。在 pH=0subscript0p_{H}=0italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0 处, 𝒢bincitysuperscriptsubscript\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT𝒢netcitysuperscriptsubscript\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT 均显示出0% 的高档化,表明当高收入代理商处于静态状态时,完全没有这种现象。在较大的模型中,向具有最小 HHitalic_H 代理迁移率 ( pH=0.01subscript0.01p_{H}=0.01italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0.01 ) 的绅士化的突然转变也很明显,两个指标均显示了相当大比例的细胞中的绅士化。随着 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 的增加,我们看到了类似的绅士化水平单调上升, 𝒢bincitysuperscriptsubscript\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT𝒢netcitysuperscriptsubscript\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT 表现出可比的趋势,但幅度略有不同,与原始模型一致。

Refer to caption
Figure S6: Gentrification dynamics in a 9x9 city model. We show gentrification levels and times across 150 simulations with fixed ϵ=20italic-ϵ20\epsilon=20italic_ϵ = 20, N=212𝑁superscript212N=2^{12}italic_N = 2 start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPT, and varying values of pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT (probability of high-income agent relocation). The left panel shows gentrification levels according to 𝒢bincitysuperscriptsubscript𝒢𝑏𝑖𝑛𝑐𝑖𝑡𝑦\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT, while the right panel displays levels according to the network-based measure 𝒢netcitysuperscriptsubscript𝒢𝑛𝑒𝑡𝑐𝑖𝑡𝑦\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT. The x-axis represents different pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT values, and the y-axis shows the percentage of gentrified cells. Box plots display the distribution of gentrification levels across simulations. The patterns observed closely resemble those in the smaller model: absence of gentrification at pH=0subscript𝑝𝐻0p_{H}=0italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0, abrupt transition to gentrification at low pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT values, and monotonic increase in gentrification levels as pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT increases.
图S6: 9x9城市模型中的绅士化动态。我们显示了具有固定 ϵ=2020\epsilon=20italic_ϵ = 20N=212superscript212N=2^{12}italic_N = 2 start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPTpHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 的变化值 (高收入代理商搬迁的概率) 的150模拟中的绅士化水平和时间。左侧面板根据 𝒢bincitysuperscriptsubscript\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT 显示绅士化级别,而右侧面板根据基于网络的度量 𝒢netcitysuperscriptsubscript\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT 显示级别。X轴表示不同的 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 值,y轴表示绅士化细胞的百分比。箱形图显示模拟中绅士化水平的分布。观察到的模式与较小模型中的模式非常相似: 在 pH=0subscript0p_{H}=0italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0 处没有绅士化,在低 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 值处突然过渡到绅士化,并且随着 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 增加,绅士化水平单调增加。

Figure S7 presents the results of simulations conducted on the 9x9 city grid, mirroring the analysis shown in Figure 5 of the main text. The gentrification patterns observed in this larger model closely resemble those of the original smaller model, providing strong evidence for the scalability and consistency of our findings.
图S7显示了在9x9城市网格上进行的模拟结果,反映了正文图5中所示的分析。在这个较大模型中观察到的绅士化模式与原始较小模型的绅士化模式非常相似,为我们发现的可扩展性和一致性提供了有力的证据。

In the bigger city model, we observe that the relationship between urban density and gentrification levels remains consistent across different values of the model parameter pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT. As in the smaller model, city-wise gentrification levels, captured by both 𝒢netcitysuperscriptsubscript𝒢𝑛𝑒𝑡𝑐𝑖𝑡𝑦\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT and 𝒢bincitysuperscriptsubscript𝒢𝑏𝑖𝑛𝑐𝑖𝑡𝑦\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT, show a clear trend: as city population density increases, so does the propensity for gentrification. This effect is amplified by the H𝐻Hitalic_H agents’ relocation rate pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT, with the curves increasing monotonically with N𝑁Nitalic_N for most values of pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT.
在更大的城市模型中,我们观察到城市密度与绅士化水平之间的关系在模型参数 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 的不同值上保持一致。与较小的模型一样, 𝒢netcitysuperscriptsubscript\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT𝒢bincitysuperscriptsubscript\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT 捕获的城市高档化水平显示出明显的趋势: 随着城市人口密度的增加,高档化的倾向也随之增加。这种影响被 HHitalic_H 代理的重新定位率 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 放大,对于 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 的大多数值,曲线随 NNitalic_N 单调增加。

The relationship between city density and the average convergence time Tdelimited-⟨⟩𝑇\langle T\rangle⟨ italic_T ⟩ in the larger model also mirrors the findings from the original model. The average convergence time increases with both N𝑁Nitalic_N and pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT, except when pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0. In extremely dense scenarios with stationary H𝐻Hitalic_H agents, the model may not reach the termination condition within the imposed step limit, consistent with the behavior observed in the smaller model.
较大模型中城市密度与平均收敛时间 Tdelimited-⟨⟩\langle T\rangle⟨ italic_T ⟩ 之间的关系也反映了原始模型的发现。平均收敛时间随 NNitalic_NpHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 的增加而增加,但 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 0时除外。在具有固定 HHitalic_H 代理的极其密集的场景中,模型可能无法在施加的步长限制内达到终止条件,与较小模型中观察到的行为一致。

Refer to caption
Figure S7: Urban density and gentrification dynamics in a 9x9 city model. We show average results across 150 simulations for varying agent populations (x-axis, logarithmic scale) and high-income agent relocation rates pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT (colors). Left panel: City-wise gentrification levels measured by 𝒢netcitysuperscriptsubscript𝒢𝑛𝑒𝑡𝑐𝑖𝑡𝑦\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT. Middle panel: City-wise gentrification levels measured by 𝒢bincitysuperscriptsubscript𝒢𝑏𝑖𝑛𝑐𝑖𝑡𝑦\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT. Right panel: Average number of simulation time steps (convergence time Tdelimited-⟨⟩𝑇\langle T\rangle⟨ italic_T ⟩). The patterns observed closely resemble those in the smaller 7x7 model: (1) gentrification levels increase with urban density for both measures, (2) this effect is amplified by higher pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT values, and (3) convergence time generally increases with both density and pHsubscript𝑝𝐻p_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT.
图S7: 9x9城市模型中的城市密度和绅士化动态。我们显示了不同代理人群体 (x轴,对数刻度) 和高收入代理人重新安置率 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT (颜色) 的150模拟的平均结果。左图: 由 𝒢netcitysuperscriptsubscript\mathcal{G}_{net}^{city}caligraphic_G start_POSTSUBSCRIPT italic_n italic_e italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT 测量的城市绅士化水平。中间面板: 以 𝒢bincitysuperscriptsubscript\mathcal{G}_{bin}^{city}caligraphic_G start_POSTSUBSCRIPT italic_b italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_i italic_t italic_y end_POSTSUPERSCRIPT 衡量的城市高档化水平。右图: 模拟时间步长的平均数 (收敛时间 Tdelimited-⟨⟩\langle T\rangle⟨ italic_T ⟩ )。观察到的模式与较小的7x7模型中的模式非常相似 :( 1) 两种措施的高档化水平均随城市密度的增加而增加,(2) 较高的 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 值会放大这种影响,(3) 收敛时间通常随密度和 pHsubscriptp_{H}italic_p start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT 的增加而增加。

These observations suggest that the fundamental dynamics of our model, including the processes driving gentrification are not artifacts of the specific grid size used in the main text. Instead, they appear to be scalable properties that emerge from the underlying mechanisms of our agent-based model.
这些观察结果表明,我们模型的基本动力学,包括驱动高档化的过程,并不是正文中使用的特定网格大小的工件。相反,它们似乎是可扩展的属性,这些属性来自我们基于代理的模型的基础机制。