Dynamic models of gentrification
绅士化的动态模型
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
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 agents, categorised into three socioeconomic groups: low-income (), middle-income (), and high-income ().
At the beginning of a simulation, each agent is assigned a fixed income , sampled from real-world data, and all agents are divided into three groups according to their assigned incomes: accounting for 38% of the population; accounting for 57% of the population; and 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 代理人填充这个网格,分为三个社会经济群体: 低收入 ( ) 、中等收入 ( ) 和高收入 ( )。在模拟开始时,每个代理被分配一个固定收入 ,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: 占人口的38%; 占人口的57%; 和 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: agents predominantly occupy the city centre, agents populate the inner areas, and 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) has a fixed maximum capacity , which limits its occupancy, that is, how many agents can stay there. When , the number of agents in the cell , reaches maximum capacity , agents are allocated to the nearest available locations.
Our model operates based on two key parameters: the parameter that corresponds to the probability that a agent relocates from its current cell to a new one where the median agent-income is increasing; the parameter that is the width of the time window over which 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 (99 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: 代理人主要占据市中心, 代理人居住在内部区域, 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) 都有一个固定的最大容量 ,这限制了它的占用率,即有多少座席可以留在那里。当 (单元格 中的座席数) 达到最大容量 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 ,它对应于 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 是 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 9个网格,81个社区),获得了类似的结果 (请参阅补充信息4)。
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 has a maximum capacity . Let denote the number of agents in cell 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 具有最大容量 。令 表示在任何给定时间单元格 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 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 agent leaving cell at time , shown in Figure LABEL:fig:fig_0a, is given by:
如果 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。 代理在时间 离开单元格 的概率,如图所示: fig_0a,由下式给出:
(1) |
where represents the agent’s relative income percentile within cell ’s income distribution at time . Upon deciding to move, agents select from the set of available cells, defined as cells where . As illustrated in Figure LABEL:fig:fig_0b, the probability of selecting cell is:
其中 表示代理在单元格 的时间 的收入分布中的相对收入百分位数。在决定移动时, 代理从可用单元的集合 中选择,定义为 的单元。如图所示: fig_0b,选择单元格 的概率为:
(2) |
where is the cell’s attractiveness score based on the agent’s prospective income percentile in cell .
其中 是单元格的吸引力分数,基于单元格 中代理的预期收入百分位数。
A agent is likely to move away when its income significantly deviates from the median income in the current cell.
The agent will move to a cell where the gap with the median income is lower. Their relocation probability, visualized in Figure LABEL:fig:fig_0c, follows::
当 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:
(3) |
where is the agent’s relative income percentile in cell . Figure LABEL:fig:fig_0d shows how their destination selection probability is determined by:
其中 是单元格 中 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:
(4) |
where scores cells based on proximity to their median income.
其中 根据与收入中位数的接近程度对单元格进行评分。
For both and agents, we choose nonlinear functional forms (square root for , quadratic for ) to ensure that small differences in percentiles lead to larger differences in probabilities when agents are far from their preferred positions. The denominators in and serve as normalisation factors, ensuring proper probability distributions.
对于 和 代理,我们都选择非线性函数形式 ( 的平方根,对于 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。 和 中的分母作为归一化因子,确保适当的概率分布。
A agent moves with a fixed probability that reflects the frequency with which profit-orientated investments are made in the city. Let denote the median income in cell at time . agents select from set of cells satisfying both and , where is the average growth rate of median income over the past time steps, as reported in (Figure LABEL:fig:fig_0e-f):
代理以固定概率 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 表示单元格 在时间 中的中值收入。 代理从满足 和 的单元格集 中选择,其中 是过去 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:
(5) |
Their destination probability is:
他们的目的地概率是:
(6) |
The simulation terminates at time when all 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 终止,此时所有 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 , , or 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 , 或 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:
(7) |
where denotes the set of nodes (grid cells) in the network, represents the layer index corresponding to , or agents, and is the adjacency matrix corresponding to the network in layer at time .
The matrix elements of correspond to weighted, directed edges connecting node pairs in layer , representing the relocation flows of agents of type between contiguous time steps of the agent-based model.
For each layer at time , we define three quantities: , the number of agents in node , and and , the in- and out-strength of node , respectively. The latter two quantify the flow of agents in moving to or from node between and :
其中 表示网络中的节点 (网格单元) 集合, 表示与 , 或 代理对应的层索引,并且 是在 时刻 层中的网络对应的邻接矩阵。 的矩阵元素对应于连接层 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 类型的代理的重定位流。对于时间 的每个层 ,我们定义三个量: ,节点中的代理数 , 和 ,分别是节点 的输入和输出强度。后两者量化了 在 和 之间往返于节点 的代理流:
(8) |
where represents the element of the weighted adjacency matrix for layer at time .
其中 表示层 在时间 的加权邻接矩阵的 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 , based on the number of agents from each socioeconomic class present in node at two time points: the current time and an earlier time . For each time point, we consider the counts of , and agents. At time , these counts are represented by , , and , respectively. Similarly, at time , the counts are denoted by , , and . The definition of 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 ,基于节点 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 和较早时间 。对于每个时间点,我们考虑 , 和 代理的计数。在时间 ,这些计数分别由 、 和 表示。类似地,在时间 ,计数由 、 和 表示。 的定义如下:
(9) |
represents the average fraction of and agents in node over the time window . We establish a significance threshold , defined as the expected node-wise fraction of and agents under uniform random distribution across the grid.
Values of exceeding indicate an over-representation of agents. Gentrification is identified when transitions from below to above this threshold. This metric functions as a node property, independent of the network edge dynamics.
表示节点 中的 和 代理在时间窗口 上的平均分数。我们建立了一个显著性阈值 ,定义为在网格上均匀随机分布下 和 代理的预期节点分数。 超过 的值表示 代理的过度表示。当 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 , representing the city-wide proportion of and agents.
To precisely capture gentrification events, we introduce the binary indicator :
为了识别绅士化,我们建立了一个临界阈值 ,代表 和 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 :
(10) |
We define , the onset of a gentrification event, as the moment when shifts from 0 to 1, indicating that a neighborhood has crossed the critical population threshold. More mathematical details about can be found in Methods.
我们将 ,即绅士化事件的开始,定义为 从0变为1的时刻,表明一个邻域已经越过临界 人口阈值。关于 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, that captures relocation patterns between neighborhoods. For any node in the network, this measure considers the net-outflow of agents, , and the net-inflow of and agents, :
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 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 ,此措施都考虑 代理, , 和 代理的净流入, :
(11) |
(12) |
where and .
其中 和 。
is high when the outflow of agents from node corresponds to a high fraction of the overall outflow of agents from , i.e., the denominator in Equation (11); is high when the inflow of and agents to node corresponds to a high fraction of the overall inflow of agents towards , i.e., the denominator in Equation (12). is therefore defined as the geometric mean of the averages of and over the last steps :
当来自节点 的 代理的流出对应于来自 的代理的总流出的高比例时, 高,即等式 (11) 中的分母; 当 和 代理向节点 的流入对应于代理向 的总流入的大部分时, 高,即,式 (12) 中的分母。因此, 被定义为 和 在最后一步的平均值的几何平均值 :
(13) |
In Equation (13), we consider only positive values of and . This approach ensures that high values of occur only when both the outflow of agents and the inflow of and agents are substantial during the same period. Including negative values would erroneously indicate gentrification in areas where agents replace and agents, which actually signals neighbourhood impoverishment.
在等式 (13) 中,我们仅考虑 和 的正值。这种方法确保只有当 代理的流出和 和 代理的流入在同一时期大量时, 才会出现高值。包括负值将错误地指示 代理取代 和 代理的地区的绅士化,这实际上标志着邻里贫困。
We define as the time of a gentrification event, corresponding to a local maximum in or the onset of a plateau after rapid growth (see Methods for details).
我们将 定义为绅士化事件的时间,对应于 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: , the percentage of nodes experiencing transitions in , and , the percentage of nodes showing peaks in (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: ,在 和 中经历转换的节点的百分比,在 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 agent mobility on gentrification dynamics. Simulations were conducted with varying agent movement probabilities (), while maintaining a fixed evaluation window of time steps for node growth rates.
我们研究了 代理移动性对绅士化动力学的影响。使用变化的 代理移动概率 ( ) 进行模拟,同时保持节点增长率的 时间步长的固定评估窗口。
Figure 2 illustrates the influence of on spatial gentrification patterns.
At , both and yield 0%, indicating complete absence of gentrification when agents are static. Introducing minimal agent mobility () triggers gentrification in 20-40% of nodes, according to both metrics. This abrupt transition highlights the critical role of agent mobility in initiating the gentrification process. As increases, we observe a monotonic rise in gentrification levels, with and showing similar trends but slightly different magnitudes.
图2说明了 对空间绅士化模式的影响。在 时, 和 均产生0%,表明当 试剂为静态时完全没有绅士化。根据两个指标,引入最小的 代理移动性 ( ) 会触发20-40% 个节点的绅士化。这种突然的转变凸显了 代理移动性在启动绅士化过程中的关键作用。随着 的增加,我们观察到绅士化水平的单调上升, 和 显示出相似的趋势,但幅度略有不同。
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 provides an intuitive measure of gentrification by capturing demographic transitions from to agent majorities, enables earlier detection by identifying patterns of coordinated movement through temporal networks, revealing gentrification dynamics before visible demographic shifts occur.
Figure 3a displays and curves for a node in a representative simulation. In this example, the node initially experiences impoverishment, as transitions from 1 to 0 at approximately , indicating a shift from over- to under-representation of and residents. However, shows a rapid increase at , followed by a plateau at , approximately 10 steps before the abrupt transition from 0 to 1 in at .
虽然 通过捕获从 到 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 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 和 曲线。在此示例中,节点最初经历贫化,因为 在大约 处从1过渡到0,表示 和 居民的代表性从高到低的转变。然而, 在 处显示快速增加,随后在 处显示平稳,在 处 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 peak for the same node (neighborhood) analyzed in Figure 3a. Red arrows indicate agent outflows, and blue arrows show inflows. At the peak (), both occur simultaneously, with more agents leaving. After the peak, inflows increase while outflows decrease. Unlike the gross flows in the visualization, 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 峰期间和之后的重定位动态。红色箭头表示 代理流出,蓝色箭头表示 流入。在峰值 ( ),两者同时发生,更多的 代理离开。在峰值之后, 流入增加,而 流出减少。与可视化中的总流量不同, 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 peaks precede transitions across our simulations, we conduct a lagged cross-correlation analysis between and 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 (see Methods for details), with significance tested against a null distribution. The highest correlation at shows that peaks systematically anticipate 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 relative to transitions disappears in the randomised version of the model (see Supplementary Information 2 for detailed analysis). This result support our main findings, demonstrating that reliably anticipates 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 峰先于 跃迁的一致性,我们在 和 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 的互相关 (详见方法),并针对空分布进行显著性测试。 处的最高相关性表明 峰系统地预测 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。 相对于 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 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 。
We analyse the relationship between and socioeconomic neighborhood dynamics in Figure 4. Nodes exhibiting peaks are colour-coded, while those where 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 peaks. A notable exception (pink curve) displays a sharp richness increase followed by a steep decline, ultimately transitioning from low to middle income. 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 to identify gentrification solely based on relocation patterns.
我们在图4中分析了 与社会经济邻域动态之间的关系。显示 峰的节点用颜色编码,而 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 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。 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 仅基于重定位模式识别高档化的能力。
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 . We conduct 150 simulations for each configuration, with fixed values of the parameters and . To model increasing urban density, we varied the number of agents () present in the grid, while keeping the grid dimension constant at 7x7 along with the capacity of individual nodes .
在图5a-c中,我们显示了模型参数 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 和 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 ( ),同时保持网格维度恒定为7x7以及各个节点的容量 。
City-wise gentrification levels, as captured by (Figure 5a) and (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 agents relocation rate , as the curves in the two figures increase monotonically with , with the exception of 0 or very low values of and low for , as shown in Figure 5a.
In Figure 5c, we show the relationship between city density and the average convergence time , the mean number of steps required to reach the termination condition over the 150 simulations, where agents can no longer move. The average convergence time increases with both and , except when = 0. In this case, where 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 .
(图5a) 和 (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 代理重定位率 放大,因为两个图中的曲线随 单调增加,除了 的 和低 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 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 代理不再移动。 平均收敛时间 随 和 而增加,但 = 0时除外。在这种情况下,在 代理存在但固定且城市极其密集的情况下,当 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。
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 ( = 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 ( = 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, and , represent two distinct approaches to quantify gentrification. The count-based measure evaluates the concentration of agents of the three types in each neighborhood over a time window . A significance threshold () is needed to detect when middle- and high-income agents are over-represented and, therefore, define the binarised version of the count-based measure, . 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 tracks the net inflow of middle- and high-income agents and the simultaneous outflow of lower-income agents over . 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. 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 are consistently and significantly observed in advance with respect to . 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 enables earlier detection of gentrification compared to count-based metrics, potentially aiding policymakers in preventing low-income displacement.
我们的措施, 和 ,代表了量化绅士化的两种不同方法。基于计数的度量 评估在时间窗口 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 ( ) 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 。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 跟踪中高收入代理商的净流入以及低收入代理商在 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。 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 中的峰值如何相对于 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 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 77 and 99 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 7和9 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 , 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 is randomly selected from within their assigned bracket.
Agents are categorized into three groups based on their assigned incomes: (low-income) agents with incomes up to $29,999.99, encompassing the 2022 poverty line for a family of four ($27,750); (high-income) agents representing the top 5% of earners; and (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 。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 。代理人根据其分配的收入分为三类: (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); (高收入) 代理人代表收入最高的5%; (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 undergoes gentrification events according to at all times where there is a binary shift from 0 to 1 in the time series throughout the simulation:
单元格 始终根据 经历绅士化事件 ,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:
(14) |
A cell undergoes gentrification events according to at all times 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 根据 在所有时间 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:
(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:
(16) |
where is the indicator function.
其中 是指标函数。
Lagged cross-correlation 滞后互相关
We compute the mean cross-correlation by calculating the lagged cross-correlation between pairs of time series of and corresponding to each cell on the grid-view of a model city, for several values of the lag , and then averaging over all cells. To compute the cross-correlation between the two time series for a cell , we transform the two time series into two binary vectors and of length , where the -th entry is if the original corresponding time series has a peak or 0-1 transition at time , 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 和 的时间序列对之间的滞后互相关来计算平均互相关 ,对于lag 的几个值,然后对所有单元进行平均。要计算单元格 的两个时间序列之间的互相关 ,我们将两个时间序列转换为两个长度为 的二进制向量 和 ,其中 -th条目是 如果原始对应的时间序列分别在时间 处具有峰值或0-1过渡:
(17) |
(18) |
For each value of the lag , we compute the lagged cross-correlation between the two vectors and obtain a value of for each cell on the grid. We then calculate by averaging, for each value of the lag , the values of over all cells :
对于滞后 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 获得 的值。然后,我们通过对滞后 的每个值对所有单元格 上的 的值求平均值来计算 :
(19) |
To establish a baseline for comparison, we generate a null distribution by computing, for each cell , the cross-correlation between pairs of randomly reshuffled versions of the two vectors and .
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 ,两个向量 和 的随机重组版本对 之间的互相关 。
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 between two time series for a cell . 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 的两个时间序列之间的互相关 。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 and . We identify key events in these time series: peaks in the continuous signal and transitions in the binary signal . These events are then encoded as binary vectors, and , 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 和 。我们识别这些时间序列中的关键事件: 连续信号 中的峰值和二进制信号 中的转变。然后将这些事件编码为二进制向量 和 ,表示相应事件发生的时间点。具体而言:
-
•
For : A peak detection algorithm is applied to the time series . If a peak is detected at a specific time , we assign ; otherwise, . Thus, the binary vector captures the occurrence of peaks in over the time period .
• 对于 : 将峰值检测算法应用于时间序列 。如果在特定时间 检测到峰值,则分配 ; 否则,分配 。因此,二进制向量 在 时间段内捕获 中的峰值的出现。 -
•
For : Similarly, we track the 0-1 transitions in the binary signal . If a transition from 0 to 1 occurs at time , we assign ; otherwise, . This forms the binary vector , encoding significant shifts in .
• 对于 : 类似地,我们跟踪二进制信号 中的0-1跃迁。如果从0到1的转换发生在时间 ,我们分配 ; 否则, 。这形成二进制向量 ,编码 中的显著移位。
The lower part of Figure S1 illustrates the resulting binary vectors, and , 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 .
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 和 。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 。
It is important to note that in our analysis, the ”left” time series (i.e., the series corresponding to negative lags) represents , while the ”right” time series (i.e., the series corresponding to positive lags) represents .
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 ,而 “右” 时间序列 (即,对应于正滞后的级数) 表示 。
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 anticipating 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 预期 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:
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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.
The peak of the cross-correlation, although not significant, occurs at . This is an expected outcome, as the measure is based on flow data, while is derived from count data. Specifically, the flow of agents moving at one time step () is reflected in the count of agents () 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 处。这是预期的结果,因为 度量基于流数据,而 从计数数据导出。具体而言,在一个时间步长 ( ) 移动的代理流反映在随后的时间步长中的代理计数 ( ) 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。
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 , the probability of high-income agents relocating. Here, we demonstrate that the observed anticipatory patterns are robust across different values of and alternative analytical measures.
在正文中,我们介绍了 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 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 values. The anticipatory relationship between and 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 值的互相关分析结果的一致性。 和 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 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 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。
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。
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 agent mobility in initiating gentrification is preserved. At , both and show 0% gentrification, indicating a complete absence of the phenomenon when high-income agents are static. The abrupt transition to gentrification with minimal agent mobility () is also evident in the larger model, with both metrics showing gentrification in a significant proportion of cells.
As increases, we see a similar monotonic rise in gentrification levels, with and exhibiting comparable trends but slightly different magnitudes, consistent with the original model.
在9x9城市模型中,我们观察到 代理移动性在启动绅士化中的关键作用得以保留。在 处, 和 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 代理迁移率 ( ) 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 的增加,我们看到了类似的绅士化水平单调上升, 和 表现出可比的趋势,但幅度略有不同,与原始模型一致。
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 . As in the smaller model, city-wise gentrification levels, captured by both and , show a clear trend: as city population density increases, so does the propensity for gentrification. This effect is amplified by the agents’ relocation rate , with the curves increasing monotonically with for most values of .
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 的不同值上保持一致。与较小的模型一样, 和 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 代理的重新定位率 放大,对于 的大多数值,曲线随 单调增加。
The relationship between city density and the average convergence time in the larger model also mirrors the findings from the original model. The average convergence time increases with both and , except when = 0. In extremely dense scenarios with stationary 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 之间的关系也反映了原始模型的发现。平均收敛时间随 和 的增加而增加,但 = 0时除外。在具有固定 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。
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。