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Abstract

This paper seeks to clarify the concept of opportunity hoarding as it applies to Black-White educational inequalities. Two prevailing interpretations stand out: a group-disparity interpretation, which treats opportunity hoarding as any process generating group differences, and an exclusionary-behaviors interpretation, which emphasizes how White actors secure advantages through exclusionary practices. I argue that the former is too broad — remaining vague about the underlying mechanisms — and the latter too narrow, overlooking what I term hoarding without hoarders, i.e., opportunity hoarding that arises even in the absence of exclusionary behaviors. I define opportunity hoarding as the relational processes that generate racial penalties in access to resources, i.e., disparities unexplained by previously formed individual differences. Using an agent-based model of racial disparities in advanced course-taking, this paper shows how network diffusion — under segregation, consolidation of race and socioeconomic status, and temporal constraints — can produce racial penalties even when behaviors are race-neutral. The framework highlights the need for scholars and policymakers to look beyond exclusionary acts in the hoarding of valuable resources.

Keywords

opportunity hoarding racial inequality course-taking social networks agent-based modeling social theory

Introduction

A recent body of sociological research emphasizes the role of opportunity hoarding in shaping Black-White educational inequalities (see Diamond and Lewis (2022); Posey-Maddox et al. (2025) for recent literature reviews). This literature demonstrates how this well-known sociological concept can be applied to explain the greater availability of resources in majority-White districts, neighborhoods, and schools (Anderson, 2010; Faber, 2021; Goetz et al., 2019; Kebede et al., 2021; Murray et al., 2019; Posey-Maddox et al., 2014; Rury and Rife, 2018; Rury and Saatcioglu, 2011; Sattin-Bajaj and Roda, 2020), as well as to capture how White families secure better educational opportunities within schools (Lewis and Diamond, 2015; Lewis-McCoy, 2014; Murray et al., 2019, 2020).

As originally proposed by Tilly (1998: 10), the concept of opportunity hoarding captures processes through which “members of a categorically bounded network acquire access to a resource that is valuable, renewable, subject to monopoly, supportive of network activities, and enhanced by the network’s modus operandi.” This concept provides an explanation for durable inequality, i.e., for how disparities across individuals from different social groups (such as race) persist over time (across one’s life course and/or across generations).

From a theoretical perspective, this explanatory purpose can be situated within the literature on cumulative advantage, which captures the processes through which disparities persist and accumulate over time (Bask and Bask, 2015; DiPrete and Eirich, 2006; Hedström and Udehn, 2011; Lynn and Espy, 2021; Merton, 1968). Specifically, opportunity hoarding aligns closely with what DiPrete and Eirich (2006) call the cumulative exposure view of cumulative advantage: membership in a given status group or social category (such as race) has persisting direct and indirect effects in an individual’s access to resources. To illustrate, let R denote race and Y_t one’s level of resource in period t. Suppose initial resources depend on race,

Y_{1} = α R + ϵ_{1},

(1)

and let future resources be expressed as

Y_{2} = β R + γ Y_{1} + ϵ_{2} .

(2)

Here, γ captures the extent to which past racial disparities with respect to Y₁ translate into racial disparities with respect to Y₂, a persisting indirect effect of race on resource access. Further, β highlights how race continues to be a factor creating disparities even net of previously formed differences, capturing a persisting direct effect of race on resource access, what education and social stratification researchers also refer to as a group-based penalty.

Using this framework to interpret Tilly’s original definition, opportunity hoarding can be seen as an umbrella term for the different micro-level processes through which members of a well-defined social category gain (direct effects α, β) or maintain (indirect effect γ) advantages in access to valuable resources across multiple domains of social life and/or across generations, thereby restricting access for others.

Because opportunity hoarding can be understood so broadly, it has inspired a wide range of interpretations of the inequality-producing (or -reproducing) mechanisms through which it operates. In this paper, I focus specifically on racial (Black–White) opportunity hoarding in the context of education. Within this context, two interpretations stand out.

Traditional interpretations of Black–White opportunity hoarding in education

First, a series of studies, largely rooted in a quantitative tradition, define opportunity hoarding in terms of a macro-level outcome: group disparities in access to valuable resources. From this perspective, any process that produces or reproduces Black–White disparities can constitute a form of opportunity hoarding. I refer to this as the group-disparity interpretation of opportunity hoarding. For example, scholars have argued that school racial segregation is a form of opportunity hoarding because it leads Black students to attend, on average, lower-quality schools than White students (Anderson, 2010; Goetz et al., 2019; Green et al., 2017; Gruijters et al., 2024; Hanselman and Fiel, 2017; Kebede et al., 2021; Reece and O’Connell, 2016; Singer and Lenhoff, 2022; Weathers and Sosina, 2022).¹ Similarly, within-school tracking has been described as a form of opportunity hoarding for it often results in the under-representation of Black students in higher-level courses (Beattie, 2017; Hanselman, 2019; Perna et al., 2015; Price, 2021).²

Second, and perhaps more commonly, several studies, largely rooted in qualitative and historical traditions, interpret opportunity hoarding in terms of well-defined micro-level processes: the ways in which White individuals (intentionally or unintentionally) engage in racial exclusionary behaviors, keeping valuable resources within their networks and excluding other groups from access — e.g., Alvord and Rauscher (2021); Castro et al. (2022); Sattin-Bajaj and Roda (2020); Rury (2020); Lewis and Diamond (2015); Lewis-McCoy (2014). I refer to this as the exclusionary-behaviors interpretation of opportunity hoarding. Within this literature, two kinds of exclusionary behaviors stand out. First, what I call administrative pressures—where White families (intentionally or unintentionally) leverage their economic and political influence to allocate resources in ways that benefit White communities (Anderson, 2022; Diamond and Lewis, 2022).³ Second, what is known as in-group favoritism, which manifests in the favoring of same-race ties in the circulation of valuable resources — such as information about employment opportunities or job recommendations (Aboud et al., 2003; Currarini and Mengel, 2016; DiTomaso, 2013; McDonald et al., 2013; Royster, 2003) — as well as in the (overt or covert) exclusion of racial minorities from spaces — such as parent-teacher organizations — that are central in shaping communities’ educational landscapes (Lewis-McCoy, 2014; Posey-Maddox, 2017).

However, these two interpretations fall short of capturing the full conceptual scope of opportunity hoarding. A general limitation is that neither distinguishes between two clearly distinct explananda: (a) how certain social groups maintain advantages — i.e., the persisting indirect effect of group membership on resource access (γ, equation (2)); or (b) how they gain new advantages, i.e., the persisting direct effect of group membership on resource access even net of previously formed individual disparities (β, equation (2)). While both these processes can be seen as legitimate explanatory targets of opportunity hoarding, they rest on distinct explanatory dynamics and should therefore be treated separately. In addition, the group-disparities interpretation is too inclusive about the mechanisms which can generate opportunity hoarding, whereas the exclusionary-behaviors interpretation is too restrictive. Note that the group-disparities interpretation is vague about the specific micro-level mechanisms with which opportunity hoarding is concerned. Because it is so expansive — both in its target and in its mechanisms — this interpretation drains the concept of explanatory relevance, allowing virtually any process that produces or reproduces group-level disparities to be labeled opportunity hoarding. Analogous to a conceptual challenge already noted with respect to cumulative advantage (DiPrete and Eirich, 2006),⁴ it effectively renders opportunity hoarding as a description, rather than an explanation, for racial disparities. In contrast, while the exclusionary-behaviors interpretation is more specific about the mechanisms of interest, it overlooks a possibility I articulate in this paper: opportunity hoarding can arise even in the absence of exclusionary behaviors.

The present study

To further the conceptual precision of this valuable sociological concept, this theoretical paper develops an account of opportunity hoarding that responds to the limitations of existing interpretations. The proposed framework focuses on one specific explanandum: how certain groups gain new advantages even net of previously formed individual differences (β in equation (2)). Further, addressing the expansiveness of the group-disparities view, it argues that opportunity hoarding is concerned with relational mechanisms. Finally, complementing the exclusionary-behaviors interpretation, I develop an agent-based model which shows that, through network diffusion — under racially segregated networks, the consolidation of race and socioeconomic status, and temporal constraints for diffusion —, opportunity hoarding can emerge even when individuals act in race-neutral ways, a process I conceptualize as hoarding without hoarders.⁵

Background

Defining opportunity hoarding

Generally, an explanation should involve two components: the explanandum (the pattern to be explained) and the mechanisms which generate it, that is, the chain of micro-level processes which, under well-defined conditions, produce the outcome of interest (Elster, 2006; Hedstrom and Swedberg, 1998; Hedström and Ylikoski, 2010).

This paper focuses on one specific explanandum: the persisting direct effects of group membership (β in equation (2)), or, to use more common terminology, a persisting group-based penalty. This focus aligns with one of Tilly’s central goals in introducing the concept of opportunity hoarding: to challenge the individualistic paradigm which emphasized the role of individual attributes — such as “human capital, ambition, educational credentials” (Tilly, 1998: 22) — in explaining group disparities (Posey-Maddox et al., 2025). His central thesis, in fact, was that “[l]arge, significant inequalities in advantages among human beings correspond mainly to categorical differences such as black/white, male/female, citizen/foreigner, or Muslim/Jew rather than to individual differences in attributes, propensities or performances” (Tilly, 1998: 7). Here, the focus on group-based penalties is clear: opportunity hoarding tries to explain group disparities in access to a valuable resource that cannot be explained by previously formed differences in individual attributes.

To define the mechanisms of interest, recall that the concept of opportunity hoarding was originally introduced to emphasize the role of social interactions in producing group-level disparities. As Tilly (1998: 236) writes, the concept highlights that “[b]onds, not essences, provide the bases of durable inequality.” Following this original definition, opportunity hoarding is concerned with relational (or network-based) micro-level processes, that is, processes that fundamentally depend on social interactions (Emirbayer, 1997).

From this perspective, racial opportunity hoarding can be defined as the relational processes that generate racial penalties in access to valuable resources, i.e., disparities unexplained by previously formed individual differences.

Network diffusion and intergroup inequality

The dynamic of hoarding without hoarders proposed here builds on a rich sociological tradition showing how the diffusion of network-based resources — e.g., information, practices, norms, and behaviors— contribute to social stratification (Coleman, 1988; Granovetter, 1973; Portes, 1998). This literature demonstrates, for instance, how network diffusion shapes inequalities in employment opportunities (Montgomery, 1992), migration patterns (DiMaggio and Garip, 2011; Garip, 2008), health outcomes (Pampel et al., 2010; Smith and Christakis, 2008), and educational choices (Manzo, 2013; McFarland and Rodan, 2009).

To unpack the mechanisms behind opportunity hoarding, the present work focuses on outlining the conditions under which network diffusion generates racial inequalities even in the absence of exclusionary behaviors. In this sense, it closely engages with studies that identify how diffusion occurs (Centola, 2015) and when it produces intergroup inequality (DiMaggio and Garip, 2011, 2012; Zhao and Garip, 2021), applying these insights to the context of educational opportunity hoarding.

The argument advanced here follows prior studies in emphasizing that network diffusion can produce intergroup inequality through the byproduct of two conditions. First, network group segregation— a condition that network scholars have long identified as a foundation for intergroup differences emerging through diffusion (DiMaggio and Garip, 2012). Second, following more recent insights, the study here shows that homophily alone is an insufficient condition. Consolidation — i.e., the correlation among traits (such as race, income, and education) in a population (Blau and Schwartz, 1997; Skvoretz, 1983) — further shapes diffusion dynamics (Centola, 2015) and constitutes an additional necessary condition for diffusion to translate into intergroup inequality (Centola, 2015; Zhao and Garip, 2021).

At the same time, by applying such insights to model opportunity hoarding, the present study departs from prior work in a key respect. Prior models (Centola, 2015; Zhao and Garip, 2021) specify group differences in adoption thresholds — i.e., the proportion of one’s social ties that must adopt a practice (or gain access to a resource) before one does so — as central determinants of network diffusion (Centola, 2015) and, further, as a condition for the production of intergroup inequality (Zhao and Garip, 2021). By contrast, the model proposed here is concerned with inequalities that are independent of previously formed individual differences, including differences in adoption thresholds. Then, as modeled here, intergroup inequality arises through a different dynamic. Following opportunity hoarding’s emphasis on resources that are “scarce” and “subject to monopoly” (Tilly, 1998: 10), the model here represents a context in which network diffusion is instrumental in the competition for a limited opportunity (i.e., spots in advanced courses). Because of scarcity and the temporal constraints of educational trajectories, there is limited time for network diffusion, with diffusion stopping when resources saturate. Therefore, beyond network segregation and consolidation, it is this competition for limited opportunities under temporal constraints — rather than heterogeneous adoption thresholds across groups — that produces the dynamic of hoarding without hoarders.

Racial penalty of interest: Advanced high school course-taking disparities between comparable Black and White students in the same school

For concreteness, the analysis here considers Black-White disparities in access to one valuable resource: advanced high school coursework, focusing on inequalities that arise between students in the same school.

From the definition of opportunity hoarding outlined above, the explanandum of interest is the emergence of a racial penalty — specifically, a Black penalty and a White premium — in access to advanced high school coursework among students in the same school. As defined earlier, this racial penalty refers to racial disparities that cannot be explained by previously formed individual differences. Accordingly, this study focuses on disparities in advanced enrollment between Black and White students who begin high school with comparable characteristics. In what follows, I review key preexisting individual differences that can account for part of the Black–White gap in advanced enrollment, evidence that such inequalities persist even net of these factors, and common relational explanations for the remaining disparities.

Individualistic explanations

Studies highlight previously formed racial differences in key individual attributes that contribute to Black-White disparities in course-taking within the same school. These differences are largely rooted in racial socioeconomic inequalities, which, on average, provide Black and White students with contrasting experiences across many domains of social life, resulting in tangible inequalities at the time they arrive in high school. First, academic background. Black students often enter high school with lower academic preparation than their White peers, leaving them less likely to be “on track” for advanced courses (Irizarry, 2021; Souto-Maior and Shroff, 2024). Second, academic aspirations. Because of differences in prior academic and social experiences, Black and White students may enter high school with distinct ambitions for advanced course-taking⁶ (Francis and Darity, 2021; Fryer and Torelli, 2010; O’Connor et al., 2011; Tyson, 2011). Finally, family-based access to information. Higher-SES parents, often disproportionately White, are better positioned to provide or secure information about the importance of advanced coursework, navigating course sequences, scheduling challenges, and teacher recommendations, drawing on their own educational backgrounds or access to private support (Calarco, 2018; Cooper and Liou, 2007; Crosnoe, 2001; Lareau, 2011; Lewis-McCoy, 2014; McDonough, 1997).

Racial disparities net of individualistic explanations

While these individual differences at the time of high school can provide a persisting indirect effect of race on access to course-taking, evidence suggests Black and White students have different course-taking patterns even net of these characteristics. Within nationally representative studies of US public high schools which introduce appropriately detailed individual-level controls and that account for between-school heterogeneity, two studies stand out. Lucas and Berends (2007) analyze differences in probability of placement in college-level track during high school. Kelly (2009) assesses Black-White disparities in levels of mathematics courses taken during high school. Both of these studies introduce detailed controls for SES factors such as parents’ income, occupation and education. Because such parental characteristics are strong determinants of racial differences in differential influence potential (Calarco, 2018; Cucchiara, 2013; Posey-Maddox et al., 2014), academic aspirations (Tyson et al., 2005), and family-based access to information resources (Bennett et al., 2012; Cooper and Liou, 2007; Lareau, 2011), such controls can account for much of pre-existing racial differences with respect to these individual-level characteristics at the time of high school entry. However, SES variables do not fully explain racial differences in academic preparation by the end of middle school (Irizarry, 2021). Then, to account for pre-existing racial differences, it is also important to control for prior academic preparedness. Accordingly, these two studies also account for prior course-taking, prior course and exam grades.⁷ Additional school-level characteristics (such as course availability) which might influence opportunities to take advanced high school courses are also taken into consideration. With such detailed controls, both studies find a lower likelihood of advanced course-taking for Black students (relative to Whites) in many US school contexts⁸ — with higher inequalities as the share of Black students increase (Kelly, 2009) and as school diversity increases (Lucas and Berends, 2007). Overall, these studies suggest that, particularly in minority-Black and racially diverse schools, Black–White disparities in advanced course-taking cannot be explained by pre-existing individual differences.

Relational explanations

Explaining this persisting racial penalty, sociologists of education highlight the importance of relational processes, thus bringing in the concept of opportunity hoarding. Within this literature, studies emphasize the role of the two forms of exclusionary behaviors detailed above: administrative pressures and in-group favoritism.

From the administrative-pressures perspective, prior studies highlight how parents, through their interactions with teachers and school administrators, can influence their children’s course placements (Lewis and Diamond, 2015; Lewis-McCoy, 2014; Tyson, 2011). Not all parents, however, observe the same success in such interactions with gatekeepers. Scholars document that, due to districts’ and schools’ financial structures, gatekeepers can be more receptive to requests from affluent families (Calarco, 2018; Cucchiara, 2013; Murray et al., 2019; Posey-Maddox et al., 2014; Sattin-Bajaj and Roda, 2020) and White families (Lewis, 2003; Lewis and Diamond, 2015; Lewis-McCoy, 2014; Warikoo, 2022). These racial differences can arise because successful pressures from affluent parents (or students) on school personnel do more than secure advantages for individual children: they can reshape institutional priorities. When these families pressure schools to customize educational settings to better serve their preferences, they can influence the organizational culture. Then, when such affluent families are disproportionately White, they (even if unintentionally) contribute to structuring schools around White norms and values — such as communication patterns, dress, parental involvement strategies (Diamond and Lewis, 2019) — thus helping to constitute the school environment as a White space (Anderson, 2022; Diamond and Lewis, 2022). In such contexts, customization requests from similar Black and White families may face different chances of success (Allen and White-Smith, 2018; Diamond, 1999; Lewis and Diamond, 2015; Lewis-McCoy, 2014), allowing White advantages to emerge.

From the in-group favoritism perspective, studies highlight how resources such as academic aspirations and information are not fully exogenous to students’ experiences and can be influenced through social interactions. For many students, the choice to take advanced courses can be a difficult social decision since moving to advanced courses can involve joining a new set of peers, leading many students to fear that they will not belong socially in this new environment (Francis and Darity, 2021; O’Connor et al., 2011; Tyson, 2011). Thus, students’ motivation to enroll in advanced coursework can be highly susceptible to the messages they receive from their friends. Further, while many families might not be able, on their own (either through their past experiences or financial resources), to gain access to all information resources necessary to navigate high school course-taking, they can rely on formal or informal interactions with other families in the school community to gain access to such information (Lewis-McCoy, 2014; Small, 2009). Since access to such resources can be a function of social interactions, scholars argue that, when students and families favor same-race ties in sharing them, intergroup inequality can arise (Finnigan and Jabbar, 2023; Lewis-McCoy, 2014; Muro, 2024; Murray et al., 2020; Murray and Hailey, 2024; Posey-Maddox, 2017).

Complementing this rich literature, this study shows that exclusionary behaviors are not necessary for relational processes to produce a racial penalty in access to advanced courses and that network diffusion, alone, can serve as an additional mechanism through which opportunity hoarding arises.

The agent-based model

Network-based processes are complex, and their resulting macro-level outcomes can be difficult to grasp (Epstein, 2006; Schelling, 1971). Thus, to demonstrate how network diffusion can produce racial penalties in access to advanced coursework in the absence of exclusionary behaviors, this paper relies on an agent-based model,⁹ a powerful computational tool to make sense of these dynamics (Bruch and Atwell, 2015; Hedstrom, 2005; Manzo, 2022; Stewart, 2023).

In what follows, I describe the model¹⁰ in detail. See Online Supplement A for a pseudo-code. For readability, I use capital letters for variables describing the simulated school environment, variable names for agent-level variables, α parameters for the structural conditions of interest, and β parameters for the mechanisms of interest.

Overview

Consider a cohort of N agents entering a high school. Each agent represents a student-parent unit. More intuitively, the agents represent family units, under the simplifying assumption of one child per family. The model represents a stylized high school and, therefore, is defined by several idealizations. First, every agent who enters high school progresses in an ideal grade-promotion trajectory—i.e., the student does not repeat grades or drop out, finishing high school in 4 years. Second, to concentrate on the dynamics between the two race groups of interest, let this high school be composed only of Black and White students with variable S capturing the share of White students in the school. Finally, this idealized high school offers only two types of courses: regular and advanced. The advanced course is a limited educational resource, available to only a fraction of students, C (capacity).

Informed by the basic structure of Souto-Maior's (2026) empirically validated model of advanced enrollment, the model captures students’ competition for advanced courses using the following logic:

(1) Initialization. Time step 0 represents the moment at which agents enter high school. At this point, agents are endowed with initial (pre–high school) characteristics and are placed within a network structure.

(2) Competition for advanced enrollment. Each time step $> 0$ represents opportunities for interactions between agents and for enrollment in the advanced course.

(3) Stopping condition. Once all available spots in the advanced course are taken, the model stops.

Variables and parameters used in the model description are summarized below (Table 1).

Table 1.

Summary of variables and parameters.

Variables/parameters	Definition	Data-generating process
Agent- level variables
Enrollment_i	Agent’s enrollment status in the advanced course	Endogenous
Black_i	Agent’s race	Exogenous; determined by share S
ses_i	Agent’s composite of SES indicators	Exogenous; for white agents: ∼ N(0, 1); for black agents: ∼ N(−α_ses-gap, 1)
ses-pctl_i	Percentile rank of ses_i	Empirical CDF, with ties assigned their mean percentile rank
acad-background_i	Agent’s composite of academic background indicators	Exogenous; = ses_i + ϵ₁, where ϵ₁ ∼ N(0, 1)
acad-background-pctl_i	Percentile rank of acad-background_i	Empirical CDF, with ties assigned their mean percentile rank
net-resource_i	Agent’s access to network-based resources	Endogenous/exogenous; time step = 0: 1 for those in top decile of the SES distribution; time step $> 0$ : Endogenous to diffusion
Characteristics of the simulated environment
N	Number of agents	Exogenous
S	Share of white agents	Exogenous
C	Capacity of the advanced course	Exogenous
α_ses-gap	Standardized mean difference in SES across black and white agents	Exogenous
α_{racial-homophily}	Coleman’s inbreeding racial homophily index for the simulated network	Exogenous; empirically calibrated
α_n-ties-b	Average number of ties for black agents	Endogenous to network-formation model; empirically calibrated
α_n-ties-w	Average number of ties for white agents	Endogenous to network-formation model; empirically calibrated
β_{net-diffusion}	Whether the network diffusion mechanisms can shape advanced enrollment	Exogenous; treatment condition
Outcome of interest
θ₁	Simulated log of black-white risk ratio of advanced enrollment	Endogenous; estimated from equation (3)

Agents

Binary variable black_i captures whether the agent is Black. Because there are only Black and White agents in this high school, it follows that if black_i = 0, the agent is White. Reflecting the individual-level attributes outlined above, each agent i, at the start of the model, is endowed with three additional characteristics.

• Socioeconomic status (SES). Agent’s socioeconomic status is defined by variable ses_i. This variable represents a standardized composite of socioeconomic indicators, such as family income and parental education, as commonly measured in education research (Fryer and Levitt, 2004; Reardon et al., 2019). For modeling purposes, I compute the percentile rank of ses_i for all agents, ses-pctl_i, which maps each observation ses_i to a value between 0 and 1. I do so using the empirical cumulative distribution function, with tied values assigned their midpoint percentile rank.¹¹

• Academic background. Agent’s academic background is defined by variable acad-background_i. This variable represents a standardized composite of prior academic trajectories and performance, including course-taking history, grades, and exam scores. See Souto-Maior and Shroff (2024) for a possible empirical estimation strategy. As done for ses_i, I also compute the percentile rank of academic background for all agents, acad-background-pctl_i, using the same approach detailed above.

• Access to network-based resources. Binary variable net-resource_i captures whether agent i, at the time of high school entry, has access to the network-based resources that facilitate the navigation of an advanced academic trajectory. For generality, a network-based resource should be interpreted as a composite of all network-based resources (such as information or academic aspirations) that are meaningful to students’ chances of advanced enrollment.

As the model progresses, binary variable enrollment_i records whether agent i is enrolled in the advanced course.

Agents’ network

Since agents represent a student-parent unit, the network structure is best understood as the union of two overlapping networks. A tie is present if either a parent-parent connection exists (e.g., parents sharing information with one another) or if a student–student connection exists (e.g., students sharing information with their peers). Because communication within families can quickly transmit information between parents and their children (Lareau, 2011; Morgan and Sørensen, 1999; Parcel and Hendrix, 2014), this can be seen as a reasonable simplification that balances two goals: (a) accounting for these two kinds of networks and (b) providing a parsimonious model. Naturally, this approach may overlook important empirical differences across networks, and thus future research can benefit from extending this framework.

Online Supplement B details the computational rules for the formation of agents’ networks. The key feature of interest is the level of network racial segregation. This is captured by parameter α_{racial-homophily} which represents Coleman’s (1958) inbreeding racial homophily index. This index captures the average tendency for same-race ties, net of opportunities for contact. An inbreeding homophily of 0 indicates that agents’ likelihood of forming same-race ties is fully explained by opportunities for contact. For instance, in a school in which Blacks represent 50% of the student body, inbreeding racial homophily for Black students is 0 if, on average, 50% of their ties are with same-race peers. Additionally, the network is defined by the average number of ties for White (α_n-ties-w) and Black (α_n-ties-b) agents.

Initial distributions

Following the empirical reality of racial inequalities at the time of high school entry (see background section), the model incorporates initial disparities across these three attributes: SES, academic background and access to network-based resources. This modeling choice reflects that students of different races, on average, do not enter high school on equal footing, but rather with differences constructed previously in their social and educational trajectories (as represented by parameter γ in equation (2)). Because racial socioeconomic inequalities are central determinants of these disparities, the model adopts a stylized context in which all initial Black–White differences are fully explained by racial gaps in SES. Accordingly, when defining the initial distribution of variables, the key structural condition of interest is the consolidation of race and SES, captured by parameter α_ses-gap.

For White agents, ses_i follows a normal distribution with mean 0 and standard deviation 1. For Black agents, ses_i follows a normal distribution with mean −α_ses-gap and standard deviation 1. In this setup, α_ses-gap represents the standardized mean difference in SES between groups, i.e., the number of standard deviations by which the average SES composite of Black agents is lower than that of White agents.

Then, these socioeconomic characteristics form for the base of the distributions of other variables. Each agent’s acad-background_i is set to ses_i + ϵ₁, where $ϵ_{1} \sim N (0,1)$ . Further, agents have initial access to network-based resources (net-resource_i = 1) if, and only if, they fall within the top decile (i.e., ses-pctl_i ≥ 0.9) of the ses_i distribution.¹²

Outcome of interest: Racial penalty in access to advanced coursework

After the simulation ends, the model records whether each agent i enrolled in the advanced course (enrollment_i). Because the model attributes all initial Black-White differences in individual characteristics to socioeconomic inequalities, we can estimate racial disparities in advanced course-taking that cannot be attributed to previously formed individual differences (the racial penalty of interest, β in equation (2)) by comparing enrollment probabilities across racial groups while controlling for SES. Specifically, I control for SES percentile rank to ensure the statistical control matches the metric through which individual differences enter the model equations (see how academic background enters the enrollment probability equation, equation (4)). Further, to better isolate the role of the diffusion process in producing inequalities, the outcome of interest is estimated for the subset of agents who were not initially endowed with access to network-based resources at the start of the model (net-resource_i = 0 at time step 0). Based on the simulation outputs, I estimate:

\Pr ({enrollment}_{i} = 1 | {race}_{i}, {s e s - p c t l}_{i}) = \exp (θ_{0} + θ_{1} {race}_{i} + θ_{2} {s e s - p c t l}_{i}),

(3)

a Poisson regression model with a log link to the binary outcome. This is a common approach in social stratification research which is particularly useful when base rates are low (Karlson and Quillian, 2023). Coefficients correspond to the log of risk ratios. For each simulation run, θ₁ captures the simulated log of Black-White risk ratio of advanced enrollment net of SES. This outcome has convenient interpretation: if θ₁ = 0, there is no racial penalty; if θ₁ < 0, we observe a Black penalty; and if θ₁ > 0, we observe a White penalty.

Computational procedures

Advanced enrollment

At each time step, advanced enrollment is governed by two procedures: (a) defining which agents are considered for enrollment, and (b) the probability of enrollment given consideration (e_i).

Consideration depends on parameter β_{net-diffusion} ∈ {TRUE, FALSE}, which captures whether network-based resources matter to placement decisions. If this parameter is TRUE, then the network diffusion mechanism plays a role in the model. In this case an agent is only considered for advanced enrollment if, and only if, net-resource_i = 1, i.e., if they have access to the network-based resource (Souto-Maior, 2026). If β_net-resource = FALSE, then network-based resources do not matter for advanced enrollment and network diffusion does not play a role in the simulations. In that case, all agents are considered for advanced enrollment.

Then, at each time step, each agent i who is taken into consideration for advanced enrollment enrolls in the advanced course with probability e_i.

e_{i} = {acad ‐ background ‐ pctl}_{i}

(4)

where percentile ranks are chosen as they capture values between 0 and 1, and more naturally map to probabilities.

If the agent is considered but does not enroll (which happens with probability 1 − e_i), then the agent is no longer eligible for future consideration.

Network diffusion

The model allows agents to gain access to the network-based resource (net-resource_i) through social interactions with other agents in the network. Since each time step represents an opportunity for social interaction, the model defines that at every time step $> 0$ , each agent i with net-resource_i = 1 randomly selects one of their ties (call it j) who does not already have access to the resource (net-resource_j = 0) and shares the resource with a 10% probability.

Base values for the simulated environment

Table 2 provides the base values for the simulated environment — i.e., unless specified otherwise, these are the values defining the environment across simulation runs. Informed by empirical descriptions of U.S. high schools, I set the number of agents (N) to 400.¹³ The capacity of the advanced course (C) is set to 25% of the student body.¹⁴ Further, I focus on a school racial composition (S) defined by 50% White agents and 50% Black agents. I focus on such a racially heterogeneous high school because these are the kinds of settings in which the outcome of interest—racial penalties in advanced enrollment — are known to be more salient (Kelly, 2009; Lucas and Berends, 2007). Online Supplement D relaxes this assumption and explores the dynamics of hoarding without hoarders across different racial compositions. Finally, I set the standardized mean difference in SES composite across Black and White agents (α_ses-gap) to 0.561, the standardized mean difference reported in the 1998 Early Childhood Longitudinal Study-Kindergarten Cohort (Fryer and Levitt, 2004).

Table 2.

Base values for the simulated environment.

Variable/parameter	Base value
N	400
S	0.5
C	0.25
α _ses-gap	0.561
α _{racial-homophily}	0.5
α _n-ties-b	6.7
α _n-ties-w	6.7
β _{net-diffusion}	TRUE

For the agents’ network, I follow estimates from Currarini et al. (2010), who provide an empirical analysis of the National Longitudinal Study of Adolescent to Adult Health (Add Health). Based on their results,¹⁵ I set α_{racial-homophily} = 0.5. Further, I set Black and White agents’ average number of ties (α_n-ties-b, α_n-ties-w) to 6.7. Online Supplement B details the calibration of the simulated network according to these values.

The results section provides explores variations in the initial values which are central to simulated outcomes. Online Supplement D provides supplementary sensitivity analyses.

Results

Hoarding without hoarders

I simulate how the network diffusion mechanism can shape the emergence of a racial penalty in advanced course-taking patterns—that is, Black–White differences in advanced enrollment that cannot be explained by previously formed individual differences (θ₁, equation (3)). I compare the distribution of simulated results under two scenarios: one in which network diffusion matters for advanced enrollment (β_{net-diffusion} = TRUE) and one in which it does not (β_{net-diffusion} = FALSE). To account for the stochasticity of the agent-based model, I run 1000 simulations for each scenario. Figure 1 presents the results.

Figure 1.

Distribution of simulated racial penalties (θ₁, equation (3)) under two scenarios: one in which network diffusion matters for advanced enrollment (β_{net-diffusion} = TRUE) and one in which it does not (β_{net-diffusion} = FALSE). Additional values defining the simulated environment follow Table 2. Each scenario is simulated 1000 times. Vertical dotted lines capture the average value of θ₁ across simulation runs. θ₁ = 0 marks the absence of any racial penalty.

So far, the model constructed here has not presumed the presence of any exclusionary behaviors — i.e., administrative pressures or in-group favoritism. Still, the results from Figure 1 show that, when network diffusion matters for advanced enrollment, a Black penalty in advanced enrollment can, on average, emerge. This result demonstrates that, in contrast to the exclusionary-behaviors interpretation of opportunity hoarding, a racial penalty in access to advanced coursework can emerge even in the absence of exclusionary behaviors. Given the stochasticity of agent-based models, the discussions below focus on average outcomes. See Online Supplement C for a discussion of where the variations in simulation outcomes come from.

To better understand this notion of hoarding without hoarders, the following section unpacks how the computational model generates this result.

Network diffusion and the emergence of hoarding without hoarders

Recall that, in the model designed here, agents’ chances of advanced enrollment are a function of two procedures: (a) determination of which agents are considered for advanced enrollment, and (b) probability of enrollment given consideration (equation (4)). Because, by design, differences in probability of enrollment under consideration are fully explained by racial socioeconomic inequalities, procedure (b) cannot explain the formation of a racial penalty. Therefore, the formation of racial penalties stems only from Black-White differences generated through procedure (a) — determination of which agents are considered for advanced enrollment.

Such consideration depends on whether the agent has access to the network-based resource needed for advanced enrollment. To understand how differences in access to this resource emerge among similar SES agents, recall that the model does not presuppose any racial penalty from the start. At time step 0, agents have access to the network-based resource if (and only if) they belong to the top 10% of the SES distribution (referred to as higher-SES agents in what follows). Those in the bottom 90% of the SES distribution (lower-SES agents) do not initially have access to the network-based resource. Differences in access to the network-based resource between lower-SES Black and White agents are, therefore, the result of the extent to which agents gain access to the resource through social interactions in community.

Given these initial conditions, lower-SES agents must interact with higher-SES agents to gain access to the network-based resource. Reflecting empirically grounded structural conditions, White agents are more likely to be higher-SES than Black agents. This structural difference, combined with the fact that agents’ networks are racially segregated (α_{racial-homophily} = 0.5), implies that lower-SES White agents have access to a more resourceful pool of network ties than their Black counterparts.¹⁶ Through the social diffusion of network-based resources, lower-SES White agents therefore tend to gain access to these resources more quickly than lower-SES Black agents, allowing them to secure enrollment in the advanced course earlier.

This dynamic highlights a known pathway through which network diffusion can produce intergroup inequality: via the byproduct of homophily and consolidation (Zhao and Garip, 2021).¹⁷ Figure 2, below, helps us visualize this process. The figure depicts the results from simulation runs initialized with based values from Table 2, but where I allow the homophily parameter (α_{racial-homophily}) to vary within {0, 0.1, …, 0.9, 1} and the consolidation parameter (α_ses-gap) to vary within {0, 0.25, 0.5, 0.75}. To account for the stochasticity, each scenario is simulated 1000 times (see Figure D.1 in Online Supplement D to better visualize variation across simulation runs).

Figure 2.

Distribution of simulated racial penalties (θ₁, equation (3)) across variations in the homophily (α_{racial-homophily}) and consolidation parameters (α_ses-gap). Additional values defining the simulated environment follow Table 2. Trend curves smoothed across 1000 simulations of each parameter combination. θ₁ = 0 marks the absence of any racial penalty. For reference, the gray dashed line marks the average θ₁ from the simulations using the base parameters (Table 2).

From these results, it is clear that the emergence of a racial penalty depends exclusively on whether lower-SES White agents have access to a more resourceful pool of network ties than lower-SES Black agents. For this to occur, both homophily and consolidation need to be present. When either structural condition is absent (α_ses-gap = 0 or α_{racial-homophily} = 0), no racial penalty emerges (θ₁ = 0).

While the joint relevance of homophily and consolidation in producing intergroup inequality has already been noted in the network diffusion literature (Zhao and Garip, 2021), the mechanism modeled here extends this work in an important way. In prior models, inequality arising from the byproducts of homophily and consolidation depends on an additional condition: group differences in adoption thresholds — that is, diffusion is modeled through complex contagion and inequality arises from differences in the proportion of one’s social ties that must adopt a practice (or gain access to a resource) before one does so (Centola, 2015; Zhao and Garip, 2021).

In contrast, because of the focus on inequalities that cannot be attributed to individual differences, the model here assumes no Black-White differences in adoption propensities. In fact, the diffusion is modeled through simple contagion — i.e., adoption can occur after exposure to a single tie. Yet, inequality still emerges. The key condition enabling this outcome is the introduction of time constraints in the diffusion process. Prior models generally examined how intergroup inequality arises in equilibrium, once diffusion has fully unfolded (Centola, 2015; Zhao and Garip, 2021). Here, however, network-based resources are instrumental in the competition for a finite good (spots in advanced courses) in a context which implies a well-defined time constraint (academic trajectories). Intuitively, academic trajectories presuppose limited windows of opportunity, during which students need to gather the needed resources (such as information and motivation) required for enrollment. After that window closes—once the enrollment period ends or high school is over—gaining access to such resources no longer makes a difference. Because timely access to network-based resources is so essential in this context, the model emphasizes time as a key dimension and allow agents to gradually fill available spots as diffusion unfolds. Intergroup inequality then arises from differences in the initial rates of resource diffusion across groups, which allow White students to secure early advantages. Figure 3 helps us visualize this dynamic.

Figure 3.

Share of not-initially-resourced (or lower-SES) agents with access to network-based resources across time steps. The simulated environment is defined by values in Table 2. Curves are smoothed over 1000 simulation runs.

Figure 3 shows lower-SES Black and White agents’ access to the network-based resource across simulated time steps. At time step 0, by design, no lower-SES (Black or White) agent has access to this resource. However, once the simulations begin, agents are allowed to share resources with their ties, and lower-SES agents gradually gain access to the resource through interactions with higher-SES agents. Because lower-SES White agents have access to a more resourceful pool of network ties than lower-SES Black agents, White agents tend, at least initially, to gain access to the resource at a faster rate — note that up until around time step 10, the trend curve for White agents is steeper. Early access to the resource is essential, as the number of available advanced course seats is limited and quickly fills up. Such early advantage, therefore, generates a higher likelihood of advanced enrollment for lower-class White agents. After approximately time step 10, the slope of the trend curve for Whites and Black agents become similar and, in fact, after this point, Black agents’ access to the network-based resource increases at a faster rate.¹⁸ However, when this occurs, most spots in the model are already taken, and White agents have already secured an advantage in advanced enrollment.

The interaction of network diffusion and exclusionary behaviors

Beyond demonstrating how racial penalties can emerge without exclusionary practices, this agent-based framework allows us to link the network diffusion mechanism emphasized here to the exclusionary-behaviors interpretation of opportunity hoarding, showing that when such practices are present, they interact with diffusion to amplify inequalities. To capture interactions, I, first, introduce administrative pressures and in-group favoritism into the model and, second, simulate how their salience shapes simulated outcomes.

In-group favoritism

To model in-group favoritism, I modify the network diffusion procedure. Before, the model specified that at each time step $> 0$ , each agent i with net-resource_i = 1 randomly selects one of their ties (call it j) who does not already have access to the resource (net-resource_j = 0) and shares the resource with a 10% probability (p = 0.1). Now, I allow such probability to depend on the nature of the tie. If the race of agent i and j is the same, agent i shares the resource with probability p = 0.1. In contrast, if the race of i and j is different, agent i shares the resource with probability = 0.1 (1 − β_{in-group-favoritism}), where parameter β_{in-group-favoritism} captures the prevalence of in-group favoritism. Parameter β_{in-group-favoritism} is continuous and can vary within [0, 1]. It captures one minus the ratio of probability of diffusion between different-race ties. When it equals 0, it captures no in-group favoritism (ratio of probability of diffusion between different-race ties is 1). As it increases, the ratio of probability of diffusion between different-race ties decreases, indicating higher levels of in-group favoritism.

Figure 4 depicts the outputs from simulations of this modified model initialized with base values from Table 2 and β_{in-group-favoritism} ∈ {0, 0.1, …, 0.9, 1}. Note that when β_{in-group-favoritism} = 0, outcomes are shaped only by the network diffusion mechanism.

Figure 4.

Distribution of simulated racial penalties (θ₁, equation (3)) under varied levels of in-group favoritism (β_{in-group-favoritism}). Additional values defining the simulated environment follow Table 2. Each scenario is simulated 1000 times. Red trend curve smoothed across all simulated outcomes from each parameter combination. θ₁ = 0 marks the absence of any racial penalty.

From these results we see that as the prevalence of in-group favoritism increases, it amplifies the racial penalty produced by network diffusion. Intuitively, when in-group favoritism shapes diffusion, it follows that even among Black and White agents with similar contact opportunities with initially resourced (higher-SES) agents, Black agents remain less likely to gain access to the resource, since higher-SES agents — who, in the assumed initial values for these simulations, are disproportionately White — tend to prioritize supporting same-race peers.

Administrative pressures

To capture the administrative pressures mechanism, I modify the original model to introduce a new agent-level variable and new parameter. Additionally, for modeling purposes, I introduce intermediate components denoted by lowercase letters.

First, the agent-level variable influence-potential_i, which captures each agent’s potential to shape placement decisions through interactions with school personnel. According to the literature reviewed above, such potential depends on one’s socioeconomic status and race. Then, I define influence-potential_i as a weighted average of the agent’s SES and a race effect (x_i):

{influence ‐ potential}_{i} = 0.5 {ses}_{i} + 0.5 x_{i} .

(5)

For White agents, x_i follows a normal distribution with mean 0 and standard deviation 1. For Black agents, x_i follows a normal distribution with mean z and standard deviation 1, where z captures the standardized mean difference in influence potential across Black and White agents. As reviewed in the background section of this paper, gatekeepers can be less receptive to customization requests from Black families when the school environment constitutes a White space — i.e., a context in which White cultural practices and behaviors are valued and rewarded at higher rates. The construction of such White spaces often occurs as affluent families pressure schools to customize educational settings to better serve their preferences (Diamond and Lewis, 2022; Lewis and Diamond, 2015; Lewis-McCoy, 2014). When these high-SES families are disproportionally White, this process has the collateral effect of structuring schools around White norms and values, thus favoring the actions of White parents and students. Because racial socioeconomic differences play such an important role in producing this race differential, I set z to equal parameter α_ses-gap. From this operationalization, when no Black–White socioeconomic disparities exist (α_ses-gap = 0), there is no race effect (x_i follows the same distribution across Black and White agents). As α_ses-gap increases, the stronger the structuring of the school as a White space, decreasing Black agents’ potential to shape gatekeepers’ decisions. For modeling purposes, I also compute influence-potential-pctl_i, the percentile rank of influence potential for all agents, using the same approach detailed above.

Second, I introduce continuous parameter β_{admin-pressures} ∈ [0, 1] to capture the extent to which administrative pressures can shape placement decisions. This parameter modifies agent’s probability of enrollment given consideration (equation (4)) such that:

\begin{array}{l} e_{i} = (1 - β_{admin-pressures}) {acad ‐ background ‐ pctl}_{i} + \end{array}

(6)

\begin{array}{l} β_{admin-pressures} {influence ‐ potential ‐ pctl}_{i} . \end{array}

As in the original model, F denotes the cumulative distribution function, and is used here to transform the standardized variables acad-background_i and influence-potential_i into values interpretable as probabilities.

Figure 5 shows the outputs from simulations of this modified model initialized with based values from Table 2 and β_{admin-pressures} ∈ {0, 0.1, …, 0.9, 1}. Note that when β_{admin-pressures} = 0, outcomes are shaped only by the network diffusion mechanism.

Figure 5.

Distribution of simulated racial penalties (θ₁, equation (3)) across levels of administrative pressures (β_{admin-pressures}). Additional values defining the simulated environment follow Table 2. Each scenario is simulated 1000 times. Red trend curve smoothed across all simulated outcomes from each parameter combination. θ₁ = 0 marks the absence of any racial penalty.

Here, we observe that as the salience of administrative pressures increases, the resulting racial penalty produced through diffusion is also amplified. Intuitively, as the role of administrative pressures on advanced enrollment increases, agents’ probability of enrollment given consideration, equation (7), places greater weight on influence-potential_i and less on acad-background_i. Because Black–White differences in influence-potential_i exceed those in acad-background_i— the latter is fully explained by racial disparities in SES, whereas the former depends on both SES and a specific race effect, equation (5)—the resulting racial penalty becomes larger.

Conclusion

This theoretical paper set out to clarify the concept of opportunity hoarding, particularly as it applies to the formation of Black-White educational inequalities. Despite its widespread use, opportunity hoarding is often loosely defined, leading to varied interpretations of the inequality-producing mechanisms it captures. I identified two prevailing interpretations. First, the group-disparity interpretation, which defines opportunity hoarding in terms of observed outcomes, allowing any micro-level process that generates group disparities to fall under the umbrella of opportunity hoarding. Second, the exclusionary-behaviors interpretation, which defines opportunity hoarding in terms of the ways in which White individuals engage in exclusionary practices — captured by the mechanisms of in-group favoritism and administrative pressures — to secure access to valuable resources.

I have argued that the group-disparities interpretation is too broad, remaining vague about the specific micro-processes through which opportunity hoarding operates, reducing the concept to a description rather than explanation of racial inequalities. Conversely, the exclusionary-behaviors interpretation is too narrow, overlooking a process I emphasize here: exclusionary behaviors are not necessary for opportunity hoarding to arise, what I conceptualize as hoarding without hoarders.

To develop this argument, I first outlined a more precise definition of opportunity hoarding, arguing that it captures the relational processes that generate racial penalties in access to valuable resources, i.e., racial disparities that cannot be explained by previously formed differences in individual attributes. Then, to demonstrate the notion of hoarding without hoarders, I constructed an agent-based model focused on the context of within-school racial disparities in access to advanced high school coursework. The model shows how network diffusion — under network racial segregation, consolidation (racial socioeconomic inequalities) and temporal constraints for diffusion — can generate racial penalties even when behaviors are race-neutral.

In reflecting on this contribution, it is important to recognize its limitations, which also suggest directions for future research. First, for conceptual precision, the definition proposed here concentrates on a specific explanatory target of opportunity hoarding: how individuals gain new advantages, even net of previously formed differences. However, as originally defined, opportunity hoarding might also be interpreted to capture how certain social groups maintain advantages — i.e., persisting indirect effects of group membership (γ in equation (2)). Thus, future work might complement the present investigation by clarifying the distinct relational processes underlying the maintenance (rather than creation) of racial inequalities. In addition, the model constructed here is highly stylized—e.g., it considers only two race groups and a fixed network structure. These idealizations were introduced to minimize unnecessary complexities, helping to provide a clear illustration of the concept of hoarding without hoarders. However, future research can build on the modeling approach presented here by examining, empirically or theoretically, the conditions under which hoarding without hoarders is more likely to arise. In particular, it would be valuable to explore variations in network structure, as the structure of agents’ networks is not exogenous to educational contexts (Kruse and Kroneberg, 2019, 2022; Small, 2009). Such extensions could provide insights into how educational organizations might counteract the emergence of hoarding without hoarders. Additionally, while this paper focused on advanced course-taking as one instance of opportunity hoarding, future research should examine the hoarding of other educational resources, such as access to well-resourced schools or districts. Finally, the general framework proposed here could also be expanded beyond education to other domains where opportunity hoarding plays a central role, including employment and housing.

Overall, this paper contributes to a more consistent and systematic conceptualization of opportunity hoarding as an explanation for racial inequalities. In particular, by proposing the notion of hoarding without hoarders, it suggests that the common emphasis on exclusionary behaviors understates the broader structural conditions under which White actors can hoard valuable educational resources. This highlights that, as scholars and policymakers seek to document, measure, and design efforts to counteract opportunity hoarding, they must not be constrained by a narrow focus on identifying exclusionary behaviors alone. Instead, they must also consider how network diffusion processes interact with structural conditions — such as network segregation and racial socioeconomic disparities — to produce the hoarding of educational opportunities across racial lines.

Supplemental material

Supplemental material - Hoarding without hoarders: Opportunity hoarding in the absence of exclusionary behaviors

Supplemental material for Hoarding without hoarders: Opportunity hoarding in the absence of exclusionary behaviors by João M. Souto-Maior in Rationality and Society

Footnotes

Acknowledgments

This paper resulted from my doctoral dissertation in the Sociology of Education program at New York University. I thank R. L’Heureux Lewis-McCoy, Samuel Lucas, Erez Hatna, Ravi Shroff, Lisa Stulberg, Marc Scott, John Skvoretz, Peter Rich, and Yasmiyn Irizarry for helpful comments and feedback. Earlier versions were presented at the 2023 Annual Conference of the International Network of Analytical Sociologists, the 2023 Annual Group Processes Conference, and the 2023 American Sociological Association Sociology of Education Roundtables, where I received valuable feedback. I am also grateful to two anonymous reviewers for a productive peer review process that improved this work, and to the NYU Agent-Based Modeling Lab for valuable training.

ORCID iD

João M. Souto-Maior

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data and code availability statement

Replication code available at: . This paper does not use external data.

Supplemental material

Supplemental material for this article is available online.

Notes

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