Knowing me, knowing you: Socio-economic status and (segregation in) peer and parental networks in primary school

Students of different socio-economic status (SES) backgrounds are sorted into different schools. While integrating schools seems an easy solution to enhance inter-group interaction, this is yet an empirical question as we know little about how networks structure along SES lines in school. We examine the tendency for friendship and parental networks in primary school to structure by SES. We furthermore explore the role of the local school context. To do so, we collected multiplex classroom network data among Dutch students in 68 classrooms (55 schools) in their final year of primary school (grade 6; age 11–12). We link these sociometric data to register data, and test our hypotheses using cross-sectional exponential random graph models and meta-analysis techniques. Findings show that the networks of primary school students and their parents display a tendency for same-SES over cross-SES ties, net of opportunity structures. We do not find evidence for SES differences in the strength of SES homophily. Descriptive analyses show SES disparities in the extent to which parents have ties with the parents of their children’s friends (i.e., intergenerational closure), but these disparities disappear when controlling for other tie-generating mechanisms using ERGMs.


Introduction
Research has repeatedly highlighted the potential benefits of bringing students from different socio-economic status (SES) backgrounds together in school.First, integrating schools may help to reduce social inequality in educational outcomes, as students from less advantaged backgrounds may gain access to educational resources through relations with more advantaged peers (Crosnoe et al., 2003;Dika and Singh, 2002;Lessard and Juvonen, 2019;Van Ewijk and Sleegers, 2010).Conversely, segregated schools may amplify educational inequalities, as this implies that the social capital embedded in peer relations will mostly accrue to advantaged students, thereby compounding individual resources (DiMaggio and Garip, 2012).Second, prior research recognizes the psycho-social benefits of friendships that cross social boundaries, as friendships with dissimilar others would expose students to other perspectives and experiences, thereby promoting mutual understanding (Lessard and Juvonen, 2019).Cross-cutting friendships are also related to feelings of safety and fewer experiences of victimization (Graham et al., 2014), which is conducive to the classroom's social climate.
Motivated by the assumed benefits associated with (diverse) peer networks, prior research investigated between-school segregation by socio-economic status (SES), demonstrating that students typically cluster in schools with students that are similar to themselves (e.g., Boterman, 2019;Brandén and Bygren, 2021).With this focus on between-school sorting, this strand of research largely ignores within-school segregation (Engzell and Raabe, 2022) i.e., the extent to which peer networks in school are segregated by SES.Though the level of between-school segregation is key in shaping opportunities for meeting (dis)similar others, having a demographically diverse student body does not necessarily lead to high levels of inter-group interaction.In fact, empirical research on ethnic and racial segregation has demonstrated that peer networks tend to be highly segregated, net of structural opportunities (e.g., McFarland et al., 2014;Moody, 2001;Smith et al., 2016).Stated differently, if networks in school are segregated by SES, policies aimed at reducing (the harmful effects of) social segregation by tackling between-school segregation may be less effective.
In this study, we examine the tendency for networks in primary school to structure by SES.We build on prior research in two main ways.First, a large body of research focuses on friendship formation along ethnic and racial lines in school (e.g., Leszczensky and Pink, 2015;Moody, 2001;Quillian and Campbell, 2003;Smith et al., 2014;Smith et al., 2016;Vermeij et al., 2009).While friendships are recognized as a fundamental domain for stratification processes, relatively little attention has been paid to the role of SES in structuring school networks (see for a notable exception Malacarne, 2017).Existing studies that include information on preferences for same-SES friends (e.g., Quillian and Campbell, 2003;Smith et al., 2014) feature students in secondary school, where opportunities for meeting dissimilar others are often lower than in primary school.Accordingly, it is far from clear how SES relates to network formation in primary school, a context where opportunities to integrate are often present.
Second, we do not only study student friendships, as most existing network studies do, but also parental ties in school.In the educational context, a crucial form of social capital is intergenerational closure (IC)i.e., relationships among parents whose children are friends.IC is theorized to enhance the spread and enforcement of pro-school norms and resources available in the network (Coleman, 1988;Geven and Van de Werfhorst, 2020).Apart from these educational benefits, relations are embedded in a larger (intergenerational) network ecology (McFarland et al., 2014;Windzio and Heiberger, 2022), and ties between students whose parents are also in contact may be more sustainable over time (Cartwright and Harary, 1956).It is important to study student friendship networks alongside parental networks: while the two are interrelated, they are also fundamentally different.That is, the costs of overcoming social boundaries depend on the type of relation: whereas children meet on a daily basis at school, relations among parents arguably require a more active effort to establish and maintain (Hunter et al., 2012;Leszczensky and Pink, 2015;Windzio and Bicer, 2013).
Aside from describing within-school segregation, we provide additional analyses to tentatively explore potential explanations for these patterns.More specifically, we relate to ongoing debates on the role of freedom of school choice in segregation (e.g., Brandén and Bygren, 2021).There is a widespread belief that parental freedom in school choice reinforces inter-group segregation.However, reducing parental school choice may also enhance tendencies for same-SES ties in school.Hence, we explore if the studied network processes are contingent on local opportunities and parental strategies for primary school choice.
We use newly collected data on multiplex classroom networks of Dutch grade 6 students in 68 classrooms (1416 students, 55 schools).We examine potential SES homogeneity in student friendship and parental contact networks at a crucial transition point in students' educational careerjust before they are allocated to different ability tracks in secondary school.These sociometric data are linked to rich register data on parental SES, spatial proximity between classmates' home addresses, and the local school context.We test our hypotheses using exponential random graph models (ERGMs) and meta-analysis techniques.

Student SES and friendship formation
School is arguably the most important context for children to form friendships.In general, three main mechanisms are theorized to drive friendship formation: (i) propinquity (opportunity structure), (ii) homophily (preferences), and (iii) relational mechanisms (balance, transitivity, third party effects).
Opportunities to meet, and form friendships with, peers from dissimilar SES backgrounds are impaired by residential segregation, and differences in school choice by SES (propinquity) (Boterman, 2019;Denessen et al., 2005).Aside from these structural opportunities, students tend to prefer friendships with peers who are similar to them, and distance themselves from dissimilar others (homophily).The core arguments for homophilic preferences are that sharing features facilitates a sense of familiarity and joint understanding, eases communication, helps to maintain a positive self-concept, and reduces uncertainty and conflict (McFarland et al., 2014;McPherson et al., 2001).In other words, interactions with similar others require less time and effort, and are more rewarding (Leszczensky and Pink, 2015).
Several related mechanisms can underlie SES homophily in student friendships.First, social identity concerns may play a role.Compared to gender, ethnicity or race, there are fewer explicit markers through which SES is made salient to children in everyday life.Nonetheless, from a young age onward, children seem capable to (i) make global SES distinctions based on wealth markers, and (ii) link (in)favorable stereotypes, such as working hard or being lazy, to SES background (Mistry et al., 2015;Vandebroeck, 2020;Weinger, 2000).Accordingly, children may notice how their housing, clothing, and possessions matches, or differs from, that of classmates during play dates, birthday parties, or other interactions, and this may affect with whom they identify most.Second, SES homophily may be a "by-product" of homophily with respect to features that correlate with SES, such as lifestyles, norms, values, achievement, or educational aspirations (see Smith et al., 2014, for an application of this hypothesis for ethnic homophily).Third, SES homophily may be an indirect consequence of shared experiences and contexts.Even if students from different backgrounds attend the same school and have opportunities to meet, same-SES students likely spend more time together, in and outside school.In school, teachers can use forms of within-class ability grouping where students are streamed to homogeneous ability groups to tailor instruction.Such organizational factors may restrain cross-SES interactions, as SES is related to academic performance (Hallinan and Smith, 1989).Outside school, same-SES students may be more likely to live close by (Kruse et al., 2016), and (therefore) engage in the same leisure activities or attend the same clubs (Hjalmarsson and Mood, 2015;Smith et al., 2014).
Next to structural opportunities and homophily, the structure of preexisting relations affects the creation of new ones.In particular, the tendency to reciprocate friendships (mutuality) and the pressure for friends of friends to become friends (transitivity) amplify potential SES homogeneity in student friendships (McFarland et al., 2014).Furthermore, third parties, such as parents, (in)directly affect children's peer relations.Children are socialized with out-group attitudes at home.Parents can also interfere in peer relations by prescribing norms that (dis)approve of relations with members of certain groups, or by directly affecting children's opportunities to spend time with (dis)similar peers (Smith et al., 2015).Social background is one of the factors (in)directly affecting parental approval of children's peers, potentially also because parents prefer their children to play with children of parents with similar norms, values, and parenting styles.
While these homophilic preferences may apply to all parents, (active) interference in children's friendships may be more prevalent among parents from advantaged SES backgrounds.Scholars suggest that high-SES parents are typically actively involved in settings where their children interact with peers to limit exposure to what they view as less desirable activities or peers, while low-SES parents tend to grant children more autonomy in how, and with whom, they spend time (Fletcher et al., 2006;Hunter et al., 2012;Lareau, 2011).
There is empirical evidence showing that children, confronted with hypothetical examples of potential friends, are most likely to prefer to befriend (imagined) same-SES peers (Weinger, 2000).Most network studies using observational data focus on ethnic homophily in secondary school, and only include parental SES as a control variable.These studies typically show that SES similarity plays some role in friendship formation, but that preferences for same-SES ties are weaker than for same-ethnic ties (Quillian and Campbell, 2003;Smith et al., 2014).However, these studies are executed in contexts with high levels of between-school segregation (i.e., U.S., or countries with between-school ability tracking in secondary school), reducing opportunities for cross-cutting SES ties and making SES differences less salient.We hypothesize: (a) Primary school students display a tendency to befriend same-SES classmates, net of opportunities, and (b) this effect is stronger among higher SES students (H1).D. Zwier and S. Geven

Parental networks and (SES disparities in) intergenerational closure
Student friendship relations are embedded in a larger (intergenerational) network ecology (McFarland et al., 2014;Windzio and Heiberger, 2022).Parental networks in primary school likely also segregate by SES, as parents also display a homophilic preference to connect with others that share certain demographic, behavioral, or status attributes (McPherson et al., 2001).Crossing SES boundaries may even require more effort for parents than for their children.While children in the same classroom meet daily for several consecutive hours, parents may only shortly cross paths when picking up children or arranging play dates.Hence, compared to students, crossing social boundaries and building relations with (dis)similar others in school may be more "costly" for parents (Leszczensky and Pink, 2015;Windzio and Bicer, 2013).As described in the previous section, forming and maintaining connections with other parents may even be a strategy for parents to monitor (the friendships of) their children.As high-SES parents tend to play a more active role in this than low-SES parents, we again expect SES differences in the strength of homophilic tendencies in these networks: (a) Parents display a tendency to associate with same-SES parents of children's peers, net of structural opportunities, and (b) this effect is stronger among higher SES parents (H2).
Next to potential segregation in parental networks, it is important to study the inter-linkages between student friendship and parental contact ties in school.Peer relations are arguably especially beneficial for educational success if ties are "closed" at the parental level (i.e., closure relationships, see Fletcher et al., 2006;Hunter et al., 2012;Windzio and Heiberger, 2022).Intergenerational closure (IC) is argued to function as a source of social control, and creates a "norm-enforcing" environment: when parents know the parents of their children's peers, it is easier to monitor and guide children's behavior.In addition, IC promotes the spread and exchange of information that is available in the network (Coleman, 1988;Geven and Van de Werfhorst, 2020), such as parenting advice, help with childcare, and information on children's school matters (Cox et al., 2021).Apart from these educational benefits, friendship ties that are embedded in parental networks may be more sustainable: from a structural balance perspective (Cartwright and Harary, 1956), ties between students whose parents are also connected (either before or after the children became friends), may be less likely to dissolve over time.
Peer networks of higher SES children arguably enjoy higher levels of IC.Parental school involvement tends to vary by SES (Lareau, 2011).In particular, higher SES parents are more likely to be in contact with teachers, to volunteer at school, and to be active in parent-teacher organizations.These activities provide ample opportunities to connect with the parents of children's school peers.While lower SES parents ascribe similar importance to education, levels of school-based involvement are typically lower (Bakker et al., 2007;Horvat et al., 2003;Kelly, 2004;Li and Fischer, 2017;Ream and Palardy, 2008).This may result in weaker network integration. 1As described, higher SES parents are also often actively involved in settings where their children interact with peers, which may cause them to meet each other while "surveilling".Accordingly, we expect: Peer networks of higher SES students enjoy higher levels of intergenerational closure (H3).

The Dutch Case
We study the tendency for peer and parental networks in primary school to structure by SES in the Netherlands.In this country, primary education starts at age 4 (compulsory from age 5), and students attend primary school for eight years.We focus on students in the final grade (Dutch group 8; international grade 6).Most primary schools are small and only have one or a few classes in the same grade level, implying that students and their parents likely know each other for years, if only by face or name.While social ties are subject to change, they have had had time to develop.
Freedom of education is a constitutional right, resulting in a differentiated educational landscape with a large share (almost 70%) of publicly-funded "private" schools based on religious or pedagogic principles, and a high degree of parental freedom in choosing a school.As both public and private schools receive state funding, financial resources play a marginal role in access to education.School choice is not constrained by catchment areas: families can in theory choose any school, though spatial proximity remains important (Borghans et al., 2015). 2 Due to the country's high population density, it is often possible to choose a school among several alternatives without moving, making residential relocation less common compared to other countries (Boterman, 2021).Home-to-school distances are short, and most students walk or bike to primary school (Goeverden and Boer, 2013).To illustrate, the students in our sample live on average 362 m from the closest school (284 m in strongly urbanized areas, 420 m in rural areas), and the median distance to the chosen school equals circa 570 m in both urban and rural areas.While most families, even in more rural areas, have multiple primary schools to choose from, the available options naturally vary both in terms of the quantity (i.e., the number of schools families can realistically pick) as well as the (diversity in) student populations at these schools.In urban areas, it is typically possible to select a school from a wide range of alternatives, while school choice is more restricted in rural areas.Furthermore, primary schools in cities tend to be more polarized in terms of SES composition, whereas suburban areas have more mixed schools (Boterman, 2021).

Data
We use data from the "transition from PRIMary to Secondary education" (PRIMS) project, including information on classroom networks of students in their final year of primary school (grade 6; age 11-12) (see for more information Zwier et al., 2023).We enrich these data with information from the Dutch registers on, i.a., family background.PRIMS data were collected in January-February 2020 (Cohort 1, Wave 1 [C1W1]) and February-March 2021 (Cohort 2, Wave 1 [C2W1]).The cohorts are pooled to have a larger sample at the classroom level.Students filled out the web-based survey of circa 45 min during regular school hours under teacher supervision. 3The research team monitored the process, and offered support through phone or e-mail to ensure that the administration of the survey went smoothly.
A two-stage sampling procedure was applied.First, primary schools were sampled from a national (stratified) sampling frame.Larger schools and schools with a larger share of students from socioeconomically disadvantaged backgrounds had a higher chance of being selected.This resulted (for both cohorts) in a sample that is representative at the school level with respect to region, level of 1 We acknowledge that this relationship is bidirectional in that being better integrated in parental networks may also contribute to higher levels of involvement (Sheldon, 2002).
2 Some primary schools in cities experience over-subscription problems.The allocation procedure in such circumstances varies across municipalities (e.g., priority rules based on proximity, (pre-)registration).Especially higher SES parents use this freedom of choice to avoid schools with a high concentration of disadvantaged students (Boterman, 2013;Karsten et al., 2003), and are better equipped at navigating the school choice process, including potential over-subscription procedures, to ensure that their children end up at the preferred school.
3 As schools were closed for in-person education due to the COVID-19 pandemic from 16 December to 8 February 2021, C2 schools were offered to have students fill out the survey at home but most students (95.6%) did not.
D. Zwier and S. Geven urbanization, SES composition, denomination, and the track recommendations students received.Second, all grade 6 students were invited to participate in the study. 4In both cohorts, about 63% of the students were granted active parental consent, yet response rates considerably varied between classrooms (range: 13.6%− 100%).The pooled data set includes 3033 students (C1W1: 1474 students, 105 classrooms, 66 schools; C2W1: 1643 students, 111 classrooms, 79 schools), of which 2986 (98.5%) could be successfully linked to their personal identifier in the registers. 5 For the purpose of this study, classrooms are selected with (i) a response rate of at least 70%; (ii) at least 15 students with no missing data on all variables of interest; (iii) no more than two students that (were) never nominated in any of the sociometric questions.These selections are necessary to well represent the network structure and to avoid convergence issues, and are similar to those applied in prior studies (cf., Huitsing et al., 2012;Kruse and Kroneberg, 2019;Kruse et al., 2016;Smith et al., 2016).This results in an analytical sample of 68 classrooms (1416 student observations, 55 schools). 6The classrooms included in our analytical sample do not significantly differ from those excluded in terms of school size and the level of urbanization; classrooms with an advantaged student population are slightly over-represented (see Appendix A, Tables A.1-A.2 for details).

Measures
Networks.Students were provided with a roster showing all classmates' names.They were allowed to nominate an unlimited number of classmates, and could tick a "Nobody" box if a question did not apply to any of them.They could also nominate classmates that did not participate but these ties are dropped, as data on student attributes is missing for non-participating students.Friendship ties were measured with the item "Which classmates are your best friends".Parental contacts were measured with the item "Whose parents do your parent(s)/caregiver(s) occasionally talk to?".Following Windzio and Heiberger (2022), we conceptualize directed ties that co-occur between both networks as IC.See Fig. 1 for a graphical illustration of the networks for one classroom.
Socio-economic status.We measure parental SES with information on parental education and household income from the registers.Parental education is measured as the highest non-missing educational degree among both registered parents.We distinguish between students with at least one college-educated parent (ISCED 5-8) and students with lower/ medium educated parents (ISCED 0-4). 7Household income is measured as disposable (equivalized) household income.Household income is corrected for household size and composition using the empirically grounded equivalence scale of Statistics Netherlands.We take the natural logarithm to correct for right-skewness.In case registered parents are separated, divorced, or not living together for other reasons, we select the household income of the parent with whom the child is living.
Table 1 displays descriptive statistics of all student-level variables.The measurement of student-level control variables (gender, grade, migrant background) can be found in Table 2.

Methods
We use a two-step procedure to test our hypotheses.First, we estimate cross-sectional directed exponential random graph models (ERGMs) to identify the importance of certain tie-generating mechanisms for peer and parental networks in each classroom (Lusher et al., 2013).ERGMs model the probability that a network is observed as a function of tie configurations.To test how important tie-generating mechanisms are for the overall network, ERGMs compare the prevalence of a tie configuration in the empirical network to its prevalence in a series of simulations.Estimated parameters indicate the extent to which a tie configuration is more or less likely to occur than expected by chance, given all structures specified.This way, the method enables us to examine SES homophily in student friendship and parental contact networks, net of structural opportunities to form ties, and network-endogenous mechanisms that we control for.Second, we use meta-analysis techniques to combine estimates across classrooms (Snijders and Baerveldt, 2003). 8or every class, we estimate multiple specifications for both SES indicators (see Table 2 for an overview of the main specifications).The set-up slightly differs depending on (the measurement level of) the SES indicator included.Given the association between both SES indicators, they are not included in the same model.All models control for other tiegenerating mechanisms identified as important in prior research, such as mutuality, transitivity, and homophily by student gender.The first two specifications assess H1-H2, and are estimated for both student friendship and parental contact networks.We capture homophily based on household income (M1.A) with a term accounting for the absolute difference in household income between ego and alter, and homophily based on parental education with a term capturing the tendency to associate with alters with the same parental education (M1.B).We estimate SES differences in SES homophily by (i) including an interaction between household income (sender) and the difference in household income (M2.A); and (ii) estimating differential homophily parameters for both parental education categories (M2.B).Since residential areas are segregated, and this may (partly) explain SES homophily (Kruse et al., 2016), we conduct supplementary analyses where we control for the spatial proximity between classmates' home addresses.
The last two models assess (SES disparities in) IC.In M3, we add the edge covariate indicating whether a tie exists among parents as a predictor for friendship networks to M1 (cf.Windzio and Heiberger, 2022).The higher this coefficient, the more co-occurring ties between student and parental networks.Note that this estimate cannot inform us about the order in which co-occurring ties were created: children may become friends because their parents know each other, yet parents may also get to know each other because their children are friends.Yet, irrespective of these dynamics (which we cannot study with our cross-sectional data), higher coefficients indicate a stronger tendency towards IC.To test for SES disparities in IC (H3), we include an interaction between household income (sender) (M4.A) or parental education (sender) (M4.B) on the 4 In multi-grade classrooms, where grade 6 students shared a classroom with grade 5 and/or grade 4 students, all students were invited to ensure a better representation of classroom networks.
5 All C1 schools were invited to stay in PRIMS; 25 schools participated in both cohorts.For this reason, a small share of students (2.8%), that were in grade 4 or 5 in 2019/20 or repeated a grade, took part in both surveys. 6The sample size in our analyses varies depending on the SES indicator.
Information on parental education is missing for more students than household income (see section 3.2).Additionally, ERGMs cannot be estimated if in-or outgroup nominations for categorical features are impossible.We only keep classrooms with at least two students with higher educated parents, and two students with low/medium educated parents in models for parental education.
The sample includes 61 classrooms (1259 student observations, 50 schools) for parental education and 68 classrooms (1403 student observations, 55 schools) for household income. 7Educational attainment is centrally registered for university programs since 1983 (wo), and for universities of applied sciences since 1986 (hbo).Educational attainment is not centrally registered for other tertiary programs.Hence, if information on parental education is missing and a student's parents are born in the Netherlands after 1966, we assume that a student's parents likely did not finish higher education.We recode these cases to low/medium (0) to reduce the share of missings, and increase the sample at the classroom level.
one hand, and parental ties on the other. 9 We present our estimates as average marginal effects (AMEs), as proposed by Duxbury (2021b), because conventional ERGM coefficients suffer from scaling issues that are well known in the context of other nonlinear probability models (see Mood, 2010).AMEs are robust to scaling, and comparable across different model specifications.Furthermore, AMEs offer a more intuitive interpretation of effect sizes than logistic estimates: AMEs can be interpret as the absolute change in the probability of observing a tie with each one-unit change in a covariate.As AMEs are still relatively new in the context of ERGMs, we also present the main results in terms of logistic estimates in Appendix A.
The AMEs form the input for univariate random effects metaanalyses to test the hypotheses across classrooms.This method weighs lower-level estimates (AMEs) by their inverse standard errors so that classes with more precise estimates contribute more to the averaged estimates. 10Only class-specific estimates where the estimation turned out successful in terms of convergence and model fit are included in the meta-analyses.In case the model does not converge well, we rerun it until all t-ratios for convergence are below |0.1| (Robins et al., 2009). 11 We examine goodness of fit (GOF) with respect to edgewise-shared partners, in-degree, out-degree, and geodesic distances.GOF-ratio's taking on values below |2| indicate satisfactory fit (Robins et al., 2009).If less than 80% of the classroom's GOF-ratios meet this criterion, the classroom is excluded.Finally, we drop classrooms with logistic coefficients or standard errors exceeding |10| or |5|, respectively, as this suggests that the model does not fit the observed network, or that the classroom is a strong outlier (cf., Kruse et al., 2016;Smith et al., 2016).SeeAppendix A, Table A.3 for the number of excluded classrooms for each criterion per model.

Descriptive results
Table 3 presents descriptive statistics of the two types of networks, summarized over all classrooms.The average number of participating students per classroom equals 20.8 (SD = 3.9).On average, students (are) nominate(d) (by) 8.7 classmates as best friends, and 65% of all nominations are reciprocated.Students indicate that their parents are, on average, in contact with 7.6 parents.The average density of the  Source: PRIMS C1W1 (2019-2020), C2W1 (2020-2021), and CBS microdata. 9We do not formally model friendships and parental ties simultaneously.Due to the increased complexity and methodological challenges, empirical studies estimating ERGMs on multiplex networks are scarce (notable exceptions include Huitsing et al., 2012;Oldenburg et al., 2018;Rambaran et al., 2021), and packages to implement such methods are not available on CRAN yet. 10Other scholars (e.g., Kruse and Kroneberg, 2019;McFarland et al., 2014) have used multivariate meta-analyses, taking the interdependencies between estimates in each network into account (see for more info An, 2015).Since we do not apply a universal model set-up, we opt for univariate meta-analyses instead. 11We re-estimate the model until all estimates fulfill this criterion, while using the values of the previous run as initial values.We apply a set-up with a Markov chain Monte Carlo (MCMC) burn-in of 15,000, a MCMC sample size of 30,000, and a maximum of 10 iterations.The MCMC burn-in and sample size are increased with a factor of 1.5 (run 2-5), and 2 (run 6-10) in repetitions.If at least one of the t-ratios still exceeds |0.1| after 10 runs, the classroom is dropped from the meta-analysis.
parental networks (0.19) is relatively high, compared to prior research using child-report information on parental networks in primary school (0.04) (Windzio and Heiberger, 2022). 12Both types of networks have relatively few students that are neither nominated by classmates nor nominate any classmates themselves (isolates).Parental contact networks have a relatively high share of sinks (i.e., students indicating that their parents are not in contact with other parents, though they receive nominations on this item by classmates).Fig. 2 displays descriptive statistics on the share of co-occurrences between friendship and parental ties among all students, and separated by parental education.On average, for 52% of the outgoing friendship ties, children indicate that their parents are connected with the parents of the nominated peer.The mean share of co-occurrences is higher among children of higher educated parents (55%), compared to children of lower/medium educated parents (47%).Among students from advantaged backgrounds, same-SES ties enjoy higher levels of IC than cross-SES ties (60% versus 49%).Such a pattern is not observed among students from more disadvantaged backgrounds (44% versus 47%).All in all, these descriptive results align with our expectations, as they point toward a substantial overlap between friendship and parental ties, and potential SES disparities in IC.Tendency to associate with (parents of) peers with a similar migration background.Migration background is derived from the registers.We distinguish between "majority" and "minority" background as there is not enough variation in our sample to consider different origin countries separately.Children are coded to have a "majority" background if they and their parents are born in the Netherlands, or migrated from a European country (excl.Turkey), the United States, Oceania, Indonesia, or Japan.This is done as children from these origin countries are considered more similar to children without a migration background than to children from other countries (Vermeij et al., 2009).If at least one parent is born in another country, students are coded to have a "minority" background.

× × × ×
Notes: a Only included in multi-grade classrooms (N = 15).b Only included in classrooms that are not too homogeneous in this respect (i.e., at least two students with a "minority" background; N = 40).c If the model does not converge or GOF-statistics indicate unsatisfactory fit when including these terms, we include the results based on a more parsimonious model (excl.these terms) in the meta-analysis.Source: PRIMS C1W1 (2019-2020), C2W1 (2020-2021), and CBS microdata.Notes: 1416 students, 68 classrooms, 55 schools.Reciprocity is measured as the ratio of the number of (directed) reciprocated ties to the total number of tie, transitivity as the number of transitive triplets divided by the number of twopaths. 12This might be explained by subtle differences in question wording, tapping into parental ties of a different strength.We ask "Whose parents do your parent (s)/caregiver(s) occasionally talk to?", while the study by Windzio and Heiberger (2022) asks "Do your parents know other classmates' parents (so that they sometimes meet up or phone)".The latter is similar to the question in the Children of Immigrants Longitudinal Survey in Four European Countries (CIL-S4EU), showing that parental networks in secondary school (where opportunities for parents to meet each other are lower than in primary school) have a mean density between 0.07 and 0.12, depending on the academic track, in the Netherlands (see Geven & Van de Werfhorst, 2020, p. 47).

D. Zwier and S. Geven
Table A.4). M1 is used to assess the general tendency for ties between students or parents from similar SES backgrounds, while controlling for other tie formation mechanisms (H1a-H2a).For student friendships, the significant negative AME for difference in household income suggests that students from dissimilar SES backgrounds are less likely to be friends: an increase of 1 SD in household income difference, reduces the tie probability on average with 0.67% points (see M1.A).Similarly, the positive coefficient for same parental education suggests that students from similar SES backgrounds tend to connecti.e., compared to a cross-SES tie, a same-SES tie is 1.27% points more likely (see M1.B). 13 Given that students on average nominate 22% of their classmates as friends (see average density friendship networks, Table 3), this effect is not negligible.These results are in line with H1a.Parental ties are more strongly driven by homophily with respect to SES (H2a): parents who differ less in terms of household income (AME = − 1.37, p < 0.001) or have a similar educational background (AME = 2.20, p < 0.001) are more likely to connect.Considering the average density of 0.19, these effects are substantial in size.
M2 tests for SES differences in the strength of SES homophily (H1b-H2b).For both the friendship and parental networks, we find no significant interaction between a student's household income (sender) and household income difference (student friendships: AME = 0.14, p > 0.10; parental contact: AME = − 0.09, p > 0.10, see M2.A).This suggests that the level of SES homophily (measured by household income) does not vary depending on household income.In M2.B, we estimate differential homophily coefficients for educational categories.These results suggest that higher educated parents are slightly more likely to associate with similar-SES parents than low/medium educated parents (AME low/medium = 1.62, p < 0.001; AME high = 2.17, p < 0.001), while the reversed pattern is found in friendship networks (AME low/me- dium = 1.22,p < 0.01; AME high = 0.86, p > 0.10).Note, however, that differences in these mean AMEs are small.Taken together, these results do not provide support for H1b-H2b.
Results for the control variables are in line with other network studies (e.g., Kruse and Kroneberg, 2019;McFarland et al., 2014).In both type of networks, we observe a tendency for mutuality and triadic closure, as indicated by the positive and significant mutual and GWESP parameters.As shown by the positive same gender coefficients, students are more likely to befriend same-gender classmates, and parents of same-gender students are more likely to be in contact.We also find evidence for homophily by migration background and grade (in multi-grade classrooms).Finally, household income and parental education are associated with a lower out-degree (negative sender effect) and a higher in-degree (positive receiver effect) in most specifications, implying that students from higher SES backgrounds receive more (friendship) nominations, while sending less.
Results for SES homophily may be (partly) explained by spatial proximity.As neighborhoods are segregated by SES, same-SES classmates may live closer to each other than to dissimilar others.As mentioned in the theory section, these recurrent opportunities to connect and to spend time together can increase chances of friendship ties (Kruse et al., 2016).Similarly, parents who live in the same neighborhood may also have more low-stakes opportunities to connect, e.g., at sport clubs or during grocery shopping.To assess if this neighborhood propinquity effect (partly) drives SES homophily, we run supplementary analyses including an edge covariate measuring the straight-line distance between the home addresses of ego and alter to M1. 14 As travel distances are generally short (median = 0.87 km, SD = 2.03), the neighborhood propinquity effect is arguably local and non-linear (i.e., travel distance may matter less as students live further away from each other).To account for such non-linearities, we also add a quadratic term. 13A formal test for moderation using AMEs, recently introduced by Duxbury (2021b), is the second difference approach.Empirical studies using this approach for pooled networks using meta-analyses do not exist yet, so we focus on interaction coefficients instead.Second differences of aggregated AMEs yield the same substantive conclusions.To illustrate, additional calculations (not displayed in table) suggest that the aggregated sender effect of parental education (1 = high) is − 2.49 for lower educated alters, and − 1.33 for higher educated alters.The second difference is 1.16, indicating that high-SES/high-SES friendships are 1.16% points more likely than high-SES/low-SES friendships, also reflecting a preference for same-SES others.
14 Missing values on spatial proximity (1.0%) are imputed with the classroom mean.

D. Zwier and S. Geven
As expected, the results (see Appendix A, Table A.5) show that students living further away from each other, are less likely to become friends (see M1.A, AME = − 0.85, p < 0.05).Furthermore, we find a negative neighborhood propinquity effect in parental contact networks that becomes significantly less negative at further travel distances (see M1.A, AME = − 1.37, p < 0.001).Accounting for spatial proximity, the results for homophily by parental education remain virtually unchanged.The AMEs for homophily by household income decrease in size (student friendships: Δ AME = − 27%; parental contact: Δ AME = − 15%) but remain negative.Together, this suggests that the neighborhood proximity mechanism is more important for explaining homophily by household income than by parental education, and cannot (fully) explain homophilic tendencies by SES in friendship and parental networks.

SES and intergenerational closure
We study (SES disparities in) IC by including the edge covariate of whether a tie exists among students' parents as a predictor of students' friendships.Table 5 displays the average coefficients and their standard errors based on meta-analyses (in AMEs, see for logistic estimates Appendix A, Table A .6).M3 shows a strong association between (directed) friendship and parental ties.More specifically, the probability of a friendship tie is about 14% points higher among students whose parents are also in contact, net of the other tie-generation mechanisms included.Though we should refrain from causal interpretations, this suggests that student friendship and parental contact networks are strongly interlinked.
To test for SES disparities in IC, M4 includes interaction terms between household income (sender) or parental education (sender) on the one hand, and parental ties on the other.While the positive direction of the interaction terms between household income and parental tie, and parental education and parental tie, is in line with our theoretical expectation, the estimates are not statistically significant.This indicates that, controlling for other tie-generating mechanisms, we find no evidence that the peer networks of students from advantaged SES backgrounds display a stronger tendency towards IC than those of less advantaged backgrounds (H3).

The role of local opportunities for school choice
Findings show that the network processes we study significantly vary across classes.To shed more light on this, we explore tentative explanations for this variation.More specifically, regional variations in school supply (see Section 2.3) may be important for the network processes we study.Research suggests that in contexts in which institutional factors already hamper the opportunity to form cross-cutting ties in school, the expression of homophilic preferences by students and parents decreases (Engzell and Raabe, 2022;Kruse, 2019).In this line of thought, the local school context may affect preference-based segregation in networks in school.If parents (can) opt for a primary school with a large share of same-SES children, they "satisfy" in-group preferences, intentionally or not, in the school choice process already, and may feel less need to steer children's relations in school.Conversely, if they face a limited school supply, it may be more difficult to satisfy certain wishesunless they relocate, or pick a school outside the residential area.In these contexts, parents may interfere more actively in children's peer relations by (dis) approving friendships with (dis)similar others, and/or showing a stronger tendency to connect with other same-SES parents in school.
To explore if the (SES disparities in) homophily and IC are contingent on a lack of school choice, we distinguish between "restrictive" and "non-restrictive" school contexts.In the Netherlands, this is not simply a matter of urbanization, as in some rural areas families can still choose between multiple schools.Hence, we use geospatial information to do so: i.e., classrooms where at least 80% of the students (i) have only one school in a 1 km distance radius from home, or (ii) whose second option is located 500 m further than the first (making one school clearly superior in terms of proximity) are coded as "restricted."This does not imply that school choice is formally restricted, but it makes it at least more difficult to satisfy parental wishes.14 out of the 68 classrooms are coded as "restricted choice" contexts.Appendix A, Table A.7 shows metaregression results with homophily and IC coefficients as dependent variables, and the class-level "restricted choice" dummy as a predictor.The findings do not suggest that SES homophily and (SES disparities in) IC are contingent on local opportunities for school choice: i.e., the "restricted choice" dummy does not significantly explain variation in any of the network processes, and most coefficients are relatively small in size compared to the intercept.
Next, for families living in areas with multiple schools to choose from, we assess if the studied network processes are contingent on attending a school with a particular student populationi.e., if SES homophily and (SES disparities in) IC are weaker if families satisfy ingroup preferences in the school choice process.We again use metaregressions to study if contextual measures (i.e., SES composition school) contribute to classroom variation in the studied network processes.We fit a quadratic function to allow for potential non-linearities (cf.Malacarne, 2017;Smith et al., 2016).Appendix A, Figure A.1 summarizes the results for parental education (results for household income similar but not reported).The results indicate that estimates are not clearly scattered around the regression lines, and the margin of error is large.This suggests that the network processes under study are not contingent on school SES composition.

Discussion and conclusion
Peer relations in school can facilitate social capital that contributes to educational success.Whilst prior research shows that students typically cluster in schools with similar others in terms of SES background, less attention has been paid to the extent to which social relations in school are segregated by SES.In this study, we examined the tendency for student friendship and parental contact networks to structure along SES lines in primary school.First, our findings demonstrate that both student and parental networks are structured by SES.Dutch late primary school students and their parents are more likely to connect with same-SES than cross-SES others in class, net of opportunity structures and homophily based on other characteristics, such as migration background.This finding is consistent across both SES indicators (i.e., household income and parental education).Moreover, these results remain after controlling for spatial proximity between classmates' home addresses, suggesting that the neighborhood proximity mechanism cannot (fully) explain SES homophily.
Against our expectation, we find no clear SES differences in the strength of SES homophily.While previous studies find that parents from higher SES backgrounds tend to be more actively involved in their children's education (Lareau, 2011), our findings suggest that this is not reflected in stronger tendencies among them or their children to connect with similar others in primary school.
Interestingly, our results do show a positive association between SES and the number of incoming nominations, suggesting that socioeconomic status markers can form socially valued qualities for tie formation for student friendships and parental contacts.This corroborates prior research in Sweden, showing that students with less economic resources tend to have fewer friends, and are more likely to experience peer rejection and social isolation (Hjalmarsson, 2018;Hjalmarsson and Mood, 2015).
Second, we observe a strong tendency towards embeddedness of student friendships and parental networks in school.Though our design does not allow for causal interpretations, this finding is in line with prior research among German late primary school students (Windzio and Heiberger, 2022).Descriptive results suggest that networks of socio-economically advantaged students enjoy higher levels of IC.In addition, among high-SES students, same-SES ties display a higher degree of overlap across friendship and parental networks than cross-SES ties, while we do not observe such a pattern for low-SES students.Prior research suggests that higher SES parents are typically better integrated in parental networks, and forge closure relations with other parents to monitor their children's peer relations (Cox et al., 2021;Fletcher et al., 2006;Horvat et al., 2003;Ream and Palardy, 2008).Yet, once we control for other tie-generating mechanisms in ERGMS, SES disparities in IC are no longer significant.This suggests that these patterns may be explained by other network mechanisms that contribute to friendship formation (e.g., students' tendency to form ties with the friends of friends, and their tendency to form friendships with same-SES others).
Third, we found that levels of SES homophily and IC varied between classes.To shed more light on this, we explored whether this variation can be explained by theoretically relevant contextual factors.More specifically, we compared the network tendencies in contexts where school choice is "restricted" to contexts where this is not the case, and explored the role of the school's SES composition.We argued that SES homophily may be more pronounced in areas where parents have less opportunities to choose a school.When parents have the opportunity to explicitly pick a school with a student population that matches their own background, parents and students may express less homophilic preferences in the relations they form in school.We did not find support for this idea.Possibly, families self-select into neighborhoods such that even in contexts with a restricted supply, the student population of the available schools meets parental wishes.Another possibility is that school choice is more often restricted in rural areas, while at the same time SES integration is easier here than in (sub)urban areas: i.e., social life is centered around a limited number of amenities (e.g., sports clubs, religious buildings, shops) and neighborhoods are less segregated, which helps to cross-cut SES boundaries.Furthermore, one may argue that families who deliberately choose for a school with many same-SES students have strong homophilic preferences, and also express these in forming ties in school.This may counteract the pattern we would expect.A fruitful direction for future research would be to further theorize and test under which conditions (preference-based) segregation is most pronounced.
This study is not free of limitations.First, our cross-sectional design does not allow us to study tie formation processes in (the interplay between) student and parental networks.We conceptualize IC as a source of social capital in the educational context, irrespective of how cooccurring ties are formed.While these dynamics are thus less crucial for the research question at hand, this issue deserves further scrutiny in future research.Furthermore, the relatively small sample size at the classroom level does not allow us to fully explore differences in the network processes across school classes.We want to note that we were the first study to explore the potential role of the local school context in network formation processes in school, as we were able to link survey data to detailed register data.However, to provide a more definitive test of this theoretically promising relationship, a larger sample of schools is needed.Third, data on parental contacts is reported by children, as time and resource constraints did not allow us to also collect data among parents.Moreover, a separate parental survey could have resulted in a smaller and more selective sample.Though a child-report measure is commonly used in research on parental networks (e.g., Geven and Van de Werfhorst, 2020;Windzio and Heiberger, 2022), it may be that some children have difficulties with indicating with whom their parents are in contact.
Our study focuses on the Netherlands, but we believe the results are of great interest to other countries as well.Most prior research on friendship formation using sociometric data is situated in secondary school.This study informs us about how a personal characteristic that is arguable less visible to children than, e.g., gender or ethnicity, plays a role in structuring peer relations, starting already in primary schoola context where opportunities for contact with dissimilar others are often present, and teachers have more power than in secondary school to stimulate inter-group interactions via seating arrangements or group work (Gremmen et al., 2018;Keller and Takács, 2019).Moreover, the network patterns we find may amplify each other and leave a long-lasting imprint on students' networks, as homophilous ties that are embedded in parental networks have arguably more potential to survive the transition to secondary school.From a policy perspective, this study stresses that school de-segregation efforts would require interventions that do not solely focus on stimulating students from dissimilar backgrounds to attend similar schools, but also on the connections that students and parents form in school.

Table 1
Descriptive statistics student-level variables.

Table 2
Table 4 presents the results of univariate meta-analyses of various ERGM set-ups (in AMEs, for logistic estimates see Appendix A, Overview ERGM terms in main model specifications.

Table 3
Descriptive statistics friendship and intergenerational classroom networks.

Table 4
Results univariate meta-analysis ERGMs, SES homophily.AMEs and standard errors are multiplied by 100 to facilitate interpretation.All models control for the baseline density (edges term) (not included in table).Household income is mean-standardized.I 2 indicates the share of variability attributable to variation between networks rather than sampling error.Same grade and same mig.bg.terms are included for part of the classrooms (see Table2), so N classes does not reflect the number of classes for which these terms are included.
Notes: AMEs and standard errors are multiplied by 100 to facilitate interpretation.All models control for the baseline density (edges term) (not included in table).Household income is mean-standardized.I 2 indicates the share of variability attributable to variation between networks rather than sampling error.+ p < 0.10 * p < 0.05, * * p < 0.01, * ** p < 0.001.