School Friendship Networks Research Articles

Central limit theorems for local network statistics

Summary Subgraph counts, in particular the number of occurrences of small shapes such as triangles, characterize properties of random networks. As a result, they have seen wide use as network summary statistics. Subgraphs are typically counted globally, making existing approaches unable to describe vertex-specific characteristics. In contrast, rooted subgraphs focus on vertex neighbourhoods, and are fundamental descriptors of local network properties. We derive the asymptotic joint distribution of rooted subgraph counts in inhomogeneous random graphs, a model that generalizes most statistical network models. This result enables a shift in the statistical analysis of graphs, from estimating network summaries to estimating models linking local network structure and vertex-specific covariates. As an example, we consider a school friendship network and show that gender and race are significant predictors of local friendship patterns.

Selection homophily and peer influence for adolescents’ smoking and vaping norms and outcomes in high and middle-income settings

The MECHANISMS study investigates how social norms for adolescent smoking and vaping are transmitted through school friendship networks, and is the first study to use behavioral economics methodology to assess smoking-related social norms. Here, we investigate the effects of selection homophily (the tendency to form friendships with similar peers) and peer influence (a social process whereby an individual’s behavior or attitudes are affected by peers acting as reference points for the individual) on experimentally measured smoking and vaping norms, and other smoking outcomes, in adolescents from high and middle-income settings. Full school year groups in six secondary schools in Northern Ireland (United Kingdom) and six secondary schools in Bogotá (Colombia) participated (n = 1344/1444, participation = 93.1%, target age 12–13 years). Over one semester, pupils received one previously tested school-based smoking prevention program (ASSIST or Dead Cool). Outcomes included experimentally measured smoking/vaping norms, self-report and objectively measured smoking behavior, and self-report smoking norms, intentions, susceptibility, attitudes, and psycho-social antecedents. We investigated selection homophily and peer influence using regressions and SIENA modeling. Regression results demonstrate lagged and contemporaneous selection homophily (odds ratios [ORs] = 0.87–1.26, p ≤ 0.01), and peer influence effects for various outcomes from average responses of friends, school classes, or school year groups (standardized coefficients [βs] = 0.07–0.55, ORs = 1.14–1.31, p ≤ 0.01). SIENA models showed that comparable proportions of smoking/vaping-based similarity between friends were due to selection homophily (32.8%) and peer influence (39.2%). A higher percentage of similarity between friends was due to selection homophily and/or peer influence for ASSIST schools compared to Dead Cool. Selection homophily was also more important in Bogotá, whilst peer influence was stronger in Northern Ireland. These findings support using social norms strategies in adolescent smoking prevention interventions. Future research should consider selection homophily and social influence jointly, and examine whether these findings translate to other high and low-middle-income settings with varying cultures and norms.

Neither Friend nor Foe: Ethnic Segregation in School Social Networks

High levels of ethnic segregation have been widely observed in school friendship networks, whereas the degree to which school bullying networks are divided along ethnic lines remains uncertain. Using data from 981 students (53% girls, 47% boys; 11–14 years of age) in British schools, we sought to visualize, quantify, and compare the degree of ethnic segregation in friendship and bullying networks. Our findings contradict the common belief that ethnic segregation in friendship networks fosters interethnic conflict; instead, we identified similarly high levels of ethnic segregation in both friendship and bullying networks. Students may therefore simultaneously avoid positive and negative interethnic relationships. The findings indicate that positive and negative networks should both be considered to provide a comprehensive assessment of interethnic relations in the school environment.

The role of school friendship stability, instability, and network size in early adolescents' social adjustment.

This study examined longitudinal associations between early adolescents' school friendship stability, instability, and network size and their perceived social adjustment. The final sample consisted of 430 early adolescents residing in seven Midwestern schools (52% female, 47% Black, 42% White, 5% Hispanic or Latinx, and 6% Other). School friendship stability, instability, and network size were assessed via students' peer nominations of their same-grade friendships collected midway through their 7th and 8th grade years. Students also self-reported on several measures of social adjustment (their social satisfaction, social confidence, and school belonging). Results indicated having new friendships and having a larger friendship network at school were each more predictive of early adolescents' perceived social adjustment than was having maintained friendships or a consistent friend group within this context. However, school friendship stability consistently predicted greater perceived school belonging, whereas having a primarily new friend group and/or having lost more or most of a prior friend group predicted lower levels of this perception. Having lost school friendships was not predictive of early adolescents' social adjustment except when losses composed a significant proportion of their school friendship network (i.e., they had lost more or most of their school friendships from the year before). These findings provide valuable insights about peer friendship dynamics and social adjustment during a life stage critical to social development. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Local linear graphon estimation using covariates

Summary We consider local linear estimation of the graphon function, which determines probabilities of pairwise edges between nodes in an unlabelled network. Real-world networks are typically characterized by node heterogeneity, with different nodes exhibiting different degrees of interaction. Existing approaches to graphon estimation are limited to local constant approximations, and are not designed to estimate heterogeneity across the full network. In this paper, we show how continuous node covariates can be employed to estimate heterogeneity in the network via a local linear graphon estimator. We derive the bias and variance of an oracle-based local linear graphon estimator, and thus obtain the mean integrated squared error optimal bandwidth rule. We also provide a plug-in bandwidth selection procedure that makes local linear estimation for unlabelled networks practically feasible. The finite-sample performance of our approach is investigated in a simulation study, and the method is applied to a school friendship network and an email network to illustrate its advantages over existing methods.

AIM against survey fraud.

ObjectivesAlthough there exists a variety of anonymous survey software, this study aimed to develop an improved system that incentivizes responses and proactively detects fraud attempts while maintaining anonymity.Materials and MethodsThe Anonymous Incentive Method (AIM) was designed to utilize a Secure Hash Algorithm, which deterministically assigned anonymous identifiers to respondents. An anonymous raffle system was established to randomly select participants for a reward. Since the system provided participants with their unique identifiers and passwords upon survey completion, participants were able to return to the survey website, input their passwords, and receive their rewards at a later date. As a case study, the validity of this novel approach was assessed in an ongoing study on vaping in high school friendship networks.ResultsAIM successfully assigned irreversible, deterministic identifiers to survey respondents. Additionally, the particular case study used to assess the efficacy of AIM verified the deterministic aspect of the identifiers.DiscussionPotential limitations, such as scammers changing the entry used to create the identifier, are acknowledged and given practical mitigation protocols. Although AIM exhibits particular usefulness for network studies, it is compatible with a wide range of applications to help preempt survey fraud and expedite study approval.ConclusionThe improvements introduced by AIM are 2-fold: (1) duplicate responses can be filtered out while maintaining anonymity and (2) the requirement for the participant to keep their identifier and password for some time before returning to the survey website to claim a reward ensures that rewards only go to actual respondents.

Ground Truth in Network Communities and Metadata-Aware Community Detection: A Case of School Friendship Network

Real-world networks are everywhere and can represent biological, technological, and social interactions. They constitute complicated structures in terms of type of things and their relations. Understanding the network requires better examination of the network structure that can be achieved at various scales including macro, meso, and micro. This research is concerned with meso scale for a student best friendship network where sub-structures in which groups of entities (students) take different functions. In this study we address the following research questions: To what extent would NeoSBM as a stochastic process underlie best friendship interaction and in -turn ground truth interactions (i.e. reported best friendship)? Do metadata such as gender or class contribute to this understanding? How can one support school managers from a meta-data aware community detection perspective? Our findings suggest that metadata aware community detection can be an effective method in supporting decision-making for class formation and group formation for in and out school activities.

A Structural Model of Homophily and Clustering in Social Networks

I develop and estimate a structural model of network formation with heterogeneous players and latent community structure, whose equilibrium homophily and clustering levels match those usually observed in real-world social networks. Players belong to communities unobserved by the econometrician and have community-specific payoffs, allowing preferences to have a bias for similar people. Players meet sequentially and decide whether to form bilateral links, after receiving a random matching shock. The model converges to a hierarchical exponential family random graph. Using school friendship network data from Add Health, I estimate the posterior distribution of parameters and unobserved heterogeneity, detecting high levels of racial homophily and payoff heterogeneity across communities. The posterior predictions of sufficient statistics show that the model is able to replicate the homophily levels and the aggregate clustering of the observed network, in contrast with standard exponential family network models without community structure.

The Problem of Scaling in Exponential Random Graph Models

This study shows that residual variation can cause problems related to scaling in exponential random graph models (ERGM). Residual variation is likely to exist when there are unmeasured variables in a model—even those uncorrelated with other predictors—or when the logistic form of the model is inappropriate. As a consequence, coefficients cannot be interpreted as effect sizes or compared between models and homophily coefficients, as well as other interaction coefficients, cannot be interpreted as substantive effects in most ERGM applications. We conduct a series of simulations considering the substantive impact of these issues, revealing that realistic levels of residual variation can have large consequences for ERGM inference. A flexible methodological framework is introduced to overcome these problems. Formal tests of mediation and moderation are also proposed. These methods are applied to revisit the relationship between selective mixing and triadic closure in a large AddHealth school friendship network. Extensions to other classes of statistical work models are discussed.

School friendship networks, homophily and multiculturalism: evidence from European countries

This paper investigates the determinants of school friendship networks among adolescents, proposing a model of network formation and estimating it using a sample of about 10,000 secondary school students in four countries: England, Germany, the Netherlands and Sweden. We test the idea that networks arise according to homophily along many characteristics (gender, school achievement and ethnic and cultural backgrounds), and assess the relative importance of each factor. In addition to gender, we find that country of origin, generational status and religion predict friendship for foreign-born students. For individuals born in the survey country, ties depend on a broader set of factors, including socioeconomic status and school achievement. In sum, homophilic preferences go considerably beyond ethnicity. Multiculturalism, which gives prominence to ethnic backgrounds, risks emphasising the differences in that dimension at the expense of affinity in others.

Parental migration and Chinese adolescents’ friendship networks in school

This study examines whether and how parental migration may affect adolescents’ friendship networks. Applying random-effects Poisson regression models to survey data collected in Shaanxi province of China in 2017, we provide new evidence that left-behind status has been stigmatized and become a salient attribute in affecting adolescents’ friendship networks. Specifically, current left-behind status has independent, negative effects on the fifth graders’ popularity (in-degree) among their classmates, above and beyond the influence of both individual characteristics and family background. Moreover, we find that the discrepancy in network sizes between left-behind children and children with no left-behind experiences is mostly the result of avoidance encountered among the subgroup of left-behind children in the classroom. Compared to girls, adolescent boys have larger friendship networks, but their friendships are also more likely to be negatively affected by their left-behind status. Left-behind boys are experiencing avoidance by their left-behind classmates, while it is not the case for girls.

GraceKao, KaraJoyner, and Kelly StamperBalistreri, The Company We Keep: Interracial Friendships and Romantic Relationships from Adolescence to Adulthood, Russell Sage Foundation, 2019. 208 p. $29.95 (paper)

One of the best measures of integration is the level of intermarriage. Among American newlyweds in 2015, 11 percent of Whites, 18 percent of Blacks, and 29 percent of Asians married someone from a different race (p. 96). Almost as good a measure of integration is the frequency of interracial friendship and dating relationships. What Kao, Joyner, and Balistreri have done is to use a unique data source, The National Longitudinal Study of Adolescent to Adult Health (Add Health), to examine these relationships among adolescents aged 12–17 and then again when they were 25–32. Add Health sampled 132 US high schools and their feeder schools in 1994–1995. On a given class period all students in a school completed a questionnaire which asked them, among other things, to list their race/ethnicity and the names of up to five classmates who were their friends. This produced school friendship networks for 90,000 7th to 12th graders, the degree to which friendship ties were reciprocated, each individual's perception of their race and Hispanic identity, and each school's race, Hispanic, and social class composition. A subsample (n = 20,745) was interviewed in their homes at multiple points in time, including Wave IV interviews (n = 15,701) in 2007–2008. The in-home interviews used computer-assisted self-interviewing to collect rosters of romantic/sexual partners. The descriptive findings are interesting in themselves, but the authors highlight their theoretical and policy relevance. The most prominent finding is that same-race friendship and romantic relationships are still the norm, happening much more frequently than what one would expect on the basis of school-based opportunities for such contacts. This is especially true for Whites and Blacks, and much less so for Hispanics and Asians. The authors see multiple race lines existing in the United States: clear lines between Whites and non-Whites, as well as between Blacks and non-Blacks, but more permeable lines for Hispanics and Asians. Gender further complicates the patterns found, especially with respect to romantic relationships. Black women have many fewer interracial romantic relationships than Black men, and young Asian men have few romantic relationships of any kind. The policy relevance of this research is clearest with respect to the effect of attending a racially diverse school and of having a different-race friend during adolescence on the likelihood of having an interracial romantic relationship in adulthood (Chapter 6). Both significantly increase this likelihood, an association the authors take as empirical support for Gordon Allport's contact theory that face-to-face interaction with members of other groups may encourage individuals to develop more favorable attitudes toward those groups. The research is summarized in Chapter 1. Chapters 2 through 6 offer in-depth treatments of particular topics. Chapter 3, for instance, covers how Hispanic relationship patterns are affected by racial, socioeconomic and immigrant status differences. Same-sex romance is examined in Chapter 5. Each chapter starts with an overview of competing theories and of past research before focusing on the authors’ findings. Both the writing and the analysis, mainly bar and line graphs, make the work accessible to undergraduate students and the general audience.

Integration through friendships in school classes – A study of pupils with a migrant background in school friendship networks at primary and secondary schools in Germany

This paper deals with the integration of migrant children in informal communication networks in primary and secondary schools in Germany. The basic assumption is that these networks can serve as an indicator for the networking and embedding that take place within the class community and therefore can be used to investigate the integration of pupils with a migrant background. The initial results of our network analytical study reveal that pupils with a migrant background are well integrated into class networks. The study discloses that the migrant background does not serve as a conclusive characteristic of the pattern of class networks. The pupils are more or less networked with each other regardless of their migrant background, though it did emerge that children and adolescents born abroad, in particular, are far more likely to have outgoing relationships that remain unreciprocated. Remarkable is, that results from primary school correspond with data from secondary school.

The importance of social environment in preventing smoking: an analysis of the Dead Cool intervention

BackgroundAn adolescent’s perceptions of their family’s and friends’ smoking attitudes and behaviour can influence their own uptake of smoking. There are two broad sources of such social influence: observing the behaviour directly, and assimilating attitudes.MethodsWe analysed data collected for the evaluation of Dead Cool, a school based smoking prevention intervention in Northern Ireland (n=480 in 20 clusters). The main analysis fits three nested logistic regression models predicting pre-intervention susceptibility to taking up smoking, as reflected in responses to three attitudinal questions. Model 1 includes only personal characteristics as explanatory factors. Model 2 adds the behaviour of friends and family that would provide an opportunity for social influence through observational learning. Model 3 adds the susceptibility of friends.ResultsEach additional group of variables improved the model fit (with reduced AIC and BIC). However, in the final model, only three variables were found to be statistically significant (p<0.05) in predicting susceptibility to smoking initiation: rebelliousness (OR [1.1,1.3]) from the personal characteristics group; and, in the observational learning group, being friends with a smoker (OR [1.0,2.9]) and frequency of being in the same room or car with someone smoking (OR [2.0,9.0] for most frequent). Adding the two measures of diffusion of susceptibility through the friendship network improved the model fit, but neither was found to be statistically significant.ConclusionsThe analysis provides additional evidence to support policies that could reduce children’s exposure to smoking behaviour, and potential subsequent smoking initiation. No conclusions could be drawn about the diffusion of smoking attitudes through the school friendship networks of children.

Social Space Diffusion: Applications of a Latent Space Model to Diffusion with Uncertain Ties.

Social networks represent two different facets of social life: (1) stable paths for diffusion, or the spread of something through a connected population, and (2) random draws from an underlying social space, which indicate the relative positions of the people in the network to one another. The dual nature of networks creates a challenge - if the observed network ties are a single random draw, is it realistic to expect that diffusion only follows the observed network ties? This study takes a first step towards integrating these two perspectives by introducing a social space diffusion model. In the model, network ties indicate positions in social space, and diffusion occurs proportionally to distance in social space. Practically, the simulation occurs in two parts. First, positions are estimated using a statistical model (in this example, a latent space model). Then, second, the predicted probabilities of a tie from that model - representing the distances in social space - or a series of networks drawn from those probabilities - representing routine churn in the network - are used as weights in a weighted averaging framework. Using longitudinal data from high school friendship networks, I explore the properties of the model. I show that the model produces smoothed diffusion results, which predict attitudes in future waves 10% better than a diffusion model using the observed network, and up to 5% better than diffusion models using alternative, non-model-based smoothing approaches.

The Co-evolution of Bullying Perpetration, Homophobic Teasing, and a School Friendship Network.

Bullying and homophobic teasing behaviors affect the lives of many school aged children, often co-occur, and tend to peak in middle school. While bullying and homophobic teasing behaviors are known to be peer group phenomena, studies typically examine the associations at the individual or school levels. An examination of these behaviors at the peer group level can aid in our understanding of the formation and maintenance of peer groups that engage in these forms of aggressive behavior (selection), and the extent to which friends and the peer group impact individual rates of these aggressive behaviors (influence). In this longitudinal study, we assess the co-evolution of friendship networks, bullying perpetration, and homophobic teasing among middle school students (n = 190) using a Stochastic Actor-Based Model (SABM) for longitudinal networks. Data were collected from 6-8th-grade students (Baseline age 12-15; 53% Female; 47% Male) across three waves of data. The sample was diverse with 58% African American, 31% White, and 11% Hispanic. Since bullying and homophobic teasing behaviors are related yet distinct forms of peer aggression, to capture the unique and combined effects of these behaviors we ran models separately and then together in a competing model. Results indicated that on average individuals with higher rates of bullying perpetration and homophobic teasing were associated with becoming increasingly popular as a friend. However, the effects were not linear, and individuals with the highest rates of bullying perpetration and homophobic teasing were less likely to receive friendship nominations. There was no evidence that bullying perpetration or homophobic teasing were associated with the number of friendship nominations made. Further, there was a preference for individuals to form or maintain friendships with peers who engaged in similar rates of homophobic name-calling; however, this effect was not found for bullying perpetration. Additionally, changes in individual rates of bullying perpetration were not found to be predicted by the bullying perpetration of their friends; however, changes in adolescent homophobic teasing were predicted by the homophobic teasing behaviors of their friends. In a competing model that combined bullying perpetration and homophobic teasing, we found no evidence that these behaviors were associated with popularity. These findings are likely due to the high association between bullying perpetration and homophobic teasing combined with the small sample size. However, friendship selection was based on homophobic name-calling, such that, there was a preference to befriend individuals with similar rates of homophobic teasing. We also examined several risk factors (dominance, traditional masculinity, impulsivity, femininity, positive attitudes of bullying, and neighborhood violence), although, impulsivity was the only covariate that was associated with higher levels of bullying perpetration and homophobic teasing. More specifically, youth with higher rates of impulsivity engaged in higher rates of bullying perpetration and homophobic teasing over time. The findings suggest bullying perpetration and homophobic teasing have important influences on friendship formation, and close friendships influence youth's engagement in homophobic teasing. Implications for prevention and intervention efforts are discussed in terms of targeting peer groups and popular peers to help reduce rates of these aggressive behaviors.

A Structural Model of Homophily and Clustering in Social Networks

Social networks display homophily and clustering, and are usually sparse. I develop and estimate a structural model of strategic network formation with heterogeneous players and latent community structure, whose equilibrium networks are sparse and exhibit homophily and clustering. Each player belongs to a community unobserved by the econometrician. Players' payoffs vary by community and depend on the composition of direct links and common neighbors, allowing preferences to have a bias for similar people. Players meet sequentially and decide whether to form bilateral links, after receiving a random matching shock. The probability of meeting people in different communities is smaller than the probability of meeting people in the same community, and it decreases with the size of the network. The model converges to an exponential family random graph, with weak dependence among links. As a consequence the equilibrium networks are sparse and the sufficient statistics of the network are asymptotically normal. The posterior distribution of structural parameters and unobserved heterogeneity is estimated with school friendship network data from Add Health, using a Bayesian exchange algorithm. The estimates detect high levels of racial homophily, and heterogeneity in both costs of links and payoffs from common friends. The posterior predictions show that the model is able to replicate the homophily levels and the aggregate clustering of the observed network, in contrast with standard exponential family network models.

Link Prediction for Partially Observed Networks

ABSTRACTLink prediction is one of the fundamental problems in network analysis. In many applications, notably in genetics, a partially observed network may not contain any negative examples, that is, edges known for certain to be absent, which creates a difficulty for existing supervised learning approaches. We develop a new method that treats the observed network as a sample of the true network with different sampling rates for positive (true edges) and negative (absent edges) examples. We obtain a relative ranking of potential links by their probabilities, using information on network topology as well as node covariates if available. The method relies on the intuitive assumption that if two pairs of nodes are similar, the probabilities of these pairs forming an edge are also similar. Empirically, the method performs well under many settings, including when the observed network is sparse. We apply the method to a protein–protein interaction network and a school friendship network.

Analytic strategies for longitudinal networks with missing data

Missing data are often problematic when analyzing complete longitudinal social network data. We review approaches for accommodating missing data when analyzing longitudinal network data with stochastic actor-based models. One common practice is to restrict analyses to participants observed at most or all time points, to achieve model convergence. We propose and evaluate an alternative, more inclusive approach to sub-setting and analyzing longitudinal network data, using data from a school friendship network observed at four waves (N=694). Compared to standard practices, our approach retained more information from partially observed participants, generated a more representative analytic sample, and led to less biased model estimates for this case study. The implications and potential applications for longitudinal network analysis are discussed.

Ethnic segregation of friendship networks in school: Testing a rational-choice argument of differences in ethnic homophily between classroom- and grade-level networks

Adolescents’ school-based friendship networks tend to be segregated along ethnic lines. But few studies have examined whether variation in network boundaries affects the degree of ethnic friendship segregation. We use rational-choice theory to argue that ethnic homophily is more pronounced for friendships between classrooms than for those within classrooms. We empirically test this hypothesis using two-wave German panel data (N=1258) and stochastic actor-oriented models (RSiena). In line with our theoretical argument, we find that the tendency to form same-ethnic friendships is indeed stronger at the grade level, which translates into stronger ethnic segregation in friendship networks at the grade level than at the classroom level. Implications for research on ethnic segregation in school-based friendship networks are discussed.