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# Structural effect of size on interracial friendship

Contributed by Yu Xie, February 28, 2013 (sent for review July 27, 2012)

## Abstract

Social contexts exert structural effects on individuals’ social relationships, including interracial friendships. In this study, we posit that, net of group composition, total context size has a distinct effect on interracial friendship. Under the assumptions of (*i*) maximization of preference in choosing a friend, (*ii*) multidimensionality of preference, and (*iii*) preference for same-race friends, we conducted analyses using microsimulation that yielded three main findings. First, increased context size decreases the likelihood of forming an interracial friendship. Second, the size effect increases with the number of preference dimensions. Third, the size effect is diluted by noise, i.e., the random component affecting friendship formation. Analysis of actual friendship data among 4,745 American high school students yielded results consistent with the main conclusion that increased context size promotes racial segregation and discourages interracial friendship.

Social scientists have long emphasized the critical role of social context for social relationship formation (1⇓⇓–4). This emphasis is based on the recognition that social context is a precondition for social relations, as it provides the milieu for social interaction and assigns individuals to different physical or social segments so that they encounter different candidates for relationships. To understand the role of a social context, we invoke a common but simplistic conceptualization that the formation of individuals’ social relations results from a combination of two mechanisms: individual preference and structural constraint. The former refers to individuals’ subjective attitudes regarding potential relationships. The latter refers to the structural constraint imposed by a social context on individuals’ exposure to candidates, which segregate people into circles that limit relationship formation to persons in contact with one another. In this study, we separate out the structural effects of social contexts from those of individuals’ preferences and model social relationship formation as being solely a function of structural variations, holding individual preferences fixed.

One long-standing issue in the research on social relationship formation is the mechanism through which social structure affects the chances of intergroup friendship. Here, in keeping with literature on the topic, we define a “social group” as multiple individuals who are different from each other but share a social attribute recognizable by others. For example, “blacks” is a social group, because persons in this group are referred to as members of a racial category “blacks.” Hence, an “intergroup friendship” refers to a friend relationship between two persons belonging to different social groups. Most social scientists assume that individuals prefer to make friends with people of similar social attributes, including race/ethnicity, relation, age, education, and social class (5⇓–7). However, any individual’s likelihood of finding a satisfactory friend in terms of group similarity is constrained by the opportunities available in the person’s social context. To examine the structural effect on intergroup friendship given individuals’ preferences for in-group members, this study builds an analytical framework that decomposes the structural constraint imposed by social context into two distinct structural components: the relative proportion of social group *g* in the context ( and the total number of persons across all social groups in the context (*N*). We call the former “group proportion” and the latter “total size.”

In the current literature, the effect of contextual group proportion on intergroup friendship is well recognized. Theoretically, the likelihood of forming intergroup ties with group *g* depends on opportunities for social contact, which increase with . Thus, the larger the relative size of a group, the more likely it is that others will befriend its members (8, 9). This theoretical prediction has been shown in prior studies, which have focused mostly on the structural effects of group composition (10⇓⇓–13). Although some earlier work has recognized the impact of context size on the presence of homophily (14), the structural effect of total size has been undertheorized and thus understudied. Although researchers have always included total size in their empirical analyses, they have treated it more or less as a statistical nuisance or, at best, a statistical control (9⇓–11). Our purpose in this study is to amend this intellectual imbalance by arguing for a structural effect of context size while holding everything else constant. That is to say, our study departs from an exclusive emphasis in the prior literature on group composition and shows a distinct effect of total size on intergroup relations. Without loss of generality, the rest of this paper focuses on interracial friendship as a particular case. We will show, with a theoretical model as well as microsimulation, that under the assumption of rational actors, multidimensional preference, and preference for same-race friends, an increase in total context size reduces the likelihood of forming interracial friendships while group composition is held unchanged.

In the remainder of this paper, we first demonstrate the mechanism through which total size matters for intergroup friendship (*Interracial Friendship and Size*) and present a formal theoretical model according to which the size effect can be derived and interpreted (*Theoretical Model*). We then report results from microsimulations (*Size Effect in Simulated Data*) and an empirical analysis (*Size Effect in Real Data*) that confirm our thesis that size reduces interracial friendship. Finally, we conclude and point out the broader implications of our model (*Conclusion*).

## Interracial Friendship and Size

To derive the effect of total size, our point of departure is the proposition that individuals’ preferences in social relations should be based on multidimensional factors. Among these preference dimensions, some are observable by both the individual actor and the researcher, such as age, race, and educational attainment, whereas others are known to the individual actor but not to the researcher, such as personality, hobbies, cultural tastes, and political ideology. Next, we consider friendship formation as a process of rational choice: An individual starts by assigning a weight to each dimension that represents its relative importance in determining the individual’s preference. Through weighting, multiple dimensions of relevant factors form a single summary measure of utility associated with choosing each candidate as a partner. We further assume that the individual chooses as a friend the candidate who yields the highest utility (i.e., the rational actor assumption). A small community, i.e., a small context size, restricts the individual’s set of possible choices, so matching on all dimensions of preference is unlikely to be realized, and some compromise is likely. If occurring, a compromise is likely to take place concerning all relevant dimensions. When total size increases, given the rationality to maximize utility, individual actors will take advantage of the enlarged choice set by forming relationships that yield the highest utility for themselves. Thus, chances of compromise decline in all relevant dimensions. In this way, total size acts to minimize compromises that go against individuals’ preexisting relationship preferences.

In the case of interracial friend making, suppose one’s preference for a friend has two dimensions: race and personality. Further suppose that individuals all prefer friends who are similar to themselves along both dimensions. Let racial segregation be our primary concern. We assume for simplicity that race is binary, white versus black, but there are many categories of personality. In a small community, the chances of finding a good match on personality within one’s own racial group are low, giving rise to the possibility of a compromise concerning either the dimension of race or the dimension of personality, in exchange for a match in the other dimension. In a larger community with the same racial composition, a person faces a larger choice set, so the chances are better for him/her to find a good match on personality within his/her own race. As a result, everything else being equal, a larger share of interracial friendships would occur in smaller communities.

Finally, it is important to note that, in reality, individuals’ friend choices are by no means determined by the multitude of preference dimensions alone. In fact, there is always some noise affecting the chooser’s decision. That is, there exists a random component in the overall utility that does not represent any meaningful dimension of individual preference that the choosers intend to maximize on. Because the noise dilutes the power of utility maximization based on real individual preference in making friends, we expect the effect of total size to be smaller when the relative contribution of noise—measured by the variance of the random component in utility function—is larger.

## Theoretical Model

Our assertion that total size matters is a theoretical statement that can be made and defended on theoretical grounds alone. In this section, we establish a formal theoretical model that shows the effect of total size on interracial friendship. We simplify the process of friendship formation so that friendship results from each agent making a choice regarding with whom to form a relationship. We call the decision maker the “agent” and the people at risk for being chosen the “candidates.” Although we realize that reciprocal ties are also important to social relations, we focus on directional ties in this study to ensure parsimony of our demonstration. This means that multiple agents can choose the same person as a friend and that the person being chosen does not have the right to reject the choice.

With this simplification of friendship formation as a choice problem, we further distinguish between latent preference and realized choice. Latent preference represents an agent’s subjective evaluation of the desirability of a candidate as a friend based on the candidates’ attributes and a set of weights the agent assigns to them. Realized choice is the actual partner whom the agent chooses out of the set of candidates. The former is unobserved, and the latter is observed. As Zeng and Xie (15) argued, the major reason for distinguishing latent preference and realized choice involves the role of structural constraints. We assume that latent preference is independent of structural constraints and invariant over the choice process, as its existence precedes one’s entrance into any structural context. Unlike latent preference, realized choice is subject to the availability constraint imposed by social structure.

Independent of structural constraints, latent preference represents the agent’s inherent assessments of other persons’ desirability as candidates for friendship. As in standard choice models in economics, we represent latent preference by a utility function. The main reason for invoking the utility function is to quantify the desirability (i.e., utility), affected by multiple dimensions, in a unidimensional space. In other words, the utility is the quantitative expression of preference. We say that the agent “prefers A over B” if and only if the utility of A exceeds that of B. The agent is “indifferent between A and B” if and only if the utilities associated with the two candidates are identical. As such, choice decision results from the comparison of the relative utilities associated with different candidates. In our model, we follow the conventional setup of models for discrete choices (16) and specify utility as a function of two main components: a systematic utility *V* and an idiosyncratic disturbance that is identically and independently distributed as the type I extreme value distribution. We further assume that the two components are additive and separable. Thus, we write the utility for “agent *i* choosing candidate *j*” as follows:

The parameterization of the systematic utility is central to our model. As we have argued earlier, the agent’s preference in social relations should be multidimensional. Therefore, should be a function of components in multiple dimensions. Suppose that the agent’s preference contains *K* dimensions, then we can represent the utility as a function of a vector . We call a “covariate” each element in , measured specifically for the pair of agent *i* and candidate *j*. We further add a parameterization of the unidimensional measure of utility as a linear function of the *K*-dimensional covariates as follows:

For the *k*th covariate in Eq. **2**, there is a corresponding for mapping the covariate onto the unidimensional utility. can be interpreted as the weight assigned to the covariate associated with the *k*th preference dimension.

Corresponding to our primary interest in interracial friendship, we parameterize a covariate ( to represent the preference for same-race friends, with if agent *i* and candidate *j* share the same race, and otherwise. Because we assume that individuals prefer same-race friends, the weight () on the racial dimension of preference will be negative.

Next, let us discuss a key assumption on latent preference in our theoretical model. We assume that the agent’s utility associated with a candidate is unaffected by other candidates in the choice set. In discrete choice models, this is called the “independence of irrelevant candidates” (IIA) assumption. In friendship choice models, the IIA assumption also means the absence of peer interferences across difference persons. That is, not only is each agent’s utility associated with a candidate determined by unique, pairwise-measured covariate values, the utility is also unaffected by the presence and preferences of other agents. Statistically, we make use of this assumption by requiring that the values in Eq. **1** are mutually independent across all *i* values and *j* values.

In our theoretical framework, we require that latent preference remain invariant across social contexts. However, latent preference is unobservable in reality. Instead, we observe actual choices realized by agents and make inferences about latent preferences through realized choices.

How does total size affect realized choice? Formally, suppose there are *N* people in a closed context, and agent *i* chooses candidate ( as his/her friend. Let us use to denote a “matched pair,” in which agent *i* chooses candidate , with its associated utility . We invoke the behavioral assumption that is essential to the size effect: the agent pursues optimal utility in social relations and chooses the candidate with the highest utility in his/her choice set:

Recall again that our model of friendship choice does not preclude different agents from choosing the same person. Thus, all agents maximize their utility in choosing their friends regardless of other agents’ choices. Let us now ask a question about the expected value of in a closed population. Holding everything else unchanged, how does the expected utility of the matched pairs in the population, expressed by vary with population size *N*? The answer is a positive relationship. To explain, we draw a distinction between the “unconditional” expectation of in the population [] and the expectation of “conditional” on the matched pairs []. The unconditional expectation is the average utility over all “potential combinations of pairs,” which add up to a total of directional pairs; the conditional expectation is the average utility over a total of *N* matched pairs as follows:

Due to our rational actor assumption, agents always make their choices at the highest utility level available. It follows that the expected utility conditional on matched pairs always exceeds the unconditional utility, i.e., . As long as the data-generating mechanism remains unchanged, unconditional utility does not change when total size *N* increases. However, the expected value of the highest utility increases with size because a larger size improves the agent’s matching quality with a larger choice set. That is, the expected utility conditional on realized matches ( increases with size.

Next, let us follow with a second question: given that increases with total size , where does the increase in utility come from? The answer is that an overall increase in utility is shared over all relevant dimensions in the preference function. To see this, recall that in Eq. **2**, the systematic component of utility ( is decomposed into additive dimensions of covariates that occupy equivalent positions in the utility function. If there is an increase in , it shall come from an increase in utility on every dimension. As a result, we expect to observe an improvement in utility in every dimension, including a higher likelihood of forming a same-race friendship, in a larger context. That is, an increase in total context size *N* decreases the share of interracial friends. In the following two sections, we will present our results using both simulated and real data.

## Size Effect in Simulated Data

In this section, we conduct two rounds of microsimulation according to our theoretical model so as to demonstrate the effect of total size under different parameter setups. Our analysis involves five steps as follows.

First, we set up the choice set. The context contains *N* people. Consider four dimensions of covariates: race, age, family background, and personality, each measured as a unidimensional variable. We assume two racial groups, with the proportion of the majority group set at 80% and that of the minority group at 20%. That is, we fix the relative group proportion as a constant. The individual attributes on the latter three dimensions are assumed to follow a continuous scale and are independent and identically distributed. In reality, if two preference dimensions are perfectly correlated, the dimensions collapse into a single one. When dimensions are correlated but not perfectly, they can always be projected into orthogonal ones through linear transformation. To preserve parsimony, we assume these covariates to be mutually independent.

Second, we set up the individual utility. As discussed earlier, we specify that all of the covariates affect friendship choice through affecting the latent utilitywhere for *k* = 1, 2, 3, 4, representing the absolute distance between an agent and a candidate’s individual attributes. is an intercept that stays the same for every *ij* pair, because we assume that individuals prefer to befriend similar peers, for *k* = 1, 2, 3, 4. Table S1 gives the covariate distribution. The random component values are independent and identically distributed as type I extreme value distribution, representing the contribution of noise in the choice model.

Third, we simulate the choice behavior of every person in the context according to our theoretical model. For each agent, his/her set of friend choices contains every other person in the population. Consistent with our assumption of rational actors, each and every agent *i* chooses friend *j* who yields the highest , which is calculated uniquely for every combination of *i* and *j*. We allow each agent to choose only one friend, so that the chosen friend may be viewed as “best friend.” Because friend choices are nonexclusive, i.e., multiple agents can choose the same friend, the sequence of agents’ choices does not matter.

Fourth, we iterate the agent-based modeling process 500 times for each setup. For each round of iterations, we calculate the share of interracial friends, which equals the number of interracial friends chosen by agents divided by the number of agents. Then we average the share of interracial friends over the 500 iterations as an estimate of the expected share of interracial friends under this setup.

Last, we repeat this process for different setups. The primary independent variable is total size *N*. We vary *N* from 50 to 1,000 by an interval of 50, with a total of 20 specifications. We then plot a curve of expected share of interracial friends by total size *N*. The slope of the curve is an indicator of the size effect.

We conduct two rounds of simulation as described above. In the first round, we test our earlier proposition that multidimensionality of individual preference is essential to the size effect. Thus, we increase the number of dimensions from 1 to 2, 3, and 4. Table S2 gives the preference coefficients for the first round of simulation. In the second round, we assess the extent to which the contribution of noise affects the effect of total size in the four-dimensional case, by changing the variance of the random component : large (), medium (), and small ( Table S3 gives the variance of random components for this round of simulation.

We present the results of the two rounds of microsimulation in Figs. 1 and 2. In both figures, the horizontal axis is total context size, and the vertical axis is the average share of interracial friends. First, let us look at the general trend. A total of eight curves in the two figures support our theoretical prediction of size effect, as they all slope downward, indicating that the expected share of interracial friends decreases with the increase in total size. Moreover, they also show that the relationship between interracial friendship and total size is nonlinear: there is a diminishing return on total size.

The four curves in Fig. 1 represent results from the first round. Curves of different colors represent results from 1, 2, 3, and 4 dimensions. The comparison of the curves reveals that not only is the share of interracial friends higher with higher preference dimensions, but also the downward slope of a curve becomes increasingly steep with higher dimensions, i.e., size effect increases with the increase in the number of preference dimensions. The four curves in Fig. 2 represent results from the second round of simulation. In this round, we vary the relative contribution of noise, i.e., the share of random components in determining overall utility. The results show that the larger the noise, the lower the level of interracial relations. In addition, the negative effect of total size on interracial friendship is the most pronounced when the influence of noise is small.

## Size Effect in Real Data

We further test our proposition about the size effect with real data pertaining to best in-school friends actually chosen by a sample of American high school students. The data came from the Wave I in-school survey of the National Longitudinal Study of Adolescent Health (Add Health), which provides a school-based sample of adolescents in grades 7–12 in the United States during the 1994–1995 school year. A total of 172 schools participated in the survey. The survey instrument asked each respondent to nominate up to five best male friends and five best female friends in order of closeness from a school roster. The roster-based nomination enables us to match the attributes of each respondent (agent) with the attributes of his/her friends. For simplicity, we focus on the agent’s choice of best same-race friend for the illustration of our model. After necessary sample restriction, our analysis is based on a sample of 4,745 students.

We focus on the interracial friendship formation of four mutually exclusive racial groups: non-Hispanic white, non-Hispanic black, non-Hispanic Asian, and Hispanic and denote them as W, B, A, and H respectively in our analysis. We break down “interracial friend” ties into a total of 12 interracial combinations. Hence, the interracial friend outcomes for a white agent are WB, WA, and WH, for a black agent are BW, BA, and BH, for an Asian agent are AW, AB, and AH, and for a Hispanic agent are HW, HB, and HA. We estimate multinomial logit regressions for each racial group. We parameterize the regression coefficients in such a way that the coefficients represent the logged odds-ratio of the agent’s likelihood of choosing an interracial friend in each racial group, with the baseline outcome category choosing a same-race friend. The independent variables include log school size, the agent’s sex, age, grade in school, indicators for the chooser’s father’s education level, the group proportion for the agent’s race in the focal school, the indicator for public or private school, and the indicator for the school’s being in an urban area. Table S4 presents the descriptive statistics of the school level variables used in our estimation, and Table S5 presents the estimated coefficients on log school size from multinomial models by the agent’s and his/her best friend’s racial group. Of the five significant coefficients, four are negative. There are many possible reasons why the size effect does not work equally well in all cases in the empirical example. For example, for some ethnic groups, preference may actually change with size, or students with particular preferences may be selective with respect to school size. We use the empirical example mainly for the purpose of illustration.

For illustration, we predict the negative impact of school size on the share of interracial friends among Asians, a group for whom the school size effect is estimated to be significantly different from zero for all three interracial friendship types (AW, AB, and AH), using the coefficients estimated from the multinomial logit regression as described above as well as the sample average of covariates. To remain consistent with our microsimulation in the previous section, we assume that each population is composed of two races: Asians and whites, Asians and blacks, and Asians and Hispanics. In Fig. 3, we present our predicted share of Asians choosing an other-race friend among all of the friends they choose. As the curves suggest, on the average, Asians are most likely to choose whites as friends, with Hispanics ranking next and blacks last. This is likely due to the difference in group proportions of the three races, as well as the differences in Asian students’ baseline race-specific preferences. The slope of the curves shows the effect of school size on Asians’ likelihood of choosing the other three races. This effect, as we have shown in our simulation, is negative and nonlinear. The size effect is the strongest for choosing a black friend and the weakest for choosing a white friend.

## Conclusion

Social scientists have long recognized the influence of social structure on individual behaviors. The main objective of this paper is to argue for the structural influence of total size in a social context on social relationship formation. Under the assumption of individuals’ preferences for friends with similar attributes, an increase in size, everything else being equal, leads to a lower share of interracial friendships. However, we believe that the contributions of this paper are not limited to the demonstration of the mere existence of a size effect. Our core result has important wider implications.

First, our model contributes to the understanding of the interaction between individual agency and social structure. Our model recognizes the individual’s “free” agency by introducing latent preference as an intrinsic disposition that is context-invariant. However, we demonstrated, with the example of the size effect on the choice of friends, that the realization of predetermined preference under “rational” individual agency is subject to structural constraints.

Second, we draw attention to multidimensional preference in relationship formation. Although existing theories have often alluded to the multidimensionality of individual preference, they have not identified its significance for the structural effects of social context. We did so by establishing that multidimensionality in preference is a necessary condition for a size effect, in that a large context size acts as a buffer against the compromises across preference dimensions.

Beyond interracial friendship formation, our model can be extended to the understanding of a variety of other social relations, such as dating, marriage, political coalition, and even cooperation between business interests. The main conclusion that size reduces social integration by allowing individuals to fully exercise their preexisting preferences, however, could be applied to social relations in general. For this reasoning, one potential negative social consequence of the internet as a social interaction medium in the ever more globalized world is to encourage social isolation and social segmentation by expanding size immensely.

## Acknowledgments

We are grateful to Elizabeth Bruch, Mark Mizruchi, Ted Mouw, Lynn Smith-Lovin, Lincoln Quillian, Zhen Zeng, and Xiang Zhou for their comments on an earlier version of the paper. The work was supported by National Institutes of Health Grant R21 NR010856.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: yuxie{at}umich.edu.

Author contributions: Y.X. designed research; S.C. performed research; and S.C. and Y.X. wrote the paper.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1303748110/-/DCSupplemental.

## References

- ↵
- Allport G

- ↵
- Blau P,
- Schwartz J

- ↵
- ↵
- ↵
- ↵
- Fischer C

- ↵
- Yamaguchi K

- ↵
- ↵
- ↵
- ↵
- ↵
- Currarini S,
- Jackson MO,
- Pin P

- ↵
- ↵
- ↵
- ↵
- McFadden D

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