We develop a model of friendship formation that sheds light on segregation patterns observed in social and economic networks. Individuals have types and see typedependent benefits from friendships. We examine the properties of a steady-state equilibrium of a matching process of friendship formation. We use the model to understand three empirical patterns of friendship formation: (i) larger groups tend to form more same-type ties and fewer other-type ties than small groups, (ii) larger groups form more ties per capita, and (iii) all groups are biased towards same-type relative to demographics, with the most extreme bias coming from middle-sized groups. We show how these empirical observations can be generated by biases in preferences and biases in meetings. We also illustrate some welfare implications of the model.

Homophily, the tendency of people to associate with others similar to themselves, is observed in many social networks, ranging from friendships to marriages to business relationships, and is based on a variety of characteristics, including race, age, gender, religion, and education. We present a technique for distinguishing two primary sources of homophily: biases in the preferences of individuals over the types of their friends and biases in the chances that people meet individuals of other types. We use this technique to analyze racial patterns in friendship networks in a set of American high schools from the Add Health dataset. Biases in preferences and biases in meeting rates are both highly significant in these data, and both types of biases differ significantly across races. Asians and Blacks are biased toward interacting with their own race at rates >7 times higher than Whites, whereas Hispanics exhibit an intermediate bias in meeting opportunities. Asians exhibit the least preference bias, valuing friendships with other types 90% as much as friendships with Asians, whereas Blacks and Hispanics value friendships with other types 55% and 65% as much as same-type friendships, respectively, and Whites fall in between, valuing other-type friendships 75% as much as friendships with Whites. Meetings are significantly more biased in large schools (>1,000 students) than in small schools (<1,000 students), and biases in preferences exhibit some significant variation with the median household income levels in the counties surrounding the schools.friendships | high schools | homophily | segregation | social networks F riendship networks from a sample of American high schools in the Add Health national survey † exhibit a strong pattern: students tend to form friendships with other students of their same ethnic group at rates that are substantially higher than their population shares ( Fig. 1) (1-4). This feature, referred to as "homophily" in the sociological literature (5), is prevalent across many applications and can have important implications for behaviors (6-9). The widespread presence of homophily indicates that friendship formation differs substantially from a process of uniformly random assortment. Two key sources of homophily are (i) biases in individual preferences for which relationships they form and (ii) biases in the rates at which individuals meet each other. It is important to identify whether homophily is primarily due to just one of these biases or to both because, for instance, this can shape policies aimed at producing more integrated high schools. In this article, we present a technique for identifying these two biases, we apply this technique to the Add Health dataset, and we estimate how preference and meeting biases differ across races.Although there is substantial evidence that race is a salient feature in how people view each other (10), such evidence does not sort out the sources of homophily, other than indicating that student preferences could be a factor. Without detailed and reliable d...

Networks describe a variety of interacting complex systems in social science, biology, and information technology. Usually the nodes of real networks are identified not only by their connections but also by some other characteristics. Examples of characteristics of nodes can be age, gender, or nationality of a person in a social network, the abundance of proteins in the cell taking part in protein-interaction networks, or the geographical position of airports that are connected by directed flights. Integrating the information on the connections of each node with the information about its characteristics is crucial to discriminating between the essential and negligible characteristics of nodes for the structure of the network. In this paper we propose a general indicator Θ, based on entropy measures, to quantify the dependence of a network's structure on a given set of features. We apply this method to social networks of friendships in U.S. schools, to the protein-interaction network of Saccharomyces cerevisiae and to the U.S. airport network, showing that the proposed measure provides information that complements other known measures.entropy | inference | social networks | communities N etworks have become a general tool for describing the structure of interaction or dependencies in such disparate systems as cell metabolism, the internet, and society (1-5). Loosely speaking, the topology of a given network can be thought of as the byproduct of chance and necessity (6), where functional aspects and structural features are selected in a stochastic evolutionary process. The issue of separating "chance" from "necessity" in networks has attracted much interest. This entails understanding random network ensembles (i.e., chance) and their inherent structural features (7-9) but also developing techniques to infer structural and functional characteristics on the basis of a given network's topology. Examples go from inference of gene function from protein-interaction networks (10) to the detection of communities in social networks (11,12). Community * detection, for example, aims at uncovering a hidden classification of nodes, and a variety of methods have been proposed relying on (i) structural properties of the network [betweenness centrality (13), modularity (14), spectral decomposition (15), cliques (16), and hierarchical structure (17)], (ii) statistical methods (18), or (iii) processes defined on the network (9, 19). Implicitly, each of these methods relies on a slightly different understanding of what a community is. Furthermore, there are intrinsic limits to detection; often the outcome depends on the algorithm and a clear assessment of the role of chance is possible in only a few cases (see, e.g., refs. 9 and 20).As a matter of fact, in several cases, a great deal of additional information, beyond the network topology, is known about the nodes. This comes in the form of attributes such as age, gender, and ethnic background in social networks or annotations of known functions for genes and proteins. Sometimes this...