Paolo Pin

An Economic Model of Friendship: Homophily, Minorities and Segregation

We develop a model of friendship formation that sheds light on segregation patterns observed in social and economic networks. Individuals come in different types and have type-dependent 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 trace each of these empirical observations to specific properties of the theoretical model and highlight the role of choice and chance in generating homophilous behavior. Finally we discuss welfare implications of the model.

Identifying the roles of race-based choice and chance in high school friendship network formation

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.

Assessing the relevance of node features for network structure

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 networks 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.

Homophily and Long-Run Integration in Social Networks

We model network formation when heterogeneous nodes enter sequentially and form connections through both random meetings and network-based search, but with type-dependent biases. We show that there is “long-run integration”, whereby the composition of types in sufficiently old nodesʼ neighborhoods approaches the global type-distribution, provided that the network-based search is unbiased. However, younger nodesʼ connections still reflect the biased meetings process. We derive the type-based degree distributions and group-level homophily patterns when there are two types and location-based biases. Finally, we illustrate aspects of the model with an empirical application to data on citations in physics journals.

Stochastic stability in best shot network games

The best shot game applied to networks is a discrete model of many processes of contribution to local public goods. It generally has a wide multiplicity of equilibria that we refine through stochastic stability. We show that, depending on how we define perturbations – i.e., possible mistakes that agents make – we can obtain very different sets of stochastically stable states. In particular and non-trivially, if we assume that the only possible source of error is that of a contributing agent that stops doing so, then the only stochastically stable states are Nash equilibria with the largest contribution.

Collaboration in social networks

The very notion of social network implies that linked individuals interact repeatedly with each other. This notion allows them not only to learn successful strategies and adapt to them, but also to condition their own behavior on the behavior of others, in a strategic forward looking manner. Game theory of repeated games shows that these circumstances are conducive to the emergence of collaboration in simple games of two players. We investigate the extension of this concept to the case where players are engaged in a local contribution game and show that rationality and credibility of threats identify a class of Nash equilibria—that we call “collaborative equilibria”—that have a precise interpretation in terms of subgraphs of the social network. For large network games, the number of such equilibria is exponentially large in the number of players. When incentives to defect are small, equilibria are supported by local structures whereas when incentives exceed a threshold they acquire a nonlocal nature, which requires a “critical mass” of more than a given fraction of the players to collaborate. Therefore, when incentives are high, an individual deviation typically causes the collapse of collaboration across the whole system. At the same time, higher incentives to defect typically support equilibria with a higher density of collaborators. The resulting picture conforms with several results in sociology and in the experimental literature on game theory, such as the prevalence of collaboration in denser groups and in the structural hubs of sparse networks.

Statistical mechanics of maximal independent sets

The graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory. A maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints. It is known that finding minimum and maximum-density maximal independent sets are hard optimization problems. In this paper, we use cavity method of statistical physics and Monte Carlo simulations to study the corresponding constraint satisfaction problem on random graphs. We obtain the entropy of maximal independent sets within the replica symmetric and one-step replica symmetry breaking frameworks, shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms. This is of particular relevance for the application to the study of strategic interactions in social and economic networks, where maximal independent sets correspond to pure Nash equilibria of a graphical game of public goods allocation.

The informational divide

We propose a model of price competition where consumers exogenously differ in the number of prices they compare. Our model can be interpreted either as a non-sequential search model or as a network model of price competition. We show that (i) if consumers who previously just sampled one firm start to compare more prices all types of consumers will expect to pay a lower price and (ii) if consumers who already sampled more than one price sample (even) more prices then there exists a threshold – the informational divide – such that all consumers comparing fewer prices than this threshold will expect to pay a higher price whereas all consumers comparing more prices will expect to pay a lower price than before. Thus increased search can create a negative externality and it is not necessarily beneficial for all consumers.

Long-Run Integration in Social Networks

We study network formation where nodes are born sequentially and form links with previously born nodes. Connections are formed through a combination of random meetings and through search, as in Jackson and Rogers (2007). A newborns random meetings of existing nodes are type-dependent and the newborns search is then by meeting the neighbors of the randomly met nodes. We study long-run integration, which requires that as a node ages sufficiently, the type distribution of the nodes connected to it approaches the overall type - distribution of the population. We show that long-run integration occurs if and only if the search part of the network formation process is unbiased, and that eventually the search process dominates in terms of the new links that an older node obtains. Integration, however, only occurs for sufficiently old nodes, and the aggregate type-distribution of connections in the network still reflects the bias of the random process. We illustrate the model with data on scientific citations in physics journals.

Opportunity and Choice in Social Networks

Our societies are heterogeneous in many dimensions such as census, education, religion, ethnic and cultural composition. The links between individuals - e.g. by friendship, marriage or collaboration - are not evenly distributed, but rather tend to be concentrated within the same group. This phenomenon, called imbreeding homophily, has been related to either (social) preference for links with own--type individuals ( choice-based homophily) or to the prevalence of individuals of her same type in the choice set of an individual ( opportunity-based homophily). We propose an indicator to distinguish between these effects for minority groups. This is based on the observation that, in environments with unbiased opportunities, as the relative size of the minority gets small, individuals of the minority rarely meet and have the chance to establish links together. Therefore the effect of choice--based homophily gets weaker and weaker as the size of the minority shrinks. We test this idea across the dimensions of race and education on data on US marriages, and across race on friendships in US schools, and find that: for what concerns education i) opportunity--based homophily is much stronger than choice--based homophily and ii) they are both remarkably stationary in time; concerning race iii) school friendships do not exhibit opportunity-based homophily, while marriages do, iv) choice-based homophily is much stronger for marriages than for friendships and v) these effects vary widely across race.