Coordination in Dynamic Social Networks Under Heterogeneity

Abstract

People often make choices or form opinions depending on the social relations they have, but they also choose their relations depending on their preferred behavior and their opinions. Most of the existing models of coevolution of networks and individual behavior assume that actors are homogeneous. In this article, we relax this assumption in a context in which actors try to coordinate their behavior with their partners. We investigate with a game-theoretic model whether social cohesion and coordination change when interests of actors are not perfectly aligned as compared to the homogeneous case. Using analytical and simulation methods we characterize the set of stable networks and examine the consequences of heterogeneity for social optimality and segregation in emerging networks.

Keywords:

• coevolution of networks and behavior
• conventions
• coordination
• game theory
• heterogeneity
• social network dynamics

Acknowledgments

We would like to thank two anonymous reviewers for their helpful suggestions. We also would like to thank Stephanie Rosenkranz, Werner Raub, Jeroen Weesie, Rense Corten, Bastian Westbrock, and Ines Lindner, as well as participants of the ISCORE seminar at the ICS in Utrecht and the 6th Workshop on Networks in Economics and Sociology: Dynamic Networks in Utrecht for useful suggestions and criticism. The research reported here was facilitated by funding through the Utrecht University 2004 High Potential grant for the project Dynamics of Cooperation, Networks, and Institutions coordinated by Vincent Buskens and Stephanie Rosenkranz.

Notes

¹Here and in the remainder of this article, we use the word complements in a way similar to how it is used in set theory. We indicate with it the two separate but exhaustive types of benefits the actors can secure by forming ties to the two types of actors A and B. This is different from the definition of complementarity in microeconomics that corresponds to a situation in which the marginal benefits of, say, ties to type A increase with the number of ties to actors of type B.

²The distinction between benefits and costs made in this article is purely technical. We decided to develop benefits and costs separately as the benefits correspond closely to behavior in the coordination problems and costs are more closely related to the social network. Both benefits and costs are integrated in a single utility function.

³“U ₁ is binding” means that it is not possible to create any further ties without breaching it, i.e., n _i  + 1 > U ₁. Although the indifference case is not really crucial, we assume here that a tie is kept or made in case actors are indifferent between having or not having this tie.

⁴Only if the number of actors in the network is odd and the number of ties everyone wants to have is odd as well there will anyway be one actor who reaches only U ₂ − 1 relations.

⁵In Buskens et al. (2008) also, other types of dynamics such as that it is predefined whether an actor can change a relation or behavior are used as well in a similar model. There it was shown that these types of changes in the dynamics do not affect the main outcomes of the simulations.

^a Fixed at 8.

^b Sampling depends on values of w and b, see text.

^c Fixed at 30.

⁶By ⌊x⌋ we denote the “floor” of x: largest integer smaller than or equal to x.

⁷The entropy of a binary variable, if measured in bits, varies between 0 and 1. Minimum is attained for distributions in which one of the types is predominant; maximum is attained for distributions in which types are equally represented (maximum heterogeneity).

⁸Formally Freeman's segregation is undefined for the homogeneous case, but by convention it is assumed to be equal to 0.

⁹We decided to model the likelihood of full segregation instead if modeling extent of segregation with usual continuous response models because of very high right-skewness of the segregation variable.

Note. Standard errors are corrected for data clustering within conditions (Huber, 1967). All variables are centered. Null model's deviance: 434961.4. Number of observations: 336000. Number of nesting conditions: 84000.

Note. Model 1: Cases in which all choosing native is optimal; Model 2: Cases for all choosing the native behavior of the majority group is optimal.