Introduction to stochastic actor-based models for network dynamics☆
Introduction
Social networks are dynamic by nature. Ties are established, they may flourish and perhaps evolve into close relationships, and they can also dissolve quietly, or suddenly turn sour and go with a bang. These relational changes may be considered the result of the structural positions of the actors within the network – e.g., when friends of friends become friends –, characteristics of the actors (‘actor covariates’), characteristics of pairs of actors (‘dyadic covariates’), and residual random influences representing unexplained influences. Social network research has in recent years paid increasing attention to network dynamics, as is shown, e.g., by the three special issues devoted to this topic in Journal of Mathematical Sociology edited by Patrick Doreian and Frans Stokman (1996, 2001, and 2003; also see Doreian and Stokman, 1997). The three issues shed light on the underlying theoretical micro-mechanisms that induce the evolution of social network structures on the macro-level. Network dynamics is important for domains ranging from friendship networks (e.g., Pearson and Michell, 2000, Burk et al., 2007) to, for example, organizational networks (see the review articles Borgatti and Foster, 2003, Brass et al., 2004).
In this article we give a tutorial introduction to what we call here stochastic actor-based models for network dynamics, which are a type of models that have the purpose to represent network dynamics on the basis of observed longitudinal data, and evaluate these according to the paradigm of statistical inference. This means that the models should be able to represent network dynamics as being driven by many different tendencies, such as the micro-mechanisms alluded to above, which could have been theoretically derived and/or empirically established in earlier research, and which may well operate simultaneously. Some examples of such tendencies are reciprocity, transitivity, homophily, and assortative matching, as will be elaborated below. In this way, the models should be able to give a good representation of the stochastic dependence between the creation, and possibly termination, of different network ties. These stochastic actor-based models allow to test hypotheses about these tendencies, and to estimate parameters expressing their strengths, while controlling for other tendencies (which in statistical terminology might be called ‘confounders’).
The literature on network dynamics has generated a large variety of mathematical models. To describe the place in the literature of stochastic actor-based models Snijders, 1996, Snijders, 2001, these models may be contrasted with other dynamic network models.
Most network dynamics models in the literature pay attention to a very specific set of micro-mechanisms – allowing detailed analyses of the properties of these models –, but lack an explicit estimation theory. Examples are models proposed by Bala and Goyal (2000), Hummon (2000), Skyrms and Pemantle (2000), and Marsili et al. (2004), all being actor-based simulation models that focus on the expression of a single social theory as reflected, e.g., by a simple utility function; those proposed by Jin et al. (2001) which represent a larger but still quite restricted number of tendencies; and models such as those proposed by Price (1976), Barabási and Albert (1999), and Jackson and Rogers (2007), which are actor-based, represent one or a restricted set of tendencies, and assume that nodes are added sequentially while existing ties cannot be deleted, which is a severe limitation to the type of longitudinal data that may be faithfully represented. Since such models do not allow to control for other influences on the network dynamics, and how to estimate and test parameters is not clear for them, they cannot be used for purposes of theory testing in a statistical model.
The earlier literature does contain some statistical dynamic network models, mainly those developed by Wasserman (1979) and Wasserman and Iacobucci (1988), but these do not allow complicated dependencies between ties such as are generated by transitive closure. Further there are papers that present an empirical analysis of network dynamics which are based on intricate and illuminating descriptions such as Holme et al. (2004) and Kossinets and Watts (2006), but which are not based on an explicit stochastic model for the network dynamics and therefore do not allow to control one tendency for other (‘confounding’) tendencies.
Distinguishing characteristics of stochastic actor-based models are flexibility, allowing to incorporate a wide variety of actor-driven micro-mechanisms influencing tie formation; and the availability of procedures for estimating and testing parameters that also allow to assess the effect of a given mechanism while controlling for the possible simultaneous operation of other mechanisms or tendencies. We assume here that the empirical data consist of two, but preferably more, repeated observations of a social network on a given set of actors; one could call this network panel data. Ties are supposed to be the dyadic constituents of relations such as friendship, trust, or cooperation, directed from one actor to another. In our examples social actors are individuals, but they could also be firms, countries, etc. The ties are supposed to be, in principle, under control of the sending actor (although this will be subject to constraints), which will exclude most types of relations where negotiations are required for a tie to come into existence. Actor covariates may be constant like sex or ethnicity, or subject to change like opinions, attitudes, or lifestyle behaviors. Actor covariates often are among the determinants of actor similarity (e.g., same sex or ethnicity) or spatial proximity between actors (e.g., same neighborhood) which influence the existence of ties. Dyadic covariates likewise may be constant, such as determined by kinship or formal status in an organization, or changing over time, like friendship between parents of children or task dependencies within organizations. This paper is organized as follows. In the next section, we present the assumptions of the actor-based model. The heart of the model is the so-called objective function, which determines probabilistically the tie changes made by the actors. One could say that it captures all theoretically relevant information the actors need to ‘evaluate’ their collection of ties. Some of the potential components of this function are structure-based (endogenous effects), such as the tendency to form reciprocal relations, others are attribute-based (exogenous effects), such as the preference for similar others. In Section 3, we discuss several statistical issues, such as data requirements and how to test and select the appropriate model. Following this we present an example about friendship dynamics, focusing on the interpretation of the parameters. Section 4 proposes some more elaborate models. In Section 5, models for the coevolution of networks and behavior are introduced and illustrated by an example. Section 6 discusses the difference between equilibrium and out-of-equilibrium situations, and how these longitudinal models relate to cross-sectional statistical modeling of social networks. Finally, in Section 7, a brief discussion is given, the Siena software is mentioned which implements these methods, and some further developments are presented.
Section snippets
Model assumptions
A dynamic network consists of ties between actors that change over time. A foundational assumption of the models discussed in this paper is that the network ties are not brief events, but can be regarded as states with a tendency to endure over time. Many relations commonly studied in network analysis naturally satisfy this requirement of gradual change, such as friendship, trust, and cooperation. Other networks more strongly resemble ‘event data’, e.g., the set of all telephone calls among a
Issues arising in statistical modeling
When employing these models, important practical issues are the question how to specify the model – boiling down mainly to the choice of the terms in the objective function – and how to interpret the results. This is treated in the current section.
More complicated models
This section treats two generalizations of the model sketched above.
Dynamics of networks and behavior
Social networks are so important also because they are relevant for behavior and other actor-level outcomes: related actors may influence one another (e.g., Friedkin, 1998), and ties will be selected in part based on the similarity between ego and potential relational partners (homophily, see McPherson et al., 2001). This means that not only is the network changing as a function of itself and of the actor variables, but likewise the actor variables are changing as a function of themselves and
Cross-sectional and longitudinal modeling
For a further understanding of this actor-based model, it may be helpful to reflect about equilibrium and out-of-equilibrium social systems. Equilibrium is understood here not as a fixed state but as dynamic equilibrium, where changes continue but may be regarded as stochastic fluctuations without a systematic trend. This can be combined with discussing the relation between cross-sectional and longitudinal statistical modeling of social networks.
For cross-sectional modeling the exponential
Discussion
This paper has given a tutorial introduction in the use of actor-based models for analyzing the dynamics of directed networks – expressed by the usual format of a directed graph – and of the joint interdependent dynamics of networks and behavior – where ‘behavior’ is an actor variable which may refer to behavior, attitudes, performance, etc., measured as an ordinal discrete variable. The purpose of these models is to be used to test hypotheses concerning network dynamics and represent the
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We are grateful to Andrea Knecht who collected the data used in the example, under the guidance of Chris Baerveldt. We also are grateful to Matthew Checkley and two reviewers for their very helpful remarks on earlier drafts.