# Simulation-Based Statistical Inference for Evolution of Social Networks

Thursday, November 20, 2003 - 1:30pm - 2:20pm

Keller 3-180

Tom Snijders (Rijksuniversiteit te Groningen)

Social networks are structures of social ties between individuals (or other social actors, e.g., companies or countries). The most common representation of a social network is a directed graph, with individuals as nodes, in which the arcs indicate for each of the ordered pairs of individuals whether the tie in question (e.g., friendship) is present or not. In addition, there can be covariate data, referring to the individuals or the dyads (ordered or unordered pairs of nodes).

Social mechanisms can be brought to light much better using longitudinal observations on social networks, than by using single observations. The most common type of longitudinal observations are repeated measures, or panel data. Repeated measures on social networks represent a complicated data structure. Computer simulation offers fruitful possibilities here, because it greatly expands the scope of modeling beyond the models for which likelihood and other functions can be analytically calculated. Continuous-time models are more appropriate for modeling longitudinal social network data than discrete-time models because of the endogenous feedback processes involved in network evolution.

Probability models for the evolution of social networks are discussed which are based on the idea of actor-oriented modeling: the nodes in the network represent actors who change their relations in a process of optimizing their objective function, plus a random component representing unexplained change. The resulting model constitutes a continuous-time Markov chain, and can be simulated in a straightforward manner. Similar models can be proposed which are tie- oriented, in which ties can change randomly as a result of optimizing an objective function. In all these models, the change in the network is modeled as the stochastic result of network effects (reciprocity, transitivity, etc.) and effects of covariates. The model specification is given by the rate function, defining the rate at which ties may change; the objective function, indicating the preferred state of the network; and the gratification function, representing change tendencies for which the drive for creation of a new tie is not the opposite of the drive for deletion of this tie when it exists. These three functions may depend on endogenous and/or exogenous actor and dyad characteristics.

The parameters of this model are weights in the rate, objective, and gratification functions, which may depend on network structure and on covariates. The parameters can be estimated using a stochastic version of the method of moments, implemented by a Robbins-Monro-type algorithm. An example is given of the evolution of the friendship network in a group of university freshmen students.

Elaborations which may be discussed include alternative model specifications and alternative estimation methods. Further information about this research is at http://stat.gamma.rug.nl/snijders/siena.html.

Social mechanisms can be brought to light much better using longitudinal observations on social networks, than by using single observations. The most common type of longitudinal observations are repeated measures, or panel data. Repeated measures on social networks represent a complicated data structure. Computer simulation offers fruitful possibilities here, because it greatly expands the scope of modeling beyond the models for which likelihood and other functions can be analytically calculated. Continuous-time models are more appropriate for modeling longitudinal social network data than discrete-time models because of the endogenous feedback processes involved in network evolution.

Probability models for the evolution of social networks are discussed which are based on the idea of actor-oriented modeling: the nodes in the network represent actors who change their relations in a process of optimizing their objective function, plus a random component representing unexplained change. The resulting model constitutes a continuous-time Markov chain, and can be simulated in a straightforward manner. Similar models can be proposed which are tie- oriented, in which ties can change randomly as a result of optimizing an objective function. In all these models, the change in the network is modeled as the stochastic result of network effects (reciprocity, transitivity, etc.) and effects of covariates. The model specification is given by the rate function, defining the rate at which ties may change; the objective function, indicating the preferred state of the network; and the gratification function, representing change tendencies for which the drive for creation of a new tie is not the opposite of the drive for deletion of this tie when it exists. These three functions may depend on endogenous and/or exogenous actor and dyad characteristics.

The parameters of this model are weights in the rate, objective, and gratification functions, which may depend on network structure and on covariates. The parameters can be estimated using a stochastic version of the method of moments, implemented by a Robbins-Monro-type algorithm. An example is given of the evolution of the friendship network in a group of university freshmen students.

Elaborations which may be discussed include alternative model specifications and alternative estimation methods. Further information about this research is at http://stat.gamma.rug.nl/snijders/siena.html.