Introduction to Statistical Models for longitudinal network data Stochastic actor-based models Kayo Fujimoto, Ph.D.

Stochastic actor-based model (Snijders 2001, 2005) Actor-oriented modeling Actor-oriented modeling –Methodological individualism Modeled as a consequence of actors: Modeled as a consequence of actors: –Making new choices –Withdrawing existing choice –Functions that actors try to maximize Continuous-time Markov chain models Continuous-time Markov chain models –Simulation models

Dependent variable Changing relation network Changing relation network –Number of changed ties between consecutive observations

Independent Variables Change in the network (DV) is modeled as the stochastic result of network effects (such as reciprocity, transitivity, etc.) and covariates Change in the network (DV) is modeled as the stochastic result of network effects (such as reciprocity, transitivity, etc.) and covariates

Model assumptions Full knowledge of the present network Full knowledge of the present network All actors control their outgoing relations All actors control their outgoing relations –A specific actor i has the opportunity to change their relations one at a time at stochastic moment t at a rate ρ m Model specification: changes of single relations Model specification: changes of single relations

Three types of effects Rate function effects Rate function effects –Models the speed by which the DV changes Objective function effects Objective function effects –Models the actors ’ satisfaction with their local network configuration Endowment function effects Endowment function effects –Model the loss of satisfaction incurred when existing network ties are dissolved

Objective function effect Determines probabilistically the tie changes made by the actors Determines probabilistically the tie changes made by the actors Defined as a function of the network Defined as a function of the network –regarded from the perspective of the focus actor Depends on parameters Depends on parameters –estimated from the data

Objective function Network evaluation function for actor i Network evaluation function for actor i –Degree of satisfaction for each actor i in relation x

Structural effects (examples) Outdegree effect (density effect) Outdegree effect (density effect) Reciprocity effect Reciprocity effect Triad effect (transitivity, cycle, balance etc.) Triad effect (transitivity, cycle, balance etc.)

Examples – transitive triplets effect

Covariate effects (V) Covariate-ego effect (sender effect) Covariate-ego effect (sender effect) –Whether actors with higher V values tend to nominate more friends and hence have a higher outdegree Covariate-alter effect (receiver effect) Covariate-alter effect (receiver effect) –Whether actors with higher V values tend to be nominated by more others and hence have higher indegrees Covariate-similarity effect (homophily) Covariate-similarity effect (homophily) –Whether ties tend to occur more often between actors with similar values o n V

Covariate effects (examples) Covariate-ego effect (covariate-related activity) Covariate-ego effect (covariate-related activity) Covariate-alter effect (covariate-related popularity) Covariate-alter effect (covariate-related popularity) Same covariate effect (homophily) Same covariate effect (homophily)

Objective function Actor i chooses alter j that maximize the value of her objective function fi(x) Actor i chooses alter j that maximize the value of her objective function fi(x) Plus random element (Gumbel dist ’ n) Plus random element (Gumbel dist ’ n) –The part of the actor ’ s preference that is not represented by the systematic component of fi(x)

Model Parameters Estimated from observed data Estimated from observed data Stochastic simulation models Stochastic simulation models –MCMC algorithm –Approximate the solution of the Method of Moment

Estimation in SIENA Choose statistics Choose statistics Obtain parameters such that the expected values of the statistics are equal to the observed values Obtain parameters such that the expected values of the statistics are equal to the observed values –Expected values are approximated as the averages over a lot of simulated network –Observed values are calculated from the dataset (target values)

Estimation in SIENA Iterative stochastic simulation algorithm Iterative stochastic simulation algorithm In phase 1: the sensitivity of the statistics to the parameters is determined In phase 1: the sensitivity of the statistics to the parameters is determined In phase 2: provisional parameter values are updated In phase 2: provisional parameter values are updated –Simulate a network based on provisional parameter values –Compute the deviations between these simulated statistics and target values –Update parameter values In phase 3: the final results of phase 2 is used and checked if the average statistic of many simulated networks are close to the targeted values In phase 3: the final results of phase 2 is used and checked if the average statistic of many simulated networks are close to the targeted values –t statistics for deviations from targets

Longitudinal network dynamic models Actor-oriented models of Snijders and colleagues Actor-oriented models of Snijders and colleagues –Assumption: network change driven by actors seeking to optimize particular structural positions Longitudinal versions of ERGM (tie- based version of the model) Longitudinal versions of ERGM (tie- based version of the model) –Assumption: network change driven by change in tie variables (particular social neighborhood of other ties)

References Snijders, T.A.B.(2001). The statistical evaluation of social network dynamics, Sociological Methodology Snijders, T.A.B.(2001). The statistical evaluation of social network dynamics, Sociological Methodology Snijders, T.A.B.(2005). Models for longitudinal network data, chapter 11 in Carrington, P., Scott, J, Wasserman S (eds), models and methods in social network analysis. New York: Cambridge University Press. Snijders, T.A.B.(2005). Models for longitudinal network data, chapter 11 in Carrington, P., Scott, J, Wasserman S (eds), models and methods in social network analysis. New York: Cambridge University Press.