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Stochastic block models have known a flowering interest in the social network literature. They provide a tool for discovering communities and identifying clusters of individuals characterized by similar social behaviors. In this framework, full maximum likelihood estimates are not achievable due to the intractability of the likelihood function. For this reason, several approximate solutions are available in the literature. In this respect, a new and more efficient approximate method for estimating model parameters is introduced. This has a hybrid nature, in the sense that it exploits different features of existing methods. The proposal is illustrated by an intensive Monte Carlo simulation study and an application to a real-world network. [ABSTRACT FROM AUTHOR]

A stochastic block model for dynamic network data is introduced, where directed relations among a set of nodes are observed at different time occasions and the blocks are represented by a sequence of latent variables following a Markov chain. Dyads are explicitly modeled conditional on the states occupied by both nodes involved in the relation. With respect to the approaches already available in the literature, the main focus is on reciprocity. In this regard, three different parameterizations are proposed in which: (i) reciprocity is allowed to depend on the blocks of the nodes in the dyad; (ii) reciprocity is assumed to be constant across blocks; and (iii) reciprocity is ruled out. The assumption of conditional independence between dyads (referred to different pairs of nodes and time occasions) given the latent blocks is always retained. Given the complexity of the model, inference on its parameters is based on a variational approach, where a lower bound of the log-likelihood function is maximized instead of the intractable full model log-likelihood. An approximate likelihood ratio test statistic is proposed which compares the value at convergence of this lower bound under different model specifications. This allows us to formally test for both the hypothesis of no reciprocity and that of constant reciprocity with respect to the latent blocks. The proposed approach is illustrated via a simulation study based on different scenarios. The application to two benchmark datasets in the social network literature is also proposed to illustrate the effectiveness of the proposal in studying reciprocity and identifying groups of nodes having a similar social behavior. [ABSTRACT FROM AUTHOR]