Settings in Social Networks: Representation by Latent Transitive Structures

A class of statistical models is proposed which aims at estimating latent transitive structures in networks. Such structures can be used e.g. to identify close-knit subsets of actors in large social networks (with up to several hundred actors). The measurement model can be sketched as follows. Given one observed network, it is assumed that this observed network was generated by latent transitive structures. The latent transitive structures are expressed by ultrametrics. It is assumed that the probability of observing an edge increases with decreasing ultrametric distance. To make statistical inference, the Maximum Likelihood principle as well as Bayesian methods are applied. The Maximum Likelihood principle is implemented using some non-greedy optimization algorithm, while Bayesian inference is implemented using a hybrid Markov chain Monte Carlo algorithm. Applications will be presented.