Variational Bayesian inference for the Latent Position Cluster Model for network data

Many recent approaches to modeling social networks have focussed on embedding the actors in a latent “social space”. Links are more likely for actors that are close in social space than for actors that are distant in social space. In particular, the Latent Position Cluster Model (LPCM) [1] allows for explicit modelling of the clustering that is exhibited in many network datasets. However, inference for the LPCM model via MCMC is cumbersome and scaling of this model to large or even medium size networks with many interacting nodes is a challenge. Variational Bayesian methods offer one solution to this problem. An approximate, closed form posterior is formed, with unknown variational parameters. These parameters are tuned to minimize the Kullback-Leibler divergence between the approximate variational posterior and the true posterior, which known only up to proportionality. The variational Bayesian approach is shown to give a computationally efficient way of fitting the LPCM. The approach is demonstrated on a number of data sets and it is shown to give a good fit. 1 The Latent Position Cluster Model Handcock et al. [1] developed the Latent Position Cluster Model (LPCM) for social network data. The model involves locating each actor in a latent social space such that actors who are close in social space have a higher probability to form links than those distant in the social space. This model extended the Latent Space Model (LSM) [2] by incorporating a Gaussian mixture model structure for the latent positions of actors in social space, to accommodate the clustering of nodes in the network. Therefore clusters are included explicitly in the model rather that found by post-hoc analysis of the network model. A strength of the latent social space model is that it automatically represents link transitivity. We develop a variational Bayesian inference procedure for approximating the posterior distribution of the parameters in the LPCM. This approach provides computational tools to facilitate the application of the LPCM to larger networks than is currently possible using the existing MCMC methodology for model fitting. In the LPCM, a binary interactions data matrix Y is modelled using logistic regression in which the probability of a link between two nodes depends on the distance between the nodes in the latent space: log-odds(yi,j = 1|zi, zj , β) = β − |zi − zj | (1) where β is an intercept parameter and |zi−zj | is the Euclidean distance between the latent positions of nodes i and j. In addition, the links are assumed to be independent conditional on the latent positions of the actors in the latent space.