Abstract
Online social networks like Twitter and Facebook are among the most popular sites on the Internet. Most online social networks involve some specific features, including reciprocity, transitivity and degree heterogeneity. Such networks are so called scale-free networks and have drawn lots of attention in research. The aim of this paper is to develop a novel methodology for directed network embedding within the latent space model (LSM) framework. It is known, the link probability between two individuals may increase as the features of each become similar, which is referred to as homophily attributes. To this end, penalized pair-specific attributes, acting as a distance measure, are introduced to provide with more powerful interpretation and improve link prediction accuracy, named penalized homophily latent space models (PHLSM). The proposed models also involve in-degree heterogeneity of directed scale-free networks by embedding with the popularity scales. We also introduce LASSO-based PHLSM to produce an accurate and sparse model for high-dimensional covariates. We make Bayesian inference using MCMC algorithms. The finite sample performance of the proposed models is evaluated by three benchmark simulation datasets and two real data examples. Our methods are competitive and interpretable, they outperform existing approaches for fitting directed networks.