Properties of Latent Variable Network Models

We derive properties of Latent Variable Models for networks, a broad class ofmodels that includes the widely-used Latent Position Models. These include theaverage degree distribution, clustering coefficient, average path length and degreecorrelations. We introduce the Gaussian Latent Position Model, and derive analyticexpressions and asymptotic approximations for its network properties. Wepay particular attention to one special case, the Gaussian Latent Position Modelswith Random Effects, and show that it can represent the heavy-tailed degree distributions,positive asymptotic clustering coefficients and small-world behaviours thatare often observed in social networks. Several real and simulated examples illustratethe ability of the models to capture important features of observed networks.