Maximum Likelihood Embedding of Logistic Random Dot Product Graphs

A latent space model for a family of random graphs assigns real-valued vectors to nodes of the graph such that edge probabilities are determined by latent positions. Latent space models provide a natural statistical framework for graph visualizing and clustering. A latent space model of particular interest is the Random Dot Product Graph (RDPG), which can be fit using an efficient spectral method; however, this method is based on a heuristic that can fail, even in simple cases. Here, we consider a closely related latent space model, the Logistic RDPG, which uses a logistic link function to map from latent positions to edge likelihoods. Over this model, we show that asymptotically exact maximum likelihood inference of latent position vectors can be achieved using an efficient spectral method. Our method involves computing top eigenvectors of a normalized adjacency matrix and scaling eigenvectors using a regression step. The novel regression scaling step is an essential part of the proposed method. In simulations, we show that our proposed method is more accurate and more robust than common practices. We also show the effectiveness of our approach over standard real networks of the karate club and political blogs.

Social Space Diffusion: Applications of a Latent Space Model to Diffusion with Uncertain Ties

Social networks represent two different facets of social life: (1) stable paths for diffusion, or the spread of something through a connected population, and (2) random draws from an underlying social space, which indicate the relative positions of the people in the network to one another. The dual nature of networks creates a challenge: if the observed network ties are a single random draw, is it realistic to expect that diffusion only follows the observed network ties? This study takes a first step toward integrating these two perspectives by introducing a social space diffusion model. In the model, network ties indicate positions in social space, and diffusion occurs proportionally to distance in social space. Practically, the simulation occurs in two parts. First, positions are estimated using a statistical model (in this example, a latent space model). Then, second, the predicted probabilities of a tie from that model—representing the distances in social space—or a series of networks drawn from those probabilities—representing routine churn in the network—are used as weights in a weighted averaging framework. Using longitudinal data from high school friendship networks, the author explores the properties of the model. The author shows that the model produces smoothed diffusion results, which predict attitudes in future waves 10 percent better than a diffusion model using the observed network and up to 5 percent better than diffusion models using alternative, non-model-based smoothing approaches.