The Geometry of Continuous Latent Space Models for Network Data

TL;DR

This paper reviews continuous latent space models for network data, emphasizing how the geometry of the latent space influences network properties and discussing spectral graph theory insights for understanding and inferring this geometry.

Contribution

It provides a geometric perspective on continuous latent space models, highlighting the impact of latent space geometry on network properties and exploring spectral methods for analysis.

Findings

Latent space geometry affects network dependence structures.

Spectral graph theory helps analyze latent space properties.

Simulation illustrates the influence of geometry on network features.

Abstract

We review the class of continuous latent space (statistical) models for network data, paying particular attention to the role of the geometry of the latent space. In these models, the presence/absence of network dyadic ties are assumed to be conditionally independent given the dyads? unobserved positions in a latent space. In this way, these models provide a probabilistic framework for embedding network nodes in a continuous space equipped with a geometry that facilitates the description of dependence between random dyadic ties. Specifically, these models naturally capture homophilous tendencies and triadic clustering, among other common properties of observed networks. In addition to reviewing the literature on continuous latent space models from a geometric perspective, we highlight the important role the geometry of the latent space plays on properties of networks arising from these models via intuition and simulation. Finally, we discuss results from spectral graph theory that allow us to explore the role of the geometry of the latent space, independent of network size. We conclude with conjectures about how these results might be used to infer the appropriate latent space geometry from observed networks.