Projective, Sparse, and Learnable Latent Position Network Models
Neil A. Spencer, Cosma Rohilla Shalizi
TL;DR
This paper introduces a new sparse, projective latent position network model by embedding node latent coordinates in an augmented space with an auxiliary dimension governed by a growing observation window. By leveraging a Poisson point process in the augmented space, the authors achieve tunable sparsity while preserving projectivity, and develop learnability results for latent positions, distances, and link probabilities under regularity conditions. They specialize to a tractable rectangular LPM in Euclidean space, proving projectivity and $n^{2-p}$ sparsity for growth parameter $p$, and provide explicit learnability conditions that depend on the link function $K$ and the growth scheme. The work also situates the framework relative to sparse graphon and graphex models, highlighting preserved transitivity and discuss limitations, with implications for robust superpopulation inference in sparse networks.
Abstract
When modeling network data using a latent position model, it is typical to assume that the nodes' positions are independently and identically distributed. However, this assumption implies the average node degree grows linearly with the number of nodes, which is inappropriate when the graph is thought to be sparse. We propose an alternative assumption -- that the latent positions are generated according to a Poisson point process -- and show that it is compatible with various levels of sparsity. Unlike other notions of sparse latent position models in the literature, our framework also defines a projective sequence of probability models, thus ensuring consistency of statistical inference across networks of different sizes. We establish conditions for consistent estimation of the latent positions, and compare our results to existing frameworks for modeling sparse networks.