Projective, Sparse, and Learnable Latent Position Network Models

TL;DR

This paper introduces a novel latent position network model using Poisson point processes to better represent sparse networks, ensuring consistent inference across different network sizes.

Contribution

It proposes a new sparse latent position model based on Poisson processes that is projective and compatible with various levels of network sparsity.

Findings

Model accommodates different sparsity levels

Ensures consistency of inference across network sizes

Provides conditions for consistent latent position estimation

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.