Intensity Dot Product Graphs

TL;DR

The paper introduces Intensity Dot Product Graphs (IDPGs), a new model extending Random Dot Product Graphs by incorporating a Poisson point process for more flexible, continuous latent structures with spectral analysis and temporal extensions.

Contribution

It develops IDPGs, linking continuous latent space with observed graphs, and provides spectral consistency results, connecting to graphon models and classical RDPGs.

Findings

Spectral consistency connects adjacency singular values to the operator spectrum.

IDPGs generalize RDPGs and graphon models with a Poisson process framework.

Temporal extensions naturally arise via partial differential equations.

Abstract

Latent-position random graph models usually treat the node set as fixed once the sample size is chosen, while graphon-based and random-measure constructions allow more randomness at the cost of weaker geometric interpretability. We introduce \emph{Intensity Dot Product Graphs} (IDPGs), which extend Random Dot Product Graphs by replacing a fixed collection of latent positions with a Poisson point process on a Euclidean latent space. This yields a model with random node populations, RDPG-style dot-product affinities, and a population-level intensity that links continuous latent structure to finite observed graphs. We define the heat map and the desire operator as continuous analogues of the probability matrix, prove a spectral consistency result connecting adjacency singular values to the operator spectrum, compare the construction with graphon and digraphon representations, and show how classical RDPGs arise in a concentrated limit. Because the model is parameterized by an evolving intensity, temporal extensions through partial differential equations arise naturally.