Intensity Dot Product Graphs
This work provides a novel graph model for researchers in network analysis and machine learning, offering improved geometric interpretability and flexibility over existing approaches, though it is incremental in extending prior random dot product graph frameworks.
The authors tackled the problem of modeling random graphs with latent positions by introducing Intensity Dot Product Graphs (IDPGs), which extend Random Dot Product Graphs to include random node populations via a Poisson point process, resulting in a model that links continuous latent structure to observed graphs and allows for natural temporal extensions.
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.