Intensity Dot Product Graphs

• URL: http://arxiv.org/abs/2604.07810v1
• Date: Thu, 09 Apr 2026 05:08:23 GMT
• Title: Intensity Dot Product Graphs
• Abstract summary: We introduce emphIntensity Dot Product Graphs (IDPGs)<n>This yields a model with random node populations, a PoissonG-style dot-product affinities, and a population-level intensity that links continuous latent structure to observed graphs.<n>Because the model is parameterized by an evolving intensity, temporal extensions through partial differential equations arise naturally.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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