Intensity Dot Product Graphs
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866908949204172800 |
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| author | Riva, Giulio Valentino Dalla Riva, Matteo Dalla |
| author_facet | Riva, Giulio Valentino Dalla Riva, Matteo Dalla |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_07810 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Intensity Dot Product Graphs Riva, Giulio Valentino Dalla Riva, Matteo Dalla Machine Learning Probability Methodology 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. |
| title | Intensity Dot Product Graphs |
| topic | Machine Learning Probability Methodology |
| url | https://arxiv.org/abs/2604.07810 |