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.

翻译：隐位置随机图模型通常在样本量确定后将节点集合视为固定，而基于图论和随机测度的构造允许更多随机性，但会削弱几何可解释性。本研究引入**强度点积图**（IDPGs），它将随机点积图进行扩展，用欧几里得隐空间上的泊松点过程替代固定的隐位置集合。该模型具有随机节点规模、RDPG风格点积亲和性，以及连接连续隐结构与有限观测图的总体强度。我们定义热力图和渴望算子作为概率矩阵的连续类比，证明连接邻接奇异值与算子谱的谱一致性结果，比较该构造与图论及有向图表示的差异，并展示经典RDPG如何在集中极限中衍生。由于模型通过演化强度参数化，基于偏微分方程的时间扩展自然成立。