Random Graphs by Product Random Measures

A natural class of models for representation of both point systems and graphs is the random counting measure. We develop results for random measures, products of themselves, and their images under transformations, including mean measures. The collection of product random measures, transformations, and non-negative test functions forms a general representation of the collection of non-negative weighted random graphs, directed or undirected, labeled or unlabeled, where (i) the composition of the test function and transformation is a non-negative edge weight function, (ii) the mean measures encode edge count/weight and vertex degree count/weight, and (iii) the mean edge weight encodes a generalized spectral representation. We develop a number of properties of these random graphs, and we give simple examples of some of their possible applications.