Networks are popular for representing complex data. In particular, differentially private synthetic networks are much in demand for method and algorithm development. The network generator should be easy to implement and should come with theoretical guarantees. Here we start with complex data as input and jointly provide a network representation as well as a synthetic network generator. Using a random connection model, we devise an effective algorithmic approach for generating attributed synthetic graphs which is ε-differentially private at the vertex level, while preserving utility under an appropriate notion of distance which we develop. We provide theoretical guarantees for the accuracy of the private synthetic graphs using the fused Gromov-Wasserstein distance, which extends the Wasserstein metric to structured data. Our method draws inspiration from the PSMM method of He et al. [2023].

In this article, we derive Stein's method for approximating a spatial random graph by a generalised random geometric graph, which has vertices given by a finite Gibbs point process and edges based on a general connection function. Our main theorems provide explicit upper bounds for integral probability metrics and, at improved rates, a recently introduced Wasserstein metric for random graph distributions. The bounds are in terms of a vertex error term based on the Papangelou kernels of the vertex point processes and two edge error terms based on conditional edge probabilities. In addition to providing new tools for spatial random graphs along the way, such as a graph-based Georgii-Nguyen-Zessin formula, we also give applications of our bounds to the percolation graph of large balls in a Boolean model and to discretising a generalised random geometric graph.

We introduce the Graph TT (GTT) and Graph OSPA (GOSPA) metrics based on optimal assignment, which allow us to compare not only the edge structures but also general vertex and edge attributes of graphs of possibly different sizes. We argue that this provides an intuitive and universal way to measure the distance between finite simple attributed graphs. Our paper discusses useful equivalences and inequalities as well as the relation of the new metrics to various existing quantifications of distance between graphs. By deriving a representation of a graph as a pair of point processes, we are able to formulate and study a new type of (finite) random graph convergence and demonstrate its applicability using general point processes of vertices with independent random edges. Computational aspects of the new metrics are studied in the form of an exact and two heuristic algorithms that are derived from previous algorithms for similar tasks. As an application, we perform a statistical test based on the GOSPA metric for functional differences in olfactory neurons of Drosophila flies.