Random geometric graphs in high dimension

Many machine learning algorithms used for dimensional reduction and manifold learning leverage on the computation of the nearest neighbors to each point of a data set to perform their tasks. These proximity relations define a so-called geometric graph, where two nodes are linked if they are sufficiently close to each other. Random geometric graphs, where the positions of nodes are randomly generated in a subset of R-d , offer a null model to study typical properties of data sets and of machine learning algorithms. Up to now, most of the literature focused on the characterization of low-dimensional random geometric graphs whereas typical data sets of interest in machine learning live in high-dimensional spaces (d >> 10(2)). In this work, we consider the infinite dimensions limit of hard and soft random geometric graphs and we show how to compute the average number of subgraphs of given finite size k , e.g., the average number of k cliques. This analysis highlights that local observables display different behaviors depending on the chosen ensemble: soft random geometric graphs with continuous activation functions converge to the naive infinite-dimensional limit provided by Erdos-Renyi graphs, whereas hard random geometric graphs can show systematic deviations from it. We present numerical evidence that our analytical results, exact in infinite dimensions, provide a good approximation also for dimension d greater than or similar to 10.