We investigate a large class of geometric random graphs defined on a Poisson point process in Euclidean space , where each vertex carries an independent random mark. On this vertex set edges are established at random, such that the class is only determined by upper and lower bounds on the connection probabilities between finitely many pairs of vertices, which depend crucially on the marks and the spatial distance of each pair of vertices. This class includes different geometric random graphs emerging from real-world network models, such as a version of spatial preferential attachment (where marks can be understood as birth times), and continuum percolation, such as the soft Boolean model, as well as a whole range of further graph models with scale-free degree distribution and edges between distant vertices.

For this class of geometric random graphs we study the occurence of short paths leading to ultrasmallness of the graphs, i.e. that the graph distance of a pair of distant vertices grows at most of doubly logarithmic order in the spatial distance of the pair. We give a sharp criterion for the absence of ultrasmallness of the graphs and in the ultrasmall regime establish a limit theorem for the chemical distance of two very distant vertices. Unlike in non-spatial scale-free network models and spatially embedded random graphs such as scale-free percolation the boundary of the ultrasmall regime and the limit theorem depend not only on the power-law exponent of the degree distribution but also on the rate of decay of the probability of an edge connecting two vertices with typical marks in terms of their Euclidean distance.

Furthermore, we study the effect of the short paths in the ultrasmall regime on the survival of the contact process on geometric random graphs in this class. We show that the non-extinction probability is positive for any positive choice of the infection rate and give precise asymptotics for it when the infection rate decays to zero. On finite spatial restrictions of the graphs we show that the extinction time is of exponential order of the size of the graphs.