Clustering means geometry in networks

Latent-space network models have been used successfully in many applications in network science and other disciplines, yet the probability distributions in the random graph ensembles that these models define tend to be highly intractable. Therefore it is usually impossible to tell if a given real network is a typical element in a graph ensemble defined by a given model, so that the model can be used for reliable predictions. It is thus desirable to identify structural properties of networks such that random graphs having these properties are guaranteed to be latent-geometric, that is, to be typical elements in a latent-space ensemble. Here we show that random graphs in which expected degrees and clustering of every node are fixed to the same values, are equivalent to random geometric graphs on the real line if clustering is sufficiently strong. Large numbers of triangles, homogeneously distributed across all nodes as in real networks, are thus a signature of their latent geometry. The methods we use to prove this are quite general and applicable to other network ensembles, geometric or not, and to certain problems in quantum gravity.