A probabilistic view of latent space graphs and phase transitions

We study random graphs with latent geometric structure, where the probability of each edge depends on the un-derlying random positions corresponding to the two endpoints. We consider the setting where this conditional probability is a general monotone increasing function of the inner product of two vectors; such a function can nat-urally be viewed as the cumulative distribution function of some independent random variable. A one-parameter family of random graphs, characterized by the variance of this random variable, that smoothly interpolates between a random dot product graph and an Erd6s-Renyi random graph, is investigated. Focusing on the dense regime, we prove phase transitions of detecting geometry in these graphs, in terms of the dimension of the underlying geomet-ric space and the variance parameter: When the dimension is high or the variance is large, the graph is similar to an Erd6s-Renyi graph with the same edge density; in other parameter regimes, there is a computationally efficient signed triangle statistic that can distinguish them. The proofs make use of information-theoretic inequalities and concentration of measure phenomena.