Soft Random Graphs in Probabilistic Metric Spaces & Inter-graph Distance

Abstract

We present a new method for learning Soft Random Geometric Graphs (SRGGs), drawn in probabilistic metric spaces, with the connection function of the graph defined as the marginal posterior probability of an edge random variable, given the correlation between the nodes connected by that edge. In fact, this inter-node correlation matrix is itself a random variable in our learning strategy, and we learn this by identifying each node as a random variable, measurements of which comprise a column of a given multivariate dataset. We undertake inference with Metropolis with a 2-block update scheme. The SRGG is shown to be generated by a non-homogeneous Poisson point process, the intensity of which is location-dependent. Given the multivariate dataset, likelihood of the inter-column correlation matrix is attained following achievement of a closed-form marginalisation over all inter-row correlation matrices. Distance between a pair of graphical models learnt given the respective datasets, offers the absolute correlation between the given datasets; such inter-graph distance computation is our ultimate objective, and is achieved using a newly introduced metric that resembles an uncertainty-normalised Hellinger distance between posterior probabilities of the two learnt SRGGs, given the respective datasets. Two sets of empirical illustrations on real data are undertaken, and application to simulated data is included to exemplify the effect of incorporating measurement noise in the learning of a graphical model.