Learning random geometric graphs, and their applications

Abstract

In the Bayesian world, referring to a variable as “random” means that this vari-able – a scalar, tensor, a network, etc. – takes a value with a probability, such that it attains a values that live in the support of this probabilistic distribution. lies in a given interval, with uncertainties. We want to make inference on the unknown parameters of a model, using the modelled probability distribution of each such unknown, given the data. In my thesis, I discuss the motivation for undertaking inference via sampling unknowns given the data, and introduce a learning of real-isations of a random graph variable. This random graph is a Random Geometric Graph (RGG) that is drawn in a probabilistic metric space. Further learning of this graph variable has also been explored in the thesis, to identify the optimal cut-off value that is imposed on the posterior probability of any edge given the (multivari-ate) dataset at hand. Such an optimal cut-off then corresponds to graph realisations that produces the most robust - to changes in the cut-off values - graph. Theoretical illustrations of the learning of this graph and the applications of such graph learning are presented using real-world data, towards: (1) scoring of severity of a disease, by computing the distance between the posterior of the learnt random graph variables, given the time series data on the physiological parameters of two patients as they suffer from the disease; (2) a method for the learning of the individualised recovery trajectory of patients who are enrolled on a physical rehabilitation programme, with the aim of regaining their lost mobility; (3) a new way of recognising critical residues of an example protein, using static and dynamic nodal degree distributions of ran-dom graphs learnt using molecular dynamical simulations of the protein. Based on my PhD research, future work is discussed.