Probabilistic Reconstruction of Spatio-Temporal Processes Over Multi-Relational Graphs

Abstract

Given nodal observations that can be limited due to sampling costs or privacy concerns, several network-science-related applications entail reconstruction of values on all network nodes by leveraging topology information. Such a semi-supervised learning (SSL) task has been tackled mainly for graphs capturing a single class of inter-dependencies (or relations) among nodal variables. Faced with multi-relational graphs (MRGs), which emerge in various real-world networks, the present work introduces a principled framework to extrapolate spatio-temporal nodal processes that could be stationary or nonstationary. Broadening the scope of graph kernel-based approaches to MRGs, stationary graph processes are modeled first using a Gaussian mixture (GM) prior, where the covariance matrix of each Gaussian component describes one of the relations in the MRG. To further cope with nonstationary nodal processes, a first-order topology-dependent Gaussian transition prior is considered per relation, what gives rise to a GM transition density that accounts for all relations. In both cases, adapting the expectation-maximization (EM) algorithm yields two novel graph-adaptive solvers that not only reconstructs nodal features over unobserved nodes, but also quantifies the contribution of each relation. To enrich expressiveness of these novel EM-based approaches, multiple kernels per relation are also explored. Experiments with real data showcase the merits of the proposed methods relative to the existing alternatives.