Barycentric subspace analysis of network-valued data

Certain data are naturally modeled by networks or weighted graphs, be they biological networks or mobility networks. When there is no canonical labeling of the nodes across the dataset, we talk about unlabeled networks. In this paper, we focus on the question of exploratory analysis of this type of data. More specifically, we address the issue of interpreting the feature subspace constructed by dimensionality reduction methods. Most existing methods for network-valued data are derived from principal component analysis (PCA) and therefore rely on subspaces generated by a set of vectors, which we identify as a major limitation in terms of interpretability. Instead, we propose to implement the method called barycentric subspace analysis (BSA), which relies on subspaces generated by a set of points, which we choose, in practice, to be samples. In order to provide a computationally feasible framework for BSA, we introduce a novel embedding for unlabeled networks where we replace their usual representation by equivalence classes of isomorphic networks with that by equivalence classes of cospectral networks. In a simulated study, we demonstrate the improved interpretability of BSA compared to tangent PCA. We then illustrate through two real-world datasets how BSA can be used both to visualize known patterns and to discover new ones in network-valued distributions.