Populations of Unlabeled Networks: Graph Space Geometry and Geodesic Principal Components

14/2020

Populations of Unlabeled Networks: Graph Space Geometry and Geodesic Principal Components

Thursday 20th February 2020

Calissano, A.; Feragen, A; Vantini, S.

Statistical analysis for populations of networks is widely applicable, but challenging as networks have strongly non-Euclidean behavior. Graph Space is an exhaustive framework for studying populations of networks which are weighted or unweighted, uni- or multi-layer, directed or undirected, labelled or unlabelled. Viewing Graph Space as the quotient of a Euclidean space with respect to a finite group action, we show that it is not a manifold, and that its curvature is unbounded from above. Motivated by these geometric properties, we define geodesic principal components, and we introduce the Align All and Compute algorithm, which allows the computation of statistics on Graph Space. The statistics and algorithm are empirically validated on one simulated study and two real datasets, showcasing the framework’s potential utility. The whole framework is implemented in a publically available GraphSpace Python package.