Statistical analysis for Unlabled Graphs: Regression Model with Unlabelled Network Response on Graph Space.

Graphs are a useful mathematical representation for different phenomena in different application fields, such as chemistry, medicine, transportation, and social science. The analysis of populations of unlabelled networks is thus a promising but challenging task. In this work, unlabelled networks with Euclidean attributes are described in Graph Space, where every equivalence class represents all the networks obtained by permuting nodes. We hereby describe the geometry of Graph Space and we introduce a Generalized Geodesic Regression to model scalar-on-network relationships. Generalized Geodesic Regression is computed using the Align All and Compute Algorithm. Two case studies are described to showcase the potential of the model: the effect of the lockdown in the usage of public transport network in Copenhagen; the player passing network as a function of the match outcome during Fifa 2018 Championship.