Characterizing Distances of Networks on the Tensor Manifold

At the core of understanding dynamical systems is the ability to maintain and control the systems behavior that includes notions of robustness, heterogeneity, and/or regime-shift detection. Recently, to explore such functional properties, a convenient representation has been to model such dynamical systems as a weighted graph consisting of a finite, but very large number of interacting agents. This said, there exists very limited relevant statistical theory that is able cope with real-life data, i.e., how does perform analysis and/or statistics over a "family" of networks as opposed to a specific network or network-to-network variation. Here, we are interested in the analysis of network families whereby each network represents a "point" on an underlying statistical manifold. To do so, we explore the Riemannian structure of the tensor manifold developed by Pennec previously applied to Diffusion Tensor Imaging (DTI) towards the problem of network analysis. In particular, while this note focuses on Pennec definition of "geodesics" amongst a family of networks, we show how it lays the foundation for future work for developing measures of network robustness for regime-shift detection. We conclude with experiments highlighting the proposed distance on synthetic networks and an application towards biological (stem-cell) systems.