Characterizing Distances of Networks on the Tensor Manifold

Abstract

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 regimeshift 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 simple 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. From this, we explore the Riemannian structure of the statistical (tensor) manifold in order to define notions of "geodesics" or shortest distance amongst such points as well as a statistical framework for time-varying complex networks for which we can utilize in higher order classification tasks.