Geometric sampling of networks

Motivated by the methods and results of manifold sampling based on Ricci curvature, we propose a similar approach for networks. To this end, we make an appeal to three types of discrete curvature, namely the graph Forman-, full Forman- and Haantjes–Ricci curvatures for edge-based and node-based sampling. The relation between the Ricci curvature of the original manifold and that of a Ricci curvature driven-discretization is studied, and we show that there is a strong connection between the Forman–Ricci curvatures of the resulting network and the Ricci curvature of the given smooth manifold. We also present the results of experiments on real-life networks, as well as for square grids arising in image processing. Moreover, we consider fitting Ricci flows, and we employ them for the detection of networks’ backbone.