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 appeal to three types of discrete curvature, namely the graph Forman-, full Forman-and Haantjes-Ricci curvatures for edgebased and node-based sampling. We 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. We also develop embedding kernels related to the Forman-Ricci curvatures and employ them for the detection of the coarse structure of networks, as well as for network visualization with applications to SVM. The relation between the Ricci curvature of the original manifold and that of a Ricci curvature driven discretization is also studied. J. Jost and E. Saucan were partly supported by the German-Israeli Foundation Grant I-1514-304.6/2019.

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.

We develop embedding kernels based on the Forman–Ricci curvature and intertwined Bochner–Laplacian and employ them for the detection of the coarse structure of networks, as well as for network visualization with applications to support-vector machines (SVMs).

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