Geometric Sampling of Networks

TL;DR

This paper introduces a geometric sampling approach for networks using discrete Ricci curvatures, with applications in backbone detection, network visualization, and understanding network structure.

Contribution

It adapts manifold sampling techniques to networks by employing discrete Ricci curvatures and develops new embedding kernels for network analysis.

Findings

Effective backbone detection in real networks

Successful network visualization using curvature-based kernels

Insights into Ricci curvature relations between manifolds and discretizations

Abstract

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 edge-based 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.