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.
翻译:根据基于Ricci 曲线的多重取样方法和结果,我们建议对网络采用类似的方法,为此,我们呼吁三种离散曲线,即Forman、Forman和Haantjes-Ricci曲线图,用于边缘和节点取样。我们介绍了在真实生活网络以及图像处理过程中产生的平方格实验的结果。此外,我们认为Ricci流适合Ricci流,我们使用它们来探测网络的骨干。我们还开发了与Forman-Ricci曲线相关的嵌入内核,并使用它们探测网络的粗糙结构,以及利用SVM应用的网络可视化。还研究了原始流体的微缩曲线与由 Riccci 曲线驱动的离散化之间的关系。