论文标题
有限加权图中的Bakry-émery曲率清晰度和曲率流。 I.理论
Bakry-Émery curvature sharpness and curvature flow in finite weighted graphs. I. Theory
论文作者
论文摘要
在这两篇论文的这一序列中，我们在（混合）加权图上引入了基于Bakry-émerycolculus的（混合）加权图。通过通过加权方案进行及时的演化来描述流动。通过调整该流程以保留马尔可夫属性，其极限结果是弯曲的。我们的目的是在最普遍的情况下呈现流动，不一定是可逆的随机步行，允许懒惰，包括消失的过渡概率沿着某些边缘（“退化”边缘）。这种方法需要将所有概念（尤其是Bakry-émery曲率相关的概念）扩展到该一般情况，并导致基本拓扑（混合组合图）和加权方案（通过过渡速率给出）之间的区别。我们介绍了有关曲率锋利顶点和加权图以及这种新曲率流的某些基本特性的各种结果。本文伴随着第二篇论文，讨论了Python中的曲率流量实施。在第二篇论文中，我们提出了示例并展示了流动的进一步特性。
In this sequence of two papers, we introduce a curvature flow on (mixed) weighted graphs which is based on the Bakry-Émery calculus. The flow is described via a time-continuous evolution through the weighting schemes. By adapting this flow to preserve the Markovian property, its limits turn out to be curvature sharp. Our aim is to present the flow in the most general case of not necessarily reversible random walks allowing laziness, including vanishing transition probabilities along some edges ("degenerate" edges). This approach requires to extend all concepts (in particular, the Bakry-Émery curvature related notions) to this general case and it leads to a distinction between the underlying topology (a mixed combinatorial graph) and the weighting scheme (given by transition rates). We present various results about curvature sharp vertices and weighted graphs as well as some fundamental properties of this new curvature flow. This paper is accompanied by a second paper discussing the curvature flow implementation in Python for practical use. In this second paper we present examples and exhibit further properties of the flow.
