Recent advances in emergent geometry and discretized approaches to quantum gravity have relied upon the notion of a discrete measure of graph curvature. We focus on the two main measures that have been studied, the so-called Ollivier-Ricci and Forman-Ricci curvatures. These two approaches have a very different origin, and both have advantages and disadvantages. In this work we study the relationship between the two measures for a class of graphs that are important in quantum gravity applications. We discover that under a specific set of circumstances they are equivalent, opening up the possibility of replacing the more fundamental Ollivier-Ricci curvature by the computationally more accessible Forman-Ricci curvature in certain applications to models of emergent spacetime and quantum gravity.

涌现几何和量子引力的离散化方法的最新进展依赖于图曲率的离散度量的概念。我们关注已研究的两个主要度量，即所谓的奥利维尔-里奇曲率和福尔曼-里奇曲率。这两种方法有着截然不同的起源，都各有优缺点。在这项工作中，我们研究了一类在量子引力应用中很重要的图的两种度量之间的关系。我们发现在一组特定的情况下它们是等效的，这开辟了在某些应用中将更基本的奥利维尔-里奇曲率替换为更容易计算的 Forman-Ricci 曲率的可能性，以用于涌现时空和量子引力模型。