Describing networks geometrically through low-dimensional latent metric
spaces has helped design efficient learning algorithms, unveil network
symmetries and study dynamical network processes. However, latent space
embeddings are limited to specific classes of networks because incompatible
metric spaces generally result in information loss. Here, we study arbitrary
networks geometrically by defining a dynamic edge curvature measuring the
similarity between pairs of dynamical network processes seeded at nearby nodes.
We show that the evolution of the curvature distribution exhibits gaps at
characteristic timescales indicating bottleneck-edges that limit information
spreading. Importantly, curvature gaps are robust to large fluctuations in node
degrees, encoding communities until the phase transition of detectability,
where spectral and node-clustering methods fail. Using this insight, we derive
geometric modularity to find multiscale communities based on deviations from
constant network curvature in generative and real-world networks, significantly
outperforming most previous methods. Our work suggests using network geometry
for studying and controlling the structure of and information spreading on
networks.

本文介绍了一种通过动态的边缘曲率来描述网络几何性质的方法，展示了网络演化中的瓶颈边缘和信息传播过程，利用该方法成功地推导出了多尺度社区结构。

使用动态 Ollivier-Ricci 曲率展开网络的多尺度结构

Unfolding the multiscale structure of networks with dynamical  Ollivier-Ricci curvature

Real networks are finite metric spaces. Yet the geometry induced by shortest
path distances in a network is definitely not its only geometry. Other forms of
network geometry are the geometry of latent spaces underlying many networks,
and the effective geometry induced by dynamical processes in networks. These
three approaches to network geometry are all intimately related, and all three
of them have been found to be exceptionally efficient in discovering
fractality, scale-invariance, self-similarity, and other forms of fundamental
symmetries in networks. Network geometry is also of great utility in a variety
of practical applications, ranging from the understanding how the brain works,
to routing in the Internet. Here, we review the most important theoretical and
practical developments dealing with these approaches to network geometry in the
last two decades, and offer perspectives on future research directions and
challenges in this novel frontier in the study of complexity.

网络几何学的三种方法包括潜在空间几何、动力学过程感应几何和最短路径距离感应几何，能够有效地发现网络的分形性、尺度不变性、自相似性等基本对称性，在理论研究和实际应用中具有很大的效用。未来的研究方向和挑战也在这个新领域，可以理解大脑是如何工作的以及在互联网中进行路由选择。

网络几何

Network Geometry

Networks representing many complex systems in nature and society share some
common structural properties like heterogeneous degree distributions and strong
clustering. Recent research on network geometry has shown that those real
networks can be adequately modeled as random geometric graphs in hyperbolic
spaces. In this paper, we present a computer program to generate such graphs.
Besides real-world-like networks, the program can generate random graphs from
other well-known graph ensembles, such as the soft configuration model, random
geometric graphs on a circle, or Erd\H{o}s-R\'enyi random graphs. The
simulations show a good match between the expected values of different network
structural properties and the corresponding empirical values measured in
generated graphs, confirming the accurate behavior of the program.

本文介绍了一种计算机程序来生成具有超几何空间中随机几何图形模型的网络，并模拟其在不同网络结构特性方面的表现。