  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.
