Defining the geometry of networks is typically associated with embedding in
low-dimensional spaces such as manifolds. This approach has helped design
efficient learning algorithms, unveil network symmetries and study dynamical
network processes. However, the choice of embedding space is network-specific,
and incompatible spaces can result in information loss. Here, we define a
dynamic edge curvature for the study of arbitrary networks measuring the
deformation between pairs of evolving dynamical network processes on different
timescales. We show that the curvature distribution exhibits gaps at
characteristic timescales indicating bottleneck-edges that limit information
spreading. Importantly, curvature gaps robustly encode communities until the
phase transition of detectability, where spectral clustering methods fail. We
use this insight to derive geometric modularity optimisation and demonstrate it
on the European power grid and the C. elegans homeobox gene regulatory network
finding previously unidentified communities on multiple scales. Our work
suggests using network geometry for studying and controlling the structure of
and information spreading on networks.