  The structure of real-world networks is usually difficult to characterize
owing to the variation of topological scales, the nondyadic complex
interactions, and the fluctuations in the network. We aim to address these
problems by introducing a general framework using a method based on topological
data analysis. By considering the diffusion process at a single specified
timescale in a network, we map the network nodes to a finite set of points that
contains the topological information of the network at a single scale.
Subsequently, we study the shape of these point sets over variable timescales
that provide scale-variant topological information, to understand the varying
topological scales and the complex interactions in the network. We conduct
experiments on synthetic and real-world data to demonstrate the effectiveness
of the proposed framework in identifying network models, classifying real-world
networks, and detecting transition points in time-evolving networks. Overall,
our study presents a unified analysis that can be applied to more complex
network structures, as in the case of multilayer and multiplex networks.
