  Most real-world networks are embedded in latent geometries. If a node in a
network is found in the vicinity of another node in the latent geometry, the
two nodes have a disproportionately high probability of being connected by a
link. The latent geometry of a complex network is a central topic of research
in network science, which has an expansive range of practical applications such
as efficient navigation, missing link prediction, and brain mapping. Despite
the important role of topology in the structures and functions of complex
systems, little to no study has been conducted to develop a method to estimate
the general unknown latent geometry of complex networks. Topological data
analysis, which has attracted extensive attention in the research community
owing to its convincing performance, can be directly implemented into complex
networks; however, even a small fraction (0.1%) of long-range links can
completely erase the topological signature of the latent geometry. Inspired by
the fact that long-range links in a network have disproportionately high loads,
we develop a set of methods that can analyze the latent geometry of a complex
network: the modified persistent homology diagram and the map of the latent
geometry. These methods successfully reveal the topological properties of the
synthetic and empirical networks used to validate the proposed methods.
