  Complex networks, which are the abstractions of many real-world systems,
present a persistent challenge across disciplines for people to decipher their
underlying information. Recently, hyperbolic geometry of latent spaces has
gained traction in network analysis, due to its ability to preserve certain
local intrinsic properties of the nodes. In this study, we explore the problem
from a much broader perspective: understanding the impact of nodes' global
topological structures on latent space placements. Our investigations reveal a
direct correlation between the topological structure of nodes and their
positioning within the latent space. Building on this deep and strong
connection between node distance and network topology, we propose a novel
embedding framework called Topology-encoded Latent Hyperbolic Geometry (TopoLa)
for analyzing complex networks. With the encoded topological information in the
latent space, TopoLa is capable of enhancing both conventional and low-rank
networks, using the singular value gap to clarify the mathematical principles
behind this enhancement. Meanwhile, we show that the equipped TopoLa distance
can also help augment pivotal deep learning models encompassing knowledge
distillation and contrastive learning.
