  Graph-structured data are widespread in real-world applications, such as
social networks, recommender systems, knowledge graphs, chemical molecules etc.
Despite the success of Euclidean space for graph-related learning tasks, its
ability to model complex patterns is essentially constrained by its
polynomially growing capacity. Recently, hyperbolic spaces have emerged as a
promising alternative for processing graph data with tree-like structure or
power-law distribution, owing to the exponential growth property. Different
from Euclidean space, which expands polynomially, the hyperbolic space grows
exponentially which makes it gains natural advantages in abstracting tree-like
or scale-free graphs with hierarchical organizations.
  In this tutorial, we aim to give an introduction to this emerging field of
graph representation learning with the express purpose of being accessible to
all audiences. We first give a brief introduction to graph representation
learning as well as some preliminary Riemannian and hyperbolic geometry. We
then comprehensively revisit the hyperbolic embedding techniques, including
hyperbolic shallow models and hyperbolic neural networks. In addition, we
introduce the technical details of the current hyperbolic graph neural networks
by unifying them into a general framework and summarizing the variants of each
component. Moreover, we further introduce a series of related applications in a
variety of fields. In the last part, we discuss several advanced topics about
hyperbolic geometry for graph representation learning, which potentially serve
as guidelines for further flourishing the non-Euclidean graph learning
community.
