  Hierarchy is an important and commonly observed topological property in
real-world graphs that indicate the relationships between supervisors and
subordinates or the organizational behavior of human groups. As hierarchy is
introduced as a new inductive bias into the Graph Neural Networks (GNNs) in
various tasks, it implies latent topological relations for attackers to improve
their inference attack performance, leading to serious privacy leakage issues.
In addition, existing privacy-preserving frameworks suffer from reduced
protection ability in hierarchical propagation due to the deficiency of
adaptive upper-bound estimation of the hierarchical perturbation boundary. It
is of great urgency to effectively leverage the hierarchical property of data
while satisfying privacy guarantees. To solve the problem, we propose the
Poincar\'e Differential Privacy framework, named PoinDP, to protect the
hierarchy-aware graph embedding based on hyperbolic geometry. Specifically,
PoinDP first learns the hierarchy weights for each entity based on the
Poincar\'e model in hyperbolic space. Then, the Personalized Hierarchy-aware
Sensitivity is designed to measure the sensitivity of the hierarchical
structure and adaptively allocate the privacy protection strength. Besides, the
Hyperbolic Gaussian Mechanism (HGM) is proposed to extend the Gaussian
mechanism in Euclidean space to hyperbolic space to realize random
perturbations that satisfy differential privacy under the hyperbolic space
metric. Extensive experiment results on five real-world datasets demonstrate
the proposed PoinDP's advantages of effective privacy protection while
maintaining good performance on the node classification task.
