  Many computer vision and machine learning problems are modelled as learning
tasks on graphs where graph neural networks GNNs have emerged as a dominant
tool for learning representations of graph structured data A key feature of
GNNs is their use of graph structures as input enabling them to exploit the
graphs inherent topological properties known as the topology awareness of GNNs
Despite the empirical successes of GNNs the influence of topology awareness on
generalization performance remains unexplored, particularly for node level
tasks that diverge from the assumption of data being independent and
identically distributed IID The precise definition and characterization of the
topology awareness of GNNs especially concerning different topological features
are still unclear This paper introduces a comprehensive framework to
characterize the topology awareness of GNNs across any topological feature
Using this framework we investigate the effects of topology awareness on GNN
generalization performance Contrary to the prevailing belief that enhancing the
topology awareness of GNNs is always advantageous our analysis reveals a
critical insight improving the topology awareness of GNNs may inadvertently
lead to unfair generalization across structural groups which might not be
desired in some scenarios Additionally we conduct a case study using the
intrinsic graph metric the shortest path distance on various benchmark datasets
The empirical results of this case study confirm our theoretical insights
Moreover we demonstrate the practical applicability of our framework by using
it to tackle the cold start problem in graph active learning
