  Graph neural network (GNN) link prediction is increasingly deployed in
citation, collaboration, and online social networks to recommend academic
literature, collaborators, and friends. While prior research has investigated
the dyadic fairness of GNN link prediction, the within-group (e.g., queer
women) fairness and "rich get richer" dynamics of link prediction remain
underexplored. However, these aspects have significant consequences for degree
and power imbalances in networks. In this paper, we shed light on how degree
bias in networks affects Graph Convolutional Network (GCN) link prediction. In
particular, we theoretically uncover that GCNs with a symmetric normalized
graph filter have a within-group preferential attachment bias. We validate our
theoretical analysis on real-world citation, collaboration, and online social
networks. We further bridge GCN's preferential attachment bias with unfairness
in link prediction and propose a new within-group fairness metric. This metric
quantifies disparities in link prediction scores within social groups, towards
combating the amplification of degree and power disparities. Finally, we
propose a simple training-time strategy to alleviate within-group unfairness,
and we show that it is effective on citation, social, and credit networks.
