  In the semi-supervised setting where labeled data are largely limited, it
remains to be a big challenge for message passing based graph neural networks
(GNNs) to learn feature representations for the nodes with the same class label
that is distributed discontinuously over the graph. To resolve the
discontinuous information transmission problem, we propose a control principle
to supervise representation learning by leveraging the prototypes (i.e., class
centers) of labeled data. Treating graph learning as a discrete dynamic process
and the prototypes of labeled data as "desired" class representations, we
borrow the pinning control idea from automatic control theory to design
learning feedback controllers for the feature learning process, attempting to
minimize the differences between message passing derived features and the class
prototypes in every round so as to generate class-relevant features.
Specifically, we equip every node with an optimal controller in each round
through learning the matching relationships between nodes and the class
prototypes, enabling nodes to rectify the aggregated information from
incompatible neighbors in a graph with strong heterophily. Our experiments
demonstrate that the proposed PCGCN model achieves better performances than
deep GNNs and other competitive heterophily-oriented methods, especially when
the graph has very few labels and strong heterophily.
