  The emergence of Graph Convolutional Network (GCN) has greatly boosted the
progress of graph learning. However, two disturbing factors, noise and
redundancy in graph data, and lack of interpretation for prediction results,
impede further development of GCN. One solution is to recognize a predictive
yet compressed subgraph to get rid of the noise and redundancy and obtain the
interpretable part of the graph. This setting of subgraph is similar to the
information bottleneck (IB) principle, which is less studied on
graph-structured data and GCN. Inspired by the IB principle, we propose a novel
subgraph information bottleneck (SIB) framework to recognize such subgraphs,
named IB-subgraph. However, the intractability of mutual information and the
discrete nature of graph data makes the objective of SIB notoriously hard to
optimize. To this end, we introduce a bilevel optimization scheme coupled with
a mutual information estimator for irregular graphs. Moreover, we propose a
continuous relaxation for subgraph selection with a connectivity loss for
stabilization. We further theoretically prove the error bound of our estimation
scheme for mutual information and the noise-invariant nature of IB-subgraph.
Extensive experiments on graph learning and large-scale point cloud tasks
demonstrate the superior property of IB-subgraph.
