  Researches on analyzing graphs with Graph Neural Networks (GNNs) have been
receiving more and more attention because of the great expressive power of
graphs. GNNs map the adjacency matrix and node features to node representations
by message passing through edges on each convolution layer. However, the
message passed through GNNs is not always beneficial for all parts in a graph.
Specifically, as the data distribution is different over the graph, the
receptive field (the farthest nodes that a node can obtain information from)
needed to gather information is also different. Existing GNNs treat all parts
of the graph uniformly, which makes it difficult to adaptively pass the most
informative message for each unique part. To solve this problem, we propose two
regularization terms that consider message passing locally: (1) Intra-Energy
Reg and (2) Inter-Energy Reg. Through experiments and theoretical discussion,
we first show that the speed of smoothing of different parts varies enormously
and the topology of each part affects the way of smoothing. With Intra-Energy
Reg, we strengthen the message passing within each part, which is beneficial
for getting more useful information. With Inter-Energy Reg, we improve the
ability of GNNs to distinguish different nodes. With the proposed two
regularization terms, GNNs are able to filter the most useful information
adaptively, learn more robustly and gain higher expressiveness. Moreover, the
proposed LEReg can be easily applied to other GNN models with plug-and-play
characteristics. Extensive experiments on several benchmarks verify that GNNs
with LEReg outperform or match the state-of-the-art methods. The effectiveness
and efficiency are also empirically visualized with elaborate experiments.
