Title: Diagonal Graph Convolutional Networks with Adaptive Neighborhood Aggregation. Abstract: Graph convolutional networks (GCNs) and their variants have generalized deep learning methods into non-Euclidean graph data, bringing a substantial improvement on many graph mining tasks. In this paper, we revisit the mathematical foundation of GCNs and study how to extend their representation capacity. We discover that their performance can be improved with an adaptive neighborhood aggregation step. The core idea is to adaptively scale the output signal for each node and automatically train a suitable nonlinear encoder for the input signal. In this work, we present a new method named Diagonal Graph Convolutional Networks (DiagGCN) based on this idea. Importantly, one of the adaptive aggregation techniques—the permutations of diagonal matrices—used in DiagGCN offers a flexible framework to design GCNs and in fact, some of the most expressive GCNs, e.g., the graph attention network, can be reformulated as a particular instance of our model. Standard experiments on open graph benchmarks show that our proposed framework can consistently improve the graph classification accuracy when compared to state-of-the-art baselines.