Finding the proper depth d of a graph convolutional network (GCN) that provides strong representation ability has drawn significant attention, yet nonetheless largely remains an open problem for the graph learning community. Although noteworthy progress has been made, the depth or the number of layers of a corresponding GCN is realized by a series of graph convolution operations, which naturally makes da positive integer (d N+). An interesting question is whether breaking the constraint of N+ by making d a real number (d R) can bring new insights into graph learning mechanisms. In this work, by redefining GCN's depth d as a trainable parameter continuously adjustable within (,+), we open a new door of controlling its signal processing capability to model graph homophily/heterophily (nodes with similar/dissimilar labels/attributes tend to be inter-connected). A simple and powerful GCN model TEDGCN, is proposed to retain the simplicity of GCN and meanwhile automatically search for the optimal d without the prior knowledge regarding whether the input graph is homophilic or heterophilic. Negative-valued dintrinsically enables high-pass frequency filtering functionality via augmented topology for graph heterophily. Extensive experiments demonstrate the superiority of TEDGCN on node classification tasks for a variety of homophilic and heterophilic graphs.

Graph Neural Networks (GNNs) extend basic Neural Networks (NNs) by using graph structures based on the relational inductive bias (homophily assumption). While GNNs have been commonly believed to outperform NNs in real-world tasks, recent work has identified a non-trivial set of datasets where their performance compared to NNs is not satisfactory. Heterophily has been considered as the main cause of this empirical observation and numerous works have been put forward to address it. In this paper, we first revisit the widely used homophily metrics and point out that their consideration of only graph-label consistency is a shortcoming. Then, we study heterophily from the perspective of post-aggregation node similarity and define new homophily metrics, which are verified to be advantageous compared to existing ones. Based on this investigation, we prove that some harmful cases of heterophily can be effectively addressed by local diversification operation. Then, we propose the Adaptive Channel Mixing (ACM), a framework to adaptively exploit aggregation, diversification and identity channels node-wisely to extract richer localized information for diverse node heterophily situations. ACM is more powerful than the commonly used uni-channel framework for node classification tasks on heterophilic graphs and is easy to be implemented in baseline GNN layers. When evaluated on 10 benchmark node classification tasks, ACM-augmented baselines consistently achieve significant performance gain, exceeding state-of-theart GNNs on most tasks without incurring significant computational burden.

Heterophily, or the tendency of connected nodes in networks to have different class labels or dissimilar features, has been identified as challenging for many Graph Neural Network (GNN) models. While the challenges of applying GNNs for node classification when class labels display strong heterophily are well understood, it is unclear how heterophily affects GNN performance in other important graph learning tasks where class labels are not available. In this work, we focus on the link prediction task and systematically analyze the impact of heterophily in node features on GNN performance. We first introduce formal definitions of homophilic and heterophilic link prediction tasks, and present a theoretical framework that highlights the different optimizations needed for the respective tasks. We then analyze how different link prediction encoders and decoders adapt to varying levels of feature homophily and introduce designs for improved performance. Based on our definitions, we identify and analyze six real-world benchmarks spanning from homophilic to heterophilic link prediction settings, with graphs containing up to 30M edges. Our empirical analysis on a variety of synthetic and real-world datasets confirms our theoretical insights and highlights the importance of adopting learnable decoders and GNN encoders with ego-and neighbor-embedding separation in message passing for link prediction tasks beyond homophily.

Recent work has shown that a much simpler model, simple graph convolution (SGC) (Wu et al., 2019),iscompetitivewithGCNs incommon graph machine learning benchmarks.

Upon increasing the depth, the over-smoothing issue arises: a GCN's