# Graph Convolutional Networks Guided by Explicitly Estimated Homophily and Heterophily Degree
###### Abstract  
Graph convolutional networks (GCNs) have been successfully applied to learning tasks on graph-structured data. However, most traditional GCNs based on graph convolutions assume homophily in graphs, which leads to a poor performance when dealing with heterophilic graphs. Although many novel methods have recently been proposed to deal with heterophily, the effect of homophily and heterophily on classifying node pairs is not clearly separated in existing approaches and inevitably influences each other. To deal with various types of graphs more accurately, in this work we propose a new GCN-based model that leverages the explicitly estimated homophily and heterophily degree between node pairs and adaptively guides the propagation and aggregation of signed messages. We also design a pre-training process to learn homophily and heterophily degree from both original node attributes that are graph-agnostic and the localized graph structure information by using Deepwalk that reflects graph topology. Extensive experiments on eight real-world benchmarks demonstrate that the new approach achieves state-of-the-art results on three homophilic graph datasets and outperforms baselines on five heterophilic graph datasets. graph convolutional networks; heterophilic graphs; node classification +
Footnote †: journal: Applied  
the opposite gender. Studies have shown that many traditional GCN models are ineffective on heterophilic graphs, especially for semi-supervised node classification tasks [18,19]. As shown in Figure 0(a), consider updating the node feature \(h_{1}\) of \(v_{1}\) that has two neighbours \(v_{2}\) and \(v_{3}\). By applying the traditional message-passing method, the updated representation of node \(v_{1}\) is obtained by adding the node features of \(v_{2}\) and \(v_{3}\) to its self-representation, respectively. After updating, the node feature of \(v_{1}\) becomes closer to that of both \(v_{2}\) and \(v_{3}\), which makes it difficult to distinguish between nodes \(v_{1}\) and \(v_{2}\) that belong to different categories. Recently, many novel GCN-based methods have been proposed to deal with heterophily. Notably, FAGCN [20] takes the perspective of graph signal processing [21] and explores both low-pass filters and high-pass filters in the model design. By connecting to the spatial domain that the process of low-pass filtering represents, the sum of node features with neighbours and the process of high-pass filtering represent the difference of node features between neighbouring nodes; it uses a self-gating mechanism to learn the signed coefficients of the messages. HOG-GCN [22] and BM-GCN [23] explore the estimated homophily degree between node pairs to guide the propagation of positive messages but do not consider how to guide the propagation of negative messages, which is often important in heterophilic graphs. Consider again the example in Figure 0(b), if we introduce signed messages during the message-passing process, the updated representation of node \(v_{1}\) is closer to that of its homophilic neighbor \(v_{3}\) and away from that of its heterophilic neighbor \(v_{2}\) as expected. We also observe that the above methods have a common limitation in that the effects of homophily and heterophily on classifying node pairs are not clearly separated in their learning approaches and inevitably influence each other. In this paper, we aim to design a new model that works well on either homophilic or heterophilic graphs for semi-supervised node classification tasks. Comparing the existing approaches and inspired by the above models, especially FAGCN and HOG-GCN, we would like to leverage explicitly the estimated homophily and heterophily degree between node pairs to adaptively guide the propagation and aggregation of signed messages, respectively. In particular, positive messages are aggregated from homophilic neighbors, and negative messages are aggregated from heterophilic neighbors.