# Graph Convolutional Networks Guided by Explicitly Estimated Homophily and Heterophily Degree
## 2 Related Work  
There are two major GCN-based approaches for tackling learning problems on heterophilic graphs. One is to mine more supplemental nodes when updating a node feature in the message-passing process, which act as those directly-connected neighbors for propagating messages to the node and participate in aggregation and updating of node representation. Geom-GCN [18] maps each node to the latent space and takes the nodes whose distance from the node is less than the setting value in the latent space as the supplemental nodes. NL-GNN [25] utilizes an attention-guided sorting method to choose the supplemental nodes. It defines a learnable calibration vector and adopts an attention mechanism to calculate the attention scores of each node representation and the calibration vector. All nodes are sorted according to their attention scores. For each node, the other nodes whose attention scores are similar are chosen as the supplemental nodes. In addition to the above methods, there are some works that choose the higher-order neighbors in the graph as supplemental nodes. They are based on the observation that for nodes in heterophilic graphs, the first-order neighbors are often heterophilic, but the higher-order neighbors are likely to be homophilic [26]. MixHop [19] chooses two-hop neighbors as the supplemental nodes for message passing. In the layers of MixHop, the messages from different hops neighbors are linearly transformed by different weights and then concatenated. H2 GCN [26] chooses higher-order neighbors as the supplemental nodes for message passing. It cancels the linear transformation and concatenates the messages aggregated from neighbors of different hops in the hidden layer, and the aggregated messages at different hidden layers are concatenated and linearly transformed in the output layer. Considering that the number of two-hop neighbors will grow exponentially with the growth of network scale, UGCN [27] limits the two-hop neighbors that have at least two different paths to the node as the supplemental nodes. Apart from the two-hop neighbors, UGCN also utilizes the cosine similarity to choose \(k\)-nearest neighbor neighbors as the supplemental nodes. However, there is a shortcoming for these methods in that mixing the messages obtained from connected nodes and supplemental nodes will damage the graph structure information which is often important in graph learning tasks. The other approach is to design a more reasonable message-passing mechanism. FAGCN [20] demonstrates the importance of high-pass signals on heterophilic graphs through experiments and illustrates that high-pass signals can be retained by propagating negative messages between the connected nodes. It uses a self-gating attention mechanism to learn the signed coefficients of neighbors for propagation, and the coefficients represent the proportions of low-pass signals and high-pass signals. However, the effects of homophily and heterophily on the learning of signed coefficients are not clearly distinguished and treated in a mixed and implicit way. GGCN [28] uses the cosine similarity function to calculate the signed coefficients of neighbors for propagation between connected nodes. CPGNN [29] defines a parameterized and learnable compatibility matrix which is used to model the probability of node pairs that belong to each category. It also adopts a pre-trained estimator to calculate the prior belief of class labels for nodes and propagates the prior belief by the compatibility matrix. HOG-GCN [22] combines the estimated homophily degree and the learnable homophily degree of node pairs to guide the positive message passing between nodes and their multi-hop neighbors. Since HOG-GCN models the homophily degree between node pairs by learnable parameters, the parameter quantity of HOG-GCN can be quite big for a large graph. BM-GCN [23] adopts the estimated soft labels to calculate a block similarity matrix and combines the estimated soft labels and block similarity matrix to guide the positive message passing.