A hierarchically structured graph comprises multiple layers of interconnected graph. Hierarchical graphs are essential for understanding the structure and dynamics of complex systems across various fields such as sociology, biology, and economics. For instance, the omics graph layered by genome, proteome, disease, the financial graph layered by individual stocks, financial derivatives, global economical indicators, and the genealogy graph stratified by ancestor-descendant relationships. Common methods like Graph Convolutional Network and Graph-based Semi- Supervised Learning (GSSL) face significant limitations when applied to hierarchical graphs. GCN fail to capture hierarchical structures and are restricted in their neighbor range, while GSSL methods do not directly incorporate node attribute information despite their ability to reflect hierarchical structures through matrix separation. To address these challenges, we propose Graph Convolutional Smoothing Networks (GCSN). GCSN integrates both structural and attribute information by employing a Smoothness Factor Matrix, which allows for the adjustment of the neighborhood range of nodes in each layer. This Smoothness Factor Matrix is designed to be learnable, enabling the optimal neighborhood range for each layer to be determined during the training process. Experiments show that the proposed method outperforms the comparison methods by taking advantage of its applicability to hierarchical graphs and its flexibility of convolutional propagation using smoothing for feature propagation.
