Title: ReD-GCN: Revisit the Depth of Graph Convolutional Network. Abstract: Finding the proper depth $d$ of a GNN that provides strong representation power 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 $d$ a positive integer ($d \in \mathbb{N}+$). An interesting question is whether breaking the constraint of $\mathbb{N}+$ by making $d$ a real number ($d \in \mathbb{R}$) can bring new insights into graph learning mechanisms. In this work, by redefining GCN's depth $d$ as a trainable parameter continuously adjustable within $(-\infty,+\infty)$, we open a new door of controlling its expressiveness on graph signal processing to model graph homophily/heterophily (nodes with similar/dissimilar labels/attributes tend to inter-connect). A simple and powerful GCN model ReD-GCN, 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 $d$ intrinsically enables high-pass frequency filtering functionality for graph heterophily. Variants extending the model flexibility/scalability are also developed. The theoretical feasibility of having a real-valued depth with explainable physical meanings is ensured via eigen-decomposition of the graph Laplacian and a properly designed transformation function from the perspective of functional calculus. Extensive experiments demonstrate the superiority of ReD-GCN on node classification tasks for a variety of graphs. Furthermore, by introducing the concept of eigengraph, a novel graph augmentation method is obtained: the optimal $d$ effectively generates a new topology through a properly weighted combination of eigengraphs, which dramatically boosts the performance even for a vanilla GCN.