Graph Convolutional Network and many of its variants are known to suffer from the dilemma between model depth and over-smoothing issues. Stacking layers of GCN usually lead to the exponential expansion of the receptive field (i.e., high-order neighbors). In order to incorporate the information from high-order neighbors to learn node representations without drastically increasing the number of graph convolution layers, we propose a simple and effective pre-processing technique to increase graph connectivity. Our approach selectively inserts connections between center nodes and informative high-order neighbors, with learnable weights to control the information flow through the connection. Experiments show that our approach improves the performance of GCN, and reduce the depth of GCNII without sacrificing its performance. Besides, our proposed homophily-based weight assignment can mitigate the effect of graph structural attacks.