The depth completion task aims to predict a dense depth map from a sparse LiDAR point cloud and an RGB image. This task is critical because an accurate depth map can be used as prior information to solve many computer vision tasks, such as downstream tasks in autonomous vehicles and robot vision. Previous deep learning methods which focus on the local affinity have achieved impressive results. However, an architecture that is directly designed to extract local affinity has not been proposed yet. In this paper, we propose multi-scaled and densely connected locally convolutional layers to learn the affinity of the neighborhood. We set a different grid factor for each step of this module, and each step consists of several convolutional layers applied only to the local area assigned from the grid factor. In addition, each step is densely connected, sequentially, to take advantage of the multi-scale receptive fields. The proposed module effectively learns the neighbor-hood's affinity in a local area with multiple scales, while keeping the network size small. As a result, our architecture achieves state-of-the-art performance compared to published works on the KITTI depth completion benchmark. On the NYU Depth V2 completion benchmark our method achieves performance comparable to state-of-the-art approaches.