Image colorization is a typical ill-posed problem. Among various colorization methods, scribble-based methods have a unique advantage that allows users to accurately resolve ambiguities and modify the colors of any objects to suit their specific tastes. However, due to the time-consuming scribble drawing process, users tend to draw sparse scribbles instead of dense and detailed scribbles, which makes it challenging for existing methods, especially for regions with no immediate scribbles. Facing the above problems, this paper proposes a novel colorization algorithm named Local and Global Affinity Net (LGA-Net) that formulates the scribble-based colorization task as an affinity propagation process at both local and global levels. Instead of predicting color values directly, our neural network learns to predict local and global affinity relationships between pixels for a given grayscale input, describing how colors should be propagated, which are independent of the scribbles. Given reliable affinity relationships, the color propagation process is formulated as a maximum a posteriori problem. Both local and global affinities are represented using a weighted graph and enabled by a graph Laplacian regularizer to ensure accurate color propagation. Extensive experiments demonstrate that LGA-Net produces state-of-the-art colorization results when using sparse scribbles.