Traditional decision and planning frameworks for self-driving vehicles (SDVs) scale poorly in new scenarios, thus they require tedious hand-tuning of rules and parameters to maintain acceptable performance in all foreseeable cases. Recently, self-driving methods based on deep learning have shown promising results with better generalization capability but less hand engineering effort. However, most of the previous learning-based methods are trained and evaluated in limited driving scenarios with scattered tasks, such as lane-following, autonomous braking, and conditional driving. In this paper, we propose a graph-based deep network to achieve scalable self-driving that can handle massive traffic scenarios. Specifically, more than 7,000 km of evaluation is conducted in a high-fidelity driving simulator, in which our method can obey the traffic rules and safely navigate the vehicle in a large variety of urban, rural, and highway environments, including unprotected left turns, narrow roads, roundabouts, and pedestrian-rich intersections. Demonstration videos are available at https: //caipeide.github.io/dignet/.