Incremental life-long learning is a main challenge towards the long-standing goal of Artificial General Intelligence. In real-life settings, learning tasks arrive in a sequence and machine learning models must continually learn to increment already acquired knowledge. Existing incremental learning approaches, fall well below the state-of-the-art cumulative models that use all training classes at once. In this paper, we propose a random path selection algorithm, called RPS-Net, that progressively chooses optimal paths for the new tasks while encouraging parameter sharing. Since the reuse of previous paths enables forward knowledge transfer, our approach requires a considerably lower computational overhead. As an added novelty, the proposed model integrates knowledge distillation and retrospection along with the path selection strategy to overcome catastrophic forgetting. In order to maintain an equilibrium between previous and newly acquired knowledge, we propose a simple controller to dynamically balance the model plasticity. Through extensive experiments, we demonstrate that the proposed method surpasses the state-of-the-art performance on incremental learning and by utilizing parallel computation this method can run in constant time with nearly the same efficiency as a conventional deep convolutional neural network.