Modulation classification, an intermediate process between signal detection and demodulation in a physical layer, is now attracting more interest to the cognitive radio field, wherein the performance is powered by artificial intelligence algorithms. However, most existing conventional approaches pose the obstacle of effectively learning weakly discriminative modulation patterns. This paper proposes a robust modulation classification method by taking advantage of deep learning to capture the meaningful information of modulation signal at multi-scale feature representations. To this end, a novel architecture of convolutional neural network, namely Chain-Net, is developed with various asymmetric kernels organized in two processing flows and associated via depth-wise concatenation and element-wise addition for optimizing feature utilization. The network is evaluated on a big dataset of 14 challenging modulation formats, including analog and high-order digital techniques. The simulation results demonstrate that Chain-Net robustly classifies the modulation of radio signals suffering from a synthetic channel deterioration and further performs better than other deep networks.