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One practice of employing deep neural networks is to apply the same architecture to all the input instances. However, a fixed architecture may not be representative enough for data with high diversity. To promote the model capacity, existing approaches usually employ larger convolutional kernels or deeper network structure, which may increase the computational cost. In this paper, we address this issue by raising the Dynamic Graph Network (DG-Net). The network learns the instance-aware connectivity, which creates different forward paths for different instances. Specifically, the network is initialized as a complete directed acyclic graph, where the nodes represent convolutional blocks and the edges represent the connection paths. We generate edge weights by a learnable module \textit{router} and select the edges whose weights are larger than a threshold, to adjust the connectivity of the neural network structure. Instead of using the same path of the network, DG-Net aggregates features dynamically in each node, which allows the network to have more representation ability. To facilitate the training, we represent the network connectivity of each sample in an adjacency matrix. The matrix is updated to aggregate features in the forward pass, cached in the memory, and used for gradient computing in the backward pass. We verify the effectiveness of our method with several static architectures, including MobileNetV2, ResNet, ResNeXt, and RegNet. Extensive experiments are performed on ImageNet classification and COCO object detection, which shows the effectiveness and generalization ability of our approach.