Networks are widely used in many fields for their powerful ability to provide vivid representations of relationships between variables. However, many of them may be corrupted by experimental noise or inappropriate network inference methods that inherently hamper the efficacy of network-based downstream analysis. Consequently, it's necessary to develop systematic methods for denoising networks, namely, improve the Signal-to-Noise Ratio (SNR) of noisy networks. In this paper, we have explored the properties of network signal and noise and proposed a novel network denoising framework called Network Refinement (NR) that adjusts the edge weights by applying a nonlinear graph operator based on a diffusion process defined by random walk on the graph. Specifically, this unified framework consists of two closely linked approaches named NR-F and NR-B, which improve the SNR of noisy input networks from two different perspectives: NR-F aims at enhancing signal strength, while NR-B aims at weakening noise strength. Users can choose from which angle to improve the SNR of the network according to the characteristics of the network itself. We show that NR can significantly refine the quality of many networks by several applications on simulated networks and typical real-world biological and social networks.