Building, managing, upgrading, and diagnosing networks becomes increasingly difficult due to the increased complexity in protocol design, enlarged network scale, rapidly-growing types of network applications, and the uncertainty of network faults and cyber attacks. It is not satisfactory to rely on network simulators/emulators for conducting these tasks because of different defects including poor fidelity, long implementing time, and weak interconnection with physical networks. To enable the interplay of networks and their virtual counterparts, this paper aims to build a Network Functions Virtualization (NFV)-empowered digital twin cyber platform (DTCP) for enabling network innovation and test with high fidelity, low complexity, and high timeliness. First, after regulating the paradigm of DTCP, a six-dimensional model is defined to profile a DTCP. Second, the architecture of the DTCP is detailed along with the key enabling technologies. Third, a prototype implementation of the DTCP is shortly introduced. Lastly, a use case based on fault diagnosis for video conference system is tested on the prototype. Results show that our DTCP prototype is a good platform for conducting different types of network operations without explicitly providing the mapping model between physical and virtual networks.