Current network technologies evolve toward the sixth generation of mobile communications (6G) systems. To achieve the performance requirements of high data rate, low latency, and massive connection, full-duplex (FD) wireless communication is one possible solution to provide high throughput and channel utilization. However, it is difficult to do so for legacy communication nodes. A distributed fairness-aware full-duplex dual-hop link scheduling algorithm, named weighted full duplex (W-FD), is proposed to achieve the efficient node selection and dynamic relay probing. The scheduling weight that considered traffic load, mutual interference, and response interval is employed to ensure link load balance. A node with a higher scheduling weight in the neighborhood is selected as the link establishment object to build a full-duplex link. Better than conventional half-duplex(HD) link scheduling, W-FD reduces link scheduling interval and ensures the access opportunities for legacy nodes. Simulation result shows that throughput and delay performance are improved. Compared with the existed full-duplex link scheduling, W-FD could better reflect the fairness of scheduling in multi-nodes and asymmetric-packets networks, and ensure the throughput and delay performance.