The complex tail dependency structure in a dynamic network with a large number of nodes is an important object to study. Here we propose a network quantile autoregression model (NQAR), which characterizes the dynamic quantile behavior. Our NQAR model consists of a system of equations, of which we relate a response to its connected nodes and node specific characteristics in a quantile autoregression process. We show the estimation of the NQAR model and the asymptotic properties with assumptions on the network structure. For this propose we develop a network Bahadur representation that gives us direct insight into the parameter asymptotics. Moreover, innovative tail-event driven impulse functions are defined. Finally, we demonstrate the usage of our model by investigating the financial contagions in the Chinese stock market accounting for shared ownership of companies. We find higher network dependency when the market is exposed to a higher volatility level.