In this work, we established a density equation for delayed airports to investigate the horizontal propagation mechanism of delays among airports in an airport network. We explored the critical conditions, steady-state features, and scale of the delay propagation, and designed a simulation system to verify the accuracy of the results. The results indicated that, due to the no-table scale-free feature of an airport network, the critical value of delay propagation is extremely small, and delays are prone to propagate among airports. Furthermore, as delay propagation reaches a steady state in an aviation network, the degree value of the node becomes highly correlated with its delay state. Hub airports with high degree values are the most prone to being affected by delay propagation. In addition, the number of airports that are initially delayed influences the time required for delay propagation to reach a steady state. Specifically, if there are fewer initially delayed airports, a longer time is required to reach a steady state. In the steady state, the delay ratios of airports with different degree values in the network converge to a balance point. The delay degree of the node is highly positively correlated with the delay propagation rate in the network, but negatively related to the degree distribution index of the network.