The distinctive characteristics of unmanned aerial vehicle networks (UAVNs), including highly dynamic network topology, high mobility, and open-air wireless environments, may make UAVNs vulnerable to attacks and threats. In this study, we propose a novel trust model for UAVNs that is based on the behavior and mobility pattern of UAV nodes and the characteristics of inter-UAV channels. The proposed trust model consists of four parts: direct trust section, indirect trust section, integrated trust section, and trust update section. Based on the trust model, the concept of a secure link in UAVNs is formulated that exists only when there is both a physical link and a trust link between two UAVs. Moreover, the metrics of both the physical connectivity probability and the secure connectivity probability between two UAVs are adopted to analyze the connectivity of UAVNs. We derive accurate and analytical expressions of both the physical connectivity probability and the secure connectivity probability using stochastic geometry with or without Doppler shift. Extensive simulations show that compared with the physical connection probability with or without malicious attacks, the proposed trust model can guarantee secure communication and reliable connectivity between UAVs and enhance network performance when UAVNs face malicious attacks and other security risks.