Quantum networks open the way to an unprecedented level of communication security. However, due to physical limitations on the distances of quantum links, current implementations of quantum networks are unavoidably equipped with trusted nodes. As a consequence, the quantum key distribution can be performed only on the links. Due to this, some new authentication and key exchange schemes must be considered to fully benefit from the unconditional security of links. One such approach uses Multiple Non-Overlapping Paths (MNOPs) for key exchange to mitigate the risk of an attack on a trusted node. The scope of the article is to perform a security analysis of this scheme for the case of both uncorrelated attacks and correlated attacks with finite resources. Furthermore, our analysis is extended to the case of Multiple Overlapping Paths (MOPs). We prove that introducing overlapping paths allows one to increase the security of the protocol, compared to the non-overlapping case with the same number of additional links added. This result may find application in optimising architectures of large-scale (hybrid) quantum networks.