  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.
