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Besides true quantum repeaters, a trusted node paradigm seems to be inevitable for practical implementations (at least in the short term), and one should consider configurations of trusted nodes as the basis for global quantum networks. In this paper, we estimate how the introduction of additional connections between trusted nodes through one, two, etc., nodes (i.e., connection density) to a quantum network with serial connections of trusted nodes affects its security. We provide proper scaling of the failure probability of authentication and quantum key distribution protocols to the level of the whole quantum network. Expressions of the failure probability dependent on the total number of connected nodes between users and the connection density for the given mean failure probability of each element are derived. The result provides an explicit trade-off between an increase of key transport security and a consequent …