  We present a detailed description of the diagrammatic technique, recently
devised in [V. V. Kuzmin et.al., npj Quantum Information 5, 115 (2019)], for
semi-analytical description of large-scale quantum-repeater networks. The
technique takes into account all essential experimental imperfections,
including dissipative Liouville dynamics of the network quantum memories and
the classical communication delays. The results obtained with the semi-analytic
method match the exact Monte Carlo simulations while the required computational
resources scale only linearly with the network size. The presented approach
opens new possibilities for the development and efficient optimization of
future quantum networks.
