  In the quantum repeater networks of the quantum Internet, the varying
stability of entangled quantum links makes dynamic topology adaption an
emerging issue. Here we define an efficient topology adaption method for
quantum repeater networks. The model assumes the random failures of entangled
links and several parallel demands from legal users. The shortest path defines
a set of entangled links for which the probability of stability is above a
critical threshold. The scheme is utilized in a base-graph of the overlay
quantum network to provide an efficient shortest path selection for the demands
of all users of the network. We study the problem of entanglement assignment in
a quantum repeater network, prove its computational complexity, and show an
optimization procedure. The results are particularly convenient for future
quantum networking, quantum-Internet, and experimental long-distance quantum
communications.
