  Networks constitute efficient tools for assessing universal features of
complex systems. In physical contexts, classical as well as quantum, networks
are used to describe a wide range of phenomena, such as phase transitions,
intricate aspects of many-body quantum systems or even characteristic features
of a future quantum internet. Random quantum networks and their associated
directed graphs are employed for capturing statistically dominant features of
complex quantum systems. Here, we develop an efficient iterative method capable
of evaluating the probability of a graph being strongly connected. It is proven
that random directed graphs with constant edge-establishing probability are
typically strongly connected, i.e. any ordered pair of vertices is connected by
a directed path. This typical topological property of directed random graphs is
exploited to demonstrate universal features of the asymptotic evolution of
large random qubit networks. These results are independent of our knowledge of
the details of the network topology. These findings suggest that also other
highly complex networks, such as a future quantum internet, may exhibit similar
universal properties.
