  Distributions of the resilience of transport networks are studied
numerically, in particular the large-deviation tails. Thus, not only typical
quantities like average or variance but the distributions over the (almost)
full support can be studied. For a proof of principle, a simple transport model
based on the edge-betweenness and three abstract yet widely studied random
network ensembles are considered here: Erdoes-Renyi random networks with finite
connectivity, small world networks and spatial networks embedded in a
two-dimensional plane. Using specific numerical large-deviation techniques,
probability densities as small as 10^(-80) are obtained here. This allows one
to study typical but also the most and the least resilient networks. The
resulting distributions fulfill the mathematical large-deviation principle,
i.e., can be well described by rate functions in the thermodynamic limit. The
analysis of the limiting rate function reveals that the resilience follows an
exponential distribution almost everywhere. An analysis of the structure of the
network shows that the most-resilient networks can be obtained, as a rule of
thumb, by minimizing the diameter of a network. Also, trivially, by including
more links a network can typically be made more resilient. On the other hand,
the least-resilient networks are very rare and characterized by one (or few)
small core(s) to which all other nodes are connected. In total, the spatial
network ensemble turns out to be most suitable for obtaining and studying
resilience of real mostly finite-dimensional networks. Studying this ensemble
in combination with the presented large-deviation approach for more realistic,
in particular dynamic transport networks appears to be very promising.
