Large-deviation properties of resilience of transportation networks

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: Erdős-Rényi 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 are obtained here. This allows 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.