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Scaling in the recovery of urban transportation systems from special events

Public transportation is a fundamental infrastructure for the daily mobility in cities. Although its capacity is prepared for the usual demand, congestion may rise when huge crowds concentrate in special events such as massive demonstrations, concerts or sport events. In this work, we study the resilience and recovery of public transportation networks from massive gatherings by means of a stylized model mimicking the mobility of individuals through the multilayer transportation network. We focus on the delays produced by the congestion in the trips of both event participants and of other citizens doing their usual traveling in the background. Our model can be solved analytically for regular lattices showing that the average delay scales with the number of event participants with an exponent equal to the inverse of the lattice dimension. We then switch to real transportation networks of eight worldwide cities, and observe that there is a whole range of exponents depending on where the event is located. These exponents are distributed around 1/2, which indicates that most of the local structure of the network is two dimensional. Yet, some of the exponents are below (above) that value, implying a local dimension higher (lower) than 2 as a consequence of the multimodality and multifractality of transportation networks. In fact, these exponents can be also obtained from the scaling of the capacity with the distance from the event. Overall, our methodology allows to dynamically probe the local dimensionality of a transportation network and identify the most vulnerable spots in cities for the celebration of massive events.