Shape and performance of fastest paths over networks with interacting selfish agents

We study the evolution of the fastest paths (FPs) in transportation networks under increasing congestion. Moving from the common edge-based to a path-based analysis, we examine the directed FPs connecting random origin-destination pairs as traffic grows. We describe their shape through effective length, detour (maximum distance of FP from a straight line), inness (signed area between FP and straight line), and their performance through a metric measuring how fast and how far an agent travels toward its destination. The entire network is characterized by analyzing the distribution of the performance metric (and its Gini coefficient) across uniformly sampled paths. The study focuses on the traffic-loading phenomenon that takes place during the morning peak hour for eight major cities: networks start with empty edges that are progressively populated by the FPs of single vehicles. As vehicle density grows, the interactions among selfish agents becomes stronger at edge level, and travel speed linearly decreases, thus optimal paths dynamically change with traffic. We fully characterize the transition to congestion and discuss the common aspects among the cities (and some peculiarities); in particular we were able to pinpoint a critical traffic level (or a sequence) for which path shape, rejection ratio, and inequality of the performance degradation show a concurrent qualitative change. For all cities we observe large peaks for both detour and inness (and their variance) in the proximity of the critical traffic level. Inness shows that paths are slightly attracted by city centers with light traffic, but switch to a strong repulsion immediately beyond the transition. Finally, our path performance metric highlighted a strongly asymmetric behavior when the city neighborhoods act as origins or destinations.