Dynamical efficiency for multimodal time-varying transportation networks.

Spatial systems that experience congestion can be modeled as weighted networks whose weights dynamically change over time with the redistribution of flows. This is particularly true for urban transportation networks. The aim of this work is to find appropriate network measures that are able to detect critical zones for traffic congestion and bottlenecks in a transportation system. We propose for both single and multi-layered networks a path-based measure, called dynamical efficiency, which computes the travel time differences under congested and free-flow conditions. The dynamical efficiency quantifies the reachability of a location embedded in the whole urban traffic condition, in lieu of a myopic description based on the average speed of single road segments. In this way, we are able to detect the formation of congestion seeds and visualize their evolution in time as well-defined clusters. Moreover, the extension to multilayer networks allows us to introduce a novel measure of centrality, which estimates the expected usage of inter-modal junctions between two different transportation means. Finally, we define the so-called dilemma factor in terms of number of alternatives that an interconnected transportation system offers to the travelers in exchange for a small increase in travel time. We find macroscopic relations between the percentage of extra-time, number of alternatives and level of congestion, useful to quantify the richness of trip choices that a city offers. As an illustrative example, we show how our methods work to study the real network of a megacity with probe traffic data.