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# Investigation of Similar Patterns in Urban Network Collapses

Urban transportation networks may be exposed to errors (heavy congestion, infrastructure failure) and attacks (terrorist attacks). From the equilibrium perspective, noticeable surge in demand or drop in supply may cause breakdowns in an urban transportation network. Due to the importance of urban transportation networks, identification of their behavior towards perturbations and fortifying them are of utmost importance. In this paper, we aim to address two questions. The first question is: Are there similar patterns among different cities throughout the world in the collapse of urban networks as the result of attacks? The second question is: Among various node importance measures, which best reveals the critical nodes of an urban network. Urban transportation networks of seven cities including Philadelphia, Chicago, Austin, Berlin, Isfahan, Gold Coast, and Birmingham are abstracted as graphs by adopting the primal approach i.e. nodes represent intersections and edges represent highways and streets. Degree, betweenness, weighted degree with weights being the capacity of incident streets, and a combination of degree and weighted degree of nodes are calculated. The nodes are sequenced based on these criteria and in each stage they are eliminated based on these orders. Moreover, in one stage the nodes are eliminated randomly to represent random failures. In each stage, after elimination of one percent of nodes, the relative size of the giant component is calculated. The nodes whose elimination have dire effects on the performance of a network and lead it to plunge are considered as critical nodes. It was found that the behavior of investigated urban networks follow an inverse sigmoid pattern. The process of network collapse can be divided to three distinct parts; in the first part for loss of 15% of nodes, the slope of collapse of a network is mild, then it experiences a severe slope, and finally by reaching loss of 35% ends to a mild slope when network is approximately devastated and collapsed. Removal of approximately 35% of nodes results in diminishing the giant component to about 20% of the initial network. Results show that in the case of minor disruptions, degree of nodes best identifies the critical nodes. As the main disruptions of urban networks cause minor node removals, nodes with higher degrees are the most critical. This implies that intersections with all two directional approaches are the most critical urban infrastructures.