In the last decades, resilience became officially the worldwide cornerstone to reduce the risk of disasters and improve preparedness, response, and recovery capacities. Although the concept of resilience is now clear, it is still under debate how to model and quantify it. The aim of this work was to quantify the resilience of a complex system, such as a densely populated and urbanized area, by modelling it with a graph, the mathematical representation of the system element and connections. We showed that the graph can account for the resilience characteristics included in its definition according to the United Nations General Assembly, considering two significant aspects of this definition in particular: (1) resilience is a property of a system and not of single entities and (2) resilience is a property of the system dynamic response. We proposed to represent the exposed elements of the system and their connections (i.e., the services they exchange) with a weighted and redundant graph. By mean of it, we assessed the systemic properties, such as authority and hub values and highlighted the centrality of some elements. Furthermore, we showed that after an external perturbation, such as a hazardous event, each element can dynamically adapt, and a new graph configuration is set up, taking advantage of the redundancy of the connections and the capacity of each element to supply lost services. Finally, we proposed a quantitative metric for resilience as the actual reduction of the impacts of events at different return periods when resilient properties of the system are activated. To illustrate step by step the proposed methodology and show its practical feasibility, we applied it to a pilot study: the city of Monza, a densely populated urban environment exposed to river and pluvial floods.
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