## 2 Interdependencies  
This paper aims to propose a definition of resilience in a multiple dimensions scale to account for the different typologies of systems that are interdependent. In this regard, interdependencies consist of the interactions between different lifelines (i.e., lifeline types such as energy, transportation, communication and water) and are fundamental to be considered since they may produce negative consequences of direct and indirect losses. In particular, the seismic resilience may be significantly underestimated if interdependence is neglected, as shown in [14; 25; 26; 27; 28]. In this regard, Ref. [29] demonstrated the importance of considering the impact of coastal hazard countermeasures on community resilience for the Tohoku region of Japan following the 2011 tsunami. In addition, Ref. [30] proposed a quantitative assessment framework and method to accurately assess the seismic resilience of interdependent lifeline networks. When an event occurs, the lifelines that enable to distribute services become ways to distribute the consequences of the loss of services along the system. Following [31], several points need to be considered. The consequences due to infrastructure interdependencies are more severe than those presented exclusively on the isolated system, because they may be responsible of cascading failures and collapses of entire regions. * Impacts due to the disaster may be reducing by considering the propagation of failures inside the network;
* The reduction of the severe impacts of disaster may be enchanted by considering network dependencies;
* As shown in [7; 24], interdependencies play a significant role in indirect losses (such as lifeline system failures and loss of services). Interdependencies may affect the resilience of the system in several ways. First, a problem or failure in one lifeline may affect the product or service delivered by the entire system, that can be damaged, degraded or interrupted. This type of interaction may occur when the cellular phone towers are not equipped with backup systems when power is lost, and a failure on individual cell towers may affect the entire electric service. Another example occurs when two lifelines are connected so that the first system provides a service to a second that supplies the first with a critical product, as in the case of failure of the electric distribution systems, which stops the delivery of coal to power plants affecting, in turn, the electric generation [31]. Figure 1 considers two typologies of failures specifically due to interconnections between lifelines. In Figure 1a, cascading that generates dependent relationships amongst lifelines and consists of a change in state in the first lifeline that propagates changes in a second, and thus in all the connected ones in turn, causes a change in state in a third lifeline. A common cause (Figure 1b) consists of disruptions of different systems due to a single event. For example, a tree blown over by the wind taking down electric distribution lines, television cables and telephone cables with it. Interdependencies may be of particular importance for those stakeholders such as public- and private-sector owners deputed to the protection of lifelines and other critical infrastructures. The disruption or the failure to one of such systems may cause cascade effects upon other lifelines. The operators need to be prepared and aware of the role of interdependencies to assess the response, prioritize recovery plans and organize emergency countermeasures. For example, the role of the interactions among the different systems needs to be considered as a significant part in designing the various component, in preventing cascade effects and in planning for redundancy. In other words, interdependency may be the difference between a controlled event and an economic disaster, due to catastrophic consequences occurring with no proper plannings. In the next section, an analytical formulation of resilience has been proposed in order to account for the functionalities of the various dimensions.