Soft and hard infrastructure: A resilience based approach for urban DRR

Raffaele Giordano and Alessandro Pagano (CNR-IRSA)

Disasters cause serious damage on structures and infrastructures, particularly in urban areas. All the lifelines (‘hard infrastructures’, e.g. water supply, transport, power supply, telecommunication) are impacted by extreme events, and their functionality is limited as a consequence of both physical damages and changes in the operating conditions. Lifelines are vulnerable elements, but also crucial assets to guarantee the safety and well-being of the impacted population. The role of hard infrastructures at urban level supports the efforts of local communities during the EM, being a key asset to cope with the disaster. As a matter of fact, society and local economy benefit from the operation of hard infrastructural systems.

Disasters cause complex short- and long-term effects on social structures as well, which are often difficult to understand and define (soft infrastructures). Community organizations and community-based networks play a key role in disaster preparedness and recovery. Local knowledge, understandings, perceptions, resources, and cooperative strategies are crucial to determine system survival and, particularly, to properly drive recovery conditions.

A strong connection exists between the two infrastructures: the reliability of physical infrastructures during the emergency management contributes to mobilize the social capital. Similarly, the existing soft infrastructures may condition the performances and the level of service provided by the hard infrastructures. More specifically, infrastructural systems play a fundamental role in keeping alive the social networks within a community in case of disasters by continuing to provide key services. The process of recovery after extreme events, is also generally supported by the availability of critical services, which significantly contribute to increase the resilience of the whole community.

Resilience based approach for DRR

Resilient systems have a reduced probability of failure, lower consequences from failures and a reduced time for recovery. At urban scale, resilience is related to the capacity of cities to cope with and recover from external shocks. Infrastructural systems of a city are linked with social and institutional systems, but also with the economic and environmental ones, and thus their resilience is necessarily connected to several intertwined dimensions.

Enhancing resilience means improving the capacity of the whole system to anticipate threats, reduce vulnerability and allow a complete recovery from impacts. Several factors contribute to increase the resilience, related to the ‘physical’ and ‘non-physical’ characteristics of the system. All these features are found to be highly influential in all phases of a disaster.

Resilience of hard and soft infrastructure

The concept of resilience tends to be strictly related to both static and dynamic components of disasters across pre and post event context. A static model of resilience identifies and organizes critical variables, whereas a dynamic model represents how and why such variables change across time and space. Resilient systems have a reduced probability of failure, lower consequences from failures and a reduced time for recovery. Referring specifically to urban environments, resilience is related to the capacity of cities to cope with and recover from external shocks. An urban system can be considered resilient if it is sustainable even during the hazard occurrence phase, the most critical period, in which the city suffers the impacts of an extreme event and tries to reconfigure both its physical and social aspects towards a new equilibrium. The infrastructural system of a city has to be conceived as linked with social and institutional systems, but also with the economic and environmental ones that are all embedded within the urban context and dynamically interacting. Physical (hard) infrastructure involves amendments to the physical surroundings and landscape to serve a given purpose (e.g. transportation, power supply, water supply, management, and treatment). Social (soft) infrastructures refers to the networks and interactions among individuals, groups, and institutions within and outside the community. The link between them is crucial, since the resilience of a system is described by its level of functionality and assuming that it directly represents the level of satisfaction of citizen.

Enhancing resilience means improving the capacity of the whole system to anticipate threats, reduce vulnerability and allow a complete recovery from impacts. Several factors contribute to increase the resilience, which might not necessarily be related to the ‘physical’ characteristics of the system. They may depend on individual conditions (e.g. well-being and survival skills) and on community characteristics (community connectedness, community infrastructure, participation in disaster response and recovery, engagement in decision making). All these features are found to be highly influential before a disaster strikes, as well as in the event of a disaster and during recovery.

Several extreme events suggested that infrastructural systems play a fundamental role in keeping alive the social networks within a community in case of disasters by continuing to provide key services. The process of recovery after extreme events, is generally supported by the availability of critical services, which significantly contributes to increase the resilience of the whole community.

The Case Study of L’Aquila supported drawing a few key conclusions:

  • Physical infrastructure provides a vital support to communities during emergency and recovery phases after a disaster. The uninterrupted availability of critical services is a requirement to guarantee the safety and the well-being of a population when a disaster occurs and speeds up the recovery: in this direction, the technical performances of the whole infrastructural system are a key asset to deal effectively with emergencies and contribute to community resilience. On the other hand, the resilience of a community affects the level of service provided by the hard infrastructural system as well: the behaviors of the users (e.g. good practices, flexibility, …), their level of knowledge along with the skills of the authorities managing the emergency and driving decision-making – in a word, their culture - have a direct influence on the response of the hard infrastructural system.
  • Infrastructural systems must directly match the needs of a community, and thus should firstly reflect the spatial distribution of the served population. Secondly, the performances of infrastructural systems should be flexible enough to evolve with time, in the aftermath of a disaster and in the recovery phase, since the needs of the whole system change according to the specific path of recovery determined by the specific strategies implemented.

Describing resilience: the resilience triangle

Referring to a wide scientific literature, the shift from risk management towards ‘resilience management’ paradigm is crucial. Resilience approach addresses the complexities of large integrated systems and the uncertainty of future threats, overcoming the main drawbacks associated to risk management: a) it allows modeling the interactions between different risks; b) it helps analyzing the gradual worsening of environmental conditions; c) it overcomes the limits associated to the use of historical/past data, focusing on the ‘unexpected’; d) it considers the new perspective on extreme events which are now becoming ‘routine’ events; e) it fosters better information to communities. The resilience approach emphasizes the system’s ability to anticipate and absorb potential disruptions, develop adaptive means to accommodate changes, and establish response behaviors aimed at either building the capacity to withstand the disruption or recover as quickly as possible after an impact. The key difference (Cimellaro 2016) is that Risk analysis is used to prioritize the mitigation strategies; Resilience analysis is used to prioritize the restoration strategies.

In its more general definition, resilience could be viewed as the ‘intrinsic capacity of a system, community or society predisposed to a shock or stress to adapt and survive by changing its non-essential attributes and rebuilding itself’. It is a multidimensional, sociotechnical issue that addresses how people, as individuals or groups, manage uncertainty. In infrastrucures and engineering field, it can be defined as the ability of systems to absorb the shocks of extreme events such as natural disasters. Resilience can be achieved by enhancing the ability of an infrastructure to perform during and after a hazard, as well as through emergency response and strategies that effectively cope with and contain losses and recovery strategies.

In order to propose a quantitative way of analyzing resilience, Bruneau et al. (2003) and Tierney and Bruneau et al (2007) suggested the use of a ‘resilience triangle’ to describe resilience, representing both the loss of functionality from damage and disruption, and the pattern of restoration and recovery over time. It is defined graphically as the normalized area underneath the performance function of a system (Cimellaro 2016). It is used to measure the functionality of a system after a disaster, and also the time it takes for a system to return to pre-disaster levels of performance (or, according to specific conditions, to reach higher or lower performance conditions). The area within the resilience triangle relates directly to the resiliency with small er areas indicating greater resilience. Actions, behaviors, and properties of social units, organizations and networks all contribute to reducing the area of the resilience triangle.Resilience-enhancing measures aim at reducing the size of the resilience triangle. Such strategies may act upon the multiple dimensions associated to resilience, which are normally included in the R4 Framework: a) robustness, the ability to withstand external actions without significant degradation or loss of performance; b) redundancy, the extent to which systems and elements are substitutable; c) resourcefulness, the ability to diagnose and prioritize problems and to initiate solutions by identifying and mobilizing resources; d) rapidity, the capacity to restore functionality in a timely way.

Graphical representation of the ‘resilience triangle’.

Resilience-enhancing measures aim at reducing the size of the resilience triangle through strategies that improve the infrastructure’s functionality and performance (the vertical axis in the figure) and that decrease the time to full recovery (the horizontal axis). For example, mitigation measures can improve both infrastructure performance and time to recovery. The time to recovery can be shortened by improving measures to restore and replace damaged infrastructure.

Clearly, identifying the features of organizations and other social units that make them resilient is difficult. The objectives of Disaster Resilience are to minimize loss of life, injuries, disruption of important services, and economic losses. Specifically, disaster resilience is characterized by (Cimellaro 2016):

  • Reduced failure probabilities– i.e., the reduced likelihood of damage and failures to critical infrastructure, systems and components;
  • Reduced consequences from failures– in terms of injuries, lives lost, damage and negative economic and social impacts;
  • Reduced time to recovery– the time required to restore a specific system or set of systems to normal or pre-disaster level of functionality.

Modeling resilience: TOSE approach

Going further into details in LS resilience, one of the the most consolidated and interesting approaches is based on the conceptualization of its four inter-related dimensions: Technical, Organizational, Social and Economic (TOSE approach). The Technical dimension of resilience refers to the ability of interconnected physical systems to perform to acceptable/desired levels. Organizational resilience is related to the organizations and institutions that manage the physical components of the systems, and is thus significantly affected by ‘culture’: it encompasses measures of capacity, planning, training, leadership, experience, and information management that may improve (or hamper) disaster-related organizational performances and problem solving. Among the influential parameters, the ability to incorporate lessons learned from past disasters, the training and the experience of personnel should be considered. The Social dimension includes population and community characteristics that render social groups either more vulnerable or more adaptable to hazards and disasters, and is strictly connected to ‘cultural’ issues as well: social indicators include for example, poverty, education, linguistic isolation, a lack of access to resources for protective action, such as evacuation. Finally, Economic resilience has been analyzed in terms of the inherent properties of local economies and in terms of their capacity for post-disaster improvisation, innovation, and resource substitution.

Resilience assessment was performed with specific reference to L’Aquila Case Study, implementing the TOSE approach on drinking water supply through SDM (see Methodological approaches: SDM & Graph Theory – System Dynamics Modeling). It is worth considering that L’Aquila case study was used to collect and structure the knowledge needed for model building and validation. Nevertheless the model and, more in general, the methodological approach, is broad enough to be implemented with minor changes and adaptations in different cases, conditions, and even on various infrastructural systems.

The global model of resilience is plotted in the following figure (full details are in Pagano et al. 2017) and in the Case Study Manual (Link). The dimensions of resilience were defined adapting the general TOSE framework to the case of drinking water supply systems. The model was built in order to deal separately with the four basic dimensions of resilience, and then the reciprocal influences among variables were identified. For instance, some capabilities reflecting organizational culture (e.g. the availability of human resources, the availability of a good knowledge concerning the infrastructure and the environment) which may have a direct influence on system resilience (i.e. they support a quick and effective response to technical issues) are explicitly included. The role of the social dimension contributing to resilience, particularly focuses on the characterization of how behaviors, attitudes and awareness of the served population may either help or hamper resilience. The model mainly aims at providing information on the water deficit during emergency and in the immediate aftermath, performing a comparison between water inflow and water demand.

Specific sub-models are identified by purple boxes and defined in order to deal with key issues contributing to resilience according to the TOSE conceptualization. Most of the sub-models are mutually interconnected, and can be run and analyzed independently. Each of the sub-models considered provides a quantitative insight into the main dynamics of the specific dimensions of the TOSE approach.

SD based resilience assessment model

The model can be also used in order to perform a scenario analysis: this may support decision-makers to understand the impact of different strategies, conditions, assumptions on the response of the system. The key outcome of modeling is the analysis of the evolution of water deficit with time, which is a way of interpreting and representing the resilience triangle. The comparative analysis of multiple scenarios helps describing the impact of different states of specific variables on the model outcomes. Starting from the Scenario 0, which reconstructs the ‘real’ evolution of the events after the earthquake, the others are built for the purpose of comparison as follows: ‘Scenario 1’ shows how a decrease in organizational skills may have a dramatic influence on system response; ‘Scenario 2’ simulates a decrease of ‘infrastructure physical vulnerability’ which, although expensive has a definitely positive impact; ‘Scenario 3’ describes instead an integrated strategy where infrastructural improvement actions are supported by also by a better ‘knowledge of critical points’, ‘training level’ of the personnel and enhancement of ‘community awareness’.

Results of the resilience assessment in terms of ‘Water deficit’ for the modeled scenarios.

The results were discussed in details with the experts, and clearly underlined that the role of ‘non-structural’ measures on soft infrastructural system might be important as well as structural ones to increase resilience. Particularly, acting on several ‘cultural’ issues related to both individual (e.g. people awareness) and organizational features (e.g. cooperation, training level, knowledge) may have a benefit comparable with the one associated to the implementation of structural measures.

L’Aquila earthquake (2009)

L’Aquila city has a long history of disastrous earthquakes (1461, 1703, 1915, 2009). The earthquake in 2009 struck L’Aquila province at 3.32 a.m. on 6 April 2009. As a consequence, 308 people died and 1500 resulted injured. Although the magnitude was moderate, the impacts were high, mainly due to the high urban vulnerability.

L’Aquila city had experienced multiple major earthquakes in the past. A disastrous earthquake occurred in 1461, with an estimated magnitude of 6.5 and approximately 150 victims. Another relevant earthquake occurred in 1703 (it was a part of a significant earthquake sequence), killing approximately 6000 people in the city and its surroundings. Referring to the recent history, an event in January 1915 killed 32500 people, including almost the whole population of Avezzano, 50 km south of L’Aquila (Alexander 2014).

The case study refers to the disastrous Magnitude 6.3 earthquake which struck L’Aquila city and its province at 3.32 a.m. on 6 April 2009. As a consequence of the event, 308 people died and 1500 were injured. Although the physical event was relatively moderate (moment magnitude 6.3), its impacts were particularly high mainly due to the very high vulnerability of lives, livelihoods, building stock and institutions in the Apennine Mountains. The physical vulnerability level of its masonry buildings (poorly maintained and not strengthened), mainly located in the historical city center, led to enormous damages. Reinforced concrete structures were affected as well. Surprisingly, more casualties were due to the collapse of reinforced concrete buildings than of the masonry ones, due to their higher vulnerability (Contreras et al. 2014).

The role of hard infrastructure (e.g. water supply infrastructures) at urban level supported efforts of local communities during the emergency phase, proving to be a key asset to cope with the disaster.

More in general, the earthquake led also to a series of scandals and controversies that lasted for years and are still ongoing, revealing the vulnerability of the ‘institutional’ framework. In the aftermath of the disaster, one of the most controversial developments was indeed related to the behavior of institutions and scientists, and their information sharing with the community (Alexander 2014).

Main features of L’Aquila earthquake in 2009. Source: USGS

Further details on the main features of the earthquake are available at gemecd.org and Google Maps.

Damage in the city center
Damage in the city center
Damage to the water supply systems

One of the most controversial issues was the trial and prosecution of seven functionaries of the Italian National Department of Civil Protection (DPC), mainly due to the kind of information (“incomplete, imprecise and contradictory on the nature, causes, dangers and future development of seismic activity in the area in question” - Il Centro 2012) shared with the community. Some citizens had acted on that information and as a consequence had lost their lives.

This had a strongly negative impact on the trust level of local community toward the emergency managers, with consequences on the acceptability of the following emergency management and recovery measures. After the earthquake, the local community was forced to abandon the city center. New towns were developed in safer places, disaggregating the original socio-cultural networks. New networks emerged after the disasters, showing different cultural aspects.

Further details on L’Aquila trial can be found at: eagris2014.com.

Hard/soft infrastructural systems in L’Aquila: water supply

Lifelines are highly vulnerable elements in emergency condition, but represent a key resource to support a quick and effective recovery. The infrastructural system of a city is linked with social and institutional systems, but also with the economic and environmental ones that are all embedded within the urban context and dynamically interacting.

After a disaster strikes, a prompt return to the status quo is needed. Nevertheless, simply rebuilding communities to pre-disaster standards would recreate the vulnerabilities that existed earlier and expose them to future disasters. Reconstruction is generally an opportunity to build back better. It is the restoration and improvement of facilities, livelihoods and living conditions of disaster-affected communities, including efforts to reduce disaster risk factors.

This “build back better” approach1 advocates for the restoration of communities and assets in a manner that makes them less vulnerable to disasters and strengthens their resilience. Disaster risk reduction measures should be included into post-disaster recovery and rehabilitation processes. Resilient recovery and reconstruction are widely recognized as imperative for sustainable development.

Recovery thus represents much more than a return to the pre-event state. Recovery actions can also promote both physical and economic resilience, and prompt or facilitate investment in infrastructure upgrades and urban revitalization. Resilient recovery and reconstruction can be realized through a variety of strategies: enhancing preparedness; relocating critical facilities to safer areas; integrating disaster risk reduction measures into infrastructure improvements; strengthening governance structures, including the development of institutional mandates for disaster risk management; using the reconstruction process to address urban planning challenges; and establishing predictable contingent financing mechanisms, including disaster risk financing.

The issue of ‘Building back better’ emerges after major disaster, like the L’Aquila earthquake in 2009. As a consequence of the earthquake and of its impacts on the built environment, L’Aquila is still undergoing a complex process of reconstruction. Particularly, on the one hand the extent of damages in the whole urban area limited the functionality of infrastructures and the accessibility for community; on the other hand, the changes in the population localization due to both temporary sheltering strategies and to the evolution of new permanent areas, forced a radical change of the performances required to the infrastructures. Particularly for the purposes of EDUCEN project, the water supply system represented the key infrastructural system to analyze.

The experience and the knowledge developed during the earthquake and in the aftermath of the disaster, provided crucial information to support the reconstruction phase. Learning from past errors and from the key criticalities encountered was a fundamental step for an innovative, sustainable, effective, safe, ‘resilient’ design. Just to provide an example, the high uncertainty of the available information and the poor accessibility of some infrastructures often limited the possibility to operate promptly during the emergency; similarly, the need to adapt the whole network to both changes in the urban pattern and specific local needs (e.g. the need to provide some buildings with water using a network with a huge number of breaks) during the reconstruction phase, caused significant stress levels for the system. The urban critical infrastructural systems were thus deeply rethought, and redesigned according to the new needs of the city, and to the experience.

One of the main damages in 2009: Gran Sasso Aqueduct

The design of the ‘SMART TUNNEL’ reflects a basic principle: electricity, gas, water and communication systems are key services supporting daily activities and the well-being of a community (http://www.sottoserviziaq.it/it/home.html). The basic idea behind the smart tunnel is simply to collect and integrate all the critical services in an ‘invisible’ shell, i.e. an underground concrete gallery, in order to protect them from external threats and make them easily accessible and repairable, both in case of disasters and in ordinary operation.

Providing safe drinking water to a community in case of disasters is one of the main commitments of emergency managers and local authorities. Particularly, the urban water distribution network of L’Aquila city, is being currently rebuilt according to innovative criteria, such as the distrectualization. The basic idea is to split the whole network into a number of subsystems characterized by spatial and functionality homogeneity in order to facilitate maintenance and management procedures. Distrectualization allows: a) controlling leakages and water losses; b) isolating single subsections of the whole network; c) implementing more effective measurements of hydraulic parameters. The distrectualization supports flexibility and adaptation capability to the evolution of the urban pattern, and thus is strongly connected to the evolution of the whole city.

The L’Aquila case study is unique and relevant also because it allows the comparative analysis of two different networks operating within the same urban pattern. The urban water distribution system was completely redesigned after the disaster and is currently being built. The same methodology was implemented to assess both infrastructural configurations, and selected metrics used to compare systems. In the following images the two systems are depicted (‘OLD’ and ‘NEW’ networks), along with their representation according to Graph Theory formalization.

Representation of the urban water distribution network: old network
Representation of the urban water distribution network: new network
Representation of the urban water distribution network according to graph theory: old network
Representation of the urban water distribution network according to graph theory: new network

The selected measures, according to graph theory computed in an undirected and unweighted version, are summarized in the following Table 1. Full details on the selected measures are available in the section ‘Methodological approaches: SDM & Graph Theory – Graph theory’. It is worth mentioning that the ‘NEW’ network is made of two independent subnetworks (‘CS’ – Centro Storico and ‘ZM’ – Zona Media), and thus the metrics are computed independently. The results generally contribute to suggest a better resilience of the new network, particularly in terms of flexibility, robustness and redundancy. Nevertheless, a comprehensive analysis should also be coupled with hydraulic models and with suitable performance indices.

Table 1. Overview of the results of Graph theory analysis

Network q k d lT Cc Rm CB Dap Dbr λ2 Δλ fc
New – CS 0.006 3 26 13.43 0.041 0.252 0.412 0.1 0.11 0.0027 0.3869 0.5
New – ZM 0.013 2.63 23 10.74 0.02 0.162 0.584 0.285 0.51 0.004 0.3798 0.387
Old 0.0006 2.15 97 32.76 0.004 37 0.455 0.391   0.00041 1.1247 0.127

Methodological approaches: SDM e Graph Theory

System Dynamics Modeling

System dynamics is a computer-aided approach to policy analysis and design. It applies to dynamic problems arising in complex social, managerial, economic, or ecological systems, which are characterized by interdependence, mutual interaction, information feedback, and circular causality (Richardson 1999).

Starting from the seminal work by Forrester on Industrial Dynamics (1961), the span of applications grew from corporate and industrial problems to include the management of research and development, urban stagnation and decay, commodity cycles, and the dynamics of growth in a finite world. SD is now applied in economics, public policy, environmental studies, engineering, defense, theory-building in social science, and other areas, as well as its home field, management.

The system dynamics approach involves:

  • Defining problems dynamically, in terms of graphs over time.
  • Focusing on the characteristics of a system that themselves generate or exacerbate the perceived problem.
  • Thinking of all concepts as continuous quantities interconnected in loops of information feedback and circular causality.
  • Identifying independent stocks or accumulations (levels) in the system and their inflows and outflows (rates).
  • Formulating a model capable of reproducing the dynamic problem of concern.
  • Deriving understandings and applicable policy insights, and implementing changes.

SDM consists of qualitative/conceptual and quantitative/numerical modelling methods. Qualitative modelling, e.g. based on causal loop diagrams, improves our conceptual system understanding. Quantitative modelling, e.g. stock-and-flow models, allows to investigate and visualize the effects of different strategies through simulation.

Causal loop diagrams and then, stock and flow diagrams, model relationships among variables which have the potential to change over time. Such models distinguish between different types of variables: there are stocks (or level or accumulation) and flows (or rate). A stock is a measurable accumulation of physical or non-physical resources, whereas a flow is the movement of something from one stock to another. Generally, stocks are graphically expressed as boxes, whereas flows are represented by arrows. It is interesting to consider that almost every business process, and its related components, can be expressed in terms of stocks and flows.

Graphical representation of stock and flow notation, with the classical example of population change.

One of the key elements in SD is the presence of loops, either reinforcing or balancing. A reinforcing loop is one in which an action produces a result which influences more of the same action thus resulting in growth or decline. A balancing loop attempts to move some current state to a desired state though some action. A balancing loop is representative of any situation where there is a goal or an objective and action is taken to achieve that goal or objective.

SDM is a particularly effective modeling solution if the system is complex and an analytical solution could be excessively time consuming or simply impossible. It shows significant capabilities for modeling decision-making processes and human behaviors, thus being particularly useful for analyzing organizational evolution. Such approach may reveal really useful in describing the way policies, delays, and structures are related, and how they influence the stability of the system. The strength of SDM also lies in its ability to account for nonlinearity in dynamics, feedbacks, and time delays.

The use of a SD model supported us in identifying and analyzing the main elements fostering or hampering resilience. The model has been used to evaluate the impact of actions and strategies for resilience improvement on the dynamic evolution of the system. Finally, it has been used to identify critical feedbacks, and to evaluate their influence on the implementation of policies aiming to enhance LS resilience, assessing their evolution with time.

An SDM project typically consists of the following phases:

  • problem definition
  • system conceptualization: Define the purpose of the model; define the model boundary and identify key variables; describe the behavior or draw the reference modes of the key variables; diagram the basic mechanisms and the feedback loops of the system
  • model formulation: Convert feedback diagrams to level and rate equations; Estimate and select parameter values.
  • model evaluation/testing: Simulate the model and test the dynamic hypothesis; test the model’s assumptions; test model behavior and sensitivity to perturbations
  • policy analysis and implementation: test the model’s response to different policies; Translate study insights to an accessible form.

The development of SD models is helpful for an improved system understanding, and the development of a tool to analyze and evaluate strategies and policies, and the testing of theories.

Suggestions for further reading: Forrester, J.W. 1961. Industrial Dynamics. Cambridge, MA: The MIT Press. Reprinted by Pegasus Communications, Waltham, MA. Sterman, J.D. 2000. Business Dynamics: Systems Thinking and Modeling for a Complex World. Boston: Irwin McGraw-Hill. Vennix, J. A. M. 1996. Group Model Building: Facilitating Team Learning Using System Dynamics. Chichester: Wiley.

Graph Theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, or lines. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another. A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or weighted graphs, are used to represent structures in which pairwise connections have some numerical values. Formally, a graph is a pair of sets (V, E), where V is the set of vertices and E is the set of edges, formed by pairs of vertices.

Graph with sets of vertices and edges

Graphs can be used to model many types of relations and processes in physical, biological, social and information systems. Many practical problems can be represented by graphs. Emphasizing their application to real-world systems, the term network is sometimes defined to mean a graph in which attributes are associated with the nodes and/or edges. Graphs are used to represent networks of communication, problems in social media, sociology, biology, and many other fields. Graph theory is also used to study molecules in chemistry and physics, in computational neuroscience , in engineering studies (e.g. to represent the micro-scale channels of porous media).

Graph Network Theory has been widely used for the analysis of engineering systems, such as infrastructures, particularly in the field of transportation or water supply (e.g. Yazdani et al. 2011, Torres et al. 2016).

The topology of critical infrastructures, such as water distribution systems, can be easily described through the definition of a network of nodes connected by links. The structure of the network itself reflects the characteristics of the underlying network of users of the service, both in terms of spatial pattern and in terms of service requirements and expected performances (e.g. the nodes reflect the water demand). The topology and the functionality are thus two strictly intertwined aspects of the operation of the whole system. This underlying pattern of users, and their specific needs, evolves with time during the whole emergency, and thus the network should be flexible enough to provide a sufficient level of service, although somehow limited, even in case of dramatic changes in some conditions (e.g. extensive damages in the network).

Such a network is typically governed by complex structures and dynamical processes, due to the large number of interconnected and interacting components. A quite simple way to model such networks is to represent the structure of the system through a mathematical graph, collecting nodes to represent specific elements and links to represent the pipes between nodes.

The study of complex networks by using graph theory helps with the classification of different networks and with the analysis of the influence of their shape and connectedness on the vulnerability, robustness and tolerance. Structural measurements may quantify the connectivity patterns among the network components. These metrics become trivial in WDNs which exhibit low redundancy and sparseness at transmission or subsystem level. The structural network measurements can be classified in several ways, but mainly in statistical and spectral forms. All the topological metrics can be used to assess the reliability of complex networks, describing the influence of the underlying structure and connectivity constrains on network behavior. The selection of a set of indicators/metrics allows identifying the key elements to support proper network operation, to prioritize actions to deal with emergency and to check whether a sub-network of users (e.g. the “critical” ones, such as the system of hospitals, shelter camps, …) can be supplied by the hard infrastructural system in variable operating scenarios.

Following the available literature, the key measures to be considered in infrastructural modeling, are summarized in the following:

  • Link density q (Network density) is the most basic indicator of the linkedness or sparseness of the structure of a network.
  • Average node degree k is a basic measure of node connectivity. It reflects the overall topological similarity of the network to perfect grids or lattice-like structures.
  • Network diameter d: captures the maximum eccentricity of nodes in the network.
  • Average path-length lT: estimates the average number of links that need to be traversed in order to reach from one point to another, representing network reachability and efficiency.
  • Clustering coefficient Cc, is used to measure the redundancy by quantifying the density of triangular loops and the degree to which junctions in a graph tend to be linked.
  • Meshed-ness coefficient Rm provides an estimation of topological redundancy by finding the number of independent loops as a percentage of the maximum possible loops.
  • Central-point dominance CB measures the concentration of the network topology around a central location. It quantifies network vulnerability against failures.
  • Density of articulation points Dap estimates the percentage of the nodes/junctions whose failure may potentially disrupt water delivery by isolating a part of the network.
  • Density of bridges Dbr estimates the percentage of the links/pipes whose failure may potentially disrupt water delivery by isolating a part of the.
  • Spectral gap Δλ is computed as the difference between the first and second eigenvalues of the adjacency matrix. Small spectral gap would probably indicate the presence of articulation points or bridges.
  • Algebraic connectivity λ2: higher values suggest better network’s structural robustness and fault tolerance.
  • Critical ratio of defragmentation fc provides a theoretical value for the critical fraction of the nodes which need to be removed for a network to lose its large scale connectivity.

The results of the implementation of Graph Theory in L’Aquila CS, with specific reference to the drinking water supply system, are described in: Hard/soft infrastructural systems in L’Aquila: water supply

Suggestions for further reading

Alexander, D. E. (2014). Communicating earthquake risk to the public: The trial of the L’Aquila Seven. Natural Hazards, 72, 1159–1173.

Contreras D., Blaschke T., Kienberger S., Zeil P. (2014). Myths and realities about the recovery of L’Aquila after the earthquake, International Journal of Disaster Risk Reduction, 8:125–142.

Forino G. (2015) Disaster recovery: narrating the resilience process in the reconstruction of L’Aquila (Italy), Geografisk Tidsskrift-Danish Journal of Geography, 115:1, 1-13, DOI: 10.1080/00167223.2014.973056

Kongar I., Esposito S., Giovinazzi S. (2015). Post-earthquake assessment and management for infrastructure systems: learning from the Canterbury (New Zealand) and L’Aquila (Italy) earthquakes, Bull Earthquake Eng DOI 10.1007/s10518-015-9761-y

Rossetto T., Peiris N., Alarcon J.E., So E., Sargeant S., Free M., Sword-Daniels V., Del Re D., Libberton C., Verrucci E., Sammonds P., Walker J.F. (2011) Field observations from the Aquila, Italy earthquake of April 6, 2009, Bulletin of Earthquake Engineering 9, 11–37, doi: 10.1007/s10518-010-9221-7

References

Alexander D. E. (2014). Communicating earthquake risk to the public: The trial of the L’Aquila Seven. Natural Hazards, 72, 1159–1173.

Bruneau, M., Chang, S. E., Eguchi, R. T., Lee, G. C., O’Rourke, T. D., Reinhorn, A. M., et al. (2003). A framework to quantitatively assess and enhance the seismic resilience of communities. Earthquake Spectra, 19(4), 733–752.

Cimellaro G.P. (2016). Urban Resilience for Emergency Response and Recovery. Fundamental Concepts and Applications. Springer International Publishing Switzerland. ISSN 1573-6059

Contreras D., Blaschke T., Kienberger S., Zeil P. (2014). Myths and realities about the recovery of L’Aquila after the earthquake. International Journal of Disaster Risk Reduction, 8, 125-142.

Jacob M. Torres, Leonardo Duenas-Osorio, Qilin Li, and Alireza Yazdani. “Exploring Topological Effects on Water Distribution System Performance Using Graph Theory and Statistical Models,” January 1, 2017. doi:10.1061/(ASCE)WR.1943-5452.0000709.

Pagano A., Pluchinotta I., Giordano R., Vurro M. (2017). Drinking water supply in resilient cities: Notes from L’Aquila earthquake case study. Sustainable Cities and Society 28: 435–449

Richardson G.P, System Dynamics. In Encyclopedia of Operations Research and Management Science, Saul Gass and Carl Harris, eds., Kluwer Academic Publishers, 1999/2011

Tierney, K., & Bruneau, M. (2007). Conceptualizing and measuring resilience: A key to disaster loss Reduction.TR NEWS 250. pp. 14–18

Yazdani, A., R. Appiah Otoo, and P. Jeffrey. “Resilience Enhancing Expansion Strategies for Water Distribution Systems: A Network Theory Approach.” Environmental Modelling & Software 26, no. 12 (December 2011): 1574–82. doi:10.1016/j.envsoft.2011.07.016.