Probabilistic Response of Interdependent Infrastructure Networks

Transcription

1 Probabilistic Response of Interdependent Infrastructure Networks Leonardo Dueñas-Osorio, James I. Craig, and Barry J. Goodno ABSTRACT Interdependent infrastructures constitute one of the most visible examples of contemporary complex networks. These networks are formed by coupling different independent networks (e.g., power grid, water distribution, gas transmission, transportation, emergency response buildings, etc.). These interdependent systems must remain functional after natural or man-made disasters for the safety and well-being of affected communities. Understanding the dynamic response of independent networks to natural or deliberate hazards has been, and still is, a subject of substantial investigations. However, understanding the dynamic response of interdependent networks represents an even grander modeling challenge. This study attempts to address this challenge by dividing the problem into analysis of static topological properties, and then analysis of the effects of those properties in dynamic response. Static characterization of networks requires knowledge of its adjacency matrix, which defines the topologic layout of the network, and the degree of coupling among its components. Different network characteristics such as global connectivity, local clustering, and global complexity are introduced. Variability in those properties represents the behavior of the network when subjected to re-edging, growth, and removal of components. A discussion of accepted theoretical network models is introduced. Also, a detailed network model is proposed in this research to capture essential features of growth and evolution within interdependent networks. Dynamic response is investigated through time-dependent properties such as network resilience and fragmentation modes. Using a small-world network model, variation of topological properties as a function of disruption severity is analyzed. Efforts are made to determine if there correlations exist among failure modes, network component removal strategies, and network topology. Different network component removal strategies are utilized, but investigation of an earthquakeinduced removal strategy is stressed and illustrated using a theoretical model of a small-scale testbed. Earthquake-induced removal is directly related to the physical vulnerability of each network component. Interdependent fragility curves, one of the novel outcomes of this work, enable synthesis of network topology with network component conditional probabilities of exceeding predefined performance levels. Finally, the potential enhancement of consequence minimization effectiveness and loss estimation accuracy is highlighted if fully coupled probabilistic interdependent network analyses are implemented. Leonardo Dueñas-Osorio, Mid-America Earthquake Center, School of Civil and Environmental Engineering, Georgia Institute of Technology, 790 Atlantic Drive, Atlanta, GA , U.S.A. James I. Craig, Mid-America Earthquake Center, School of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Drive, Atlanta, GA , U.S.A. Barry J. Goodno, Mid-America Earthquake Center, School of Civil and Environmental Engineering, Georgia Institute of Technology, 790 Atlantic Drive, Atlanta, GA , U.S.A.

2 INTRODUCTION Critical infrastructures in the United States have become highly interconnected and mutually dependent in complex ways, both physically and through a host of information and communication technologies. Such interconnectedness creates a false sense of redundancy because while it is true that redundancy increases, its tractability is greatly reduced. This lost tractability may lead to hidden and unforeseen interactions among infrastructures that may result in cascading and escalating failures (DHS, 2003). Examples of interdependent infrastructures include interconnected civil networks such as power and water distribution grids, gas transmission lines, transportation systems, and emergency response buildings, among others. Generally, natural disasters and man-made hazards are the kinds of disruptive events that reveal the inherent interconnectedness among critical infrastructures and expose their vulnerabilities. In particular earthquakes can trigger a strong geographical and physical correlation capable of generating unexpected failure modes and are able to expose weaknesses in analysis, design and construction of networks based upon best-practice procedures. In order to understand the response and failure evolution of interdependent infrastructures, this study proposes a model for constructing interdependent networks. The model is able to incorporate properties of geographical extent, structural vulnerability, network component coupling, demand and supply capacities, population growth, and regional development plans. Dynamic analyses are also performed using a simplified network model. It allows quantification of impacts due to deletion of network components, and reveals possible correlation among network topology, strategy of network component removal, and failure modes. GRAPH THEORY A network, or in more mathematical terms, a graph, is a set of components referred to as vertices with connections between them called edges. These components can have different properties within the network in terms of their function, ability to connect with others, preferential attachment, and importance. They also have different probabilities of failure given natural or manmade hazards (see Figure 1). Network research has increased in recent years, with the focus shifting away from the analysis of single small networks and the properties of individual vertices or edges within such networks, to consideration of large-scale statistical properties of graphs (Newman, 2003). Vertex Edge Figure 1. Fundamental components of generic networks. The current body of theory has three major objectives: (1) to find and highlight statistical properties, such as path lengths, degree distributions, long-range connections, and resilience, that characterize the structure and behavior of networks; (2) to create graph models that reproduce essential properties of real networks, allowing understanding of statistical properties; and (3) to

3 predict what the behavior of networks would be on the basis of measured topologic properties, external disruptions, and local physical rules governing vertices and edges. A more rigorous definition of a graph is taken from Wilson and Watkins (1990): a graph G consists of a nonempty set of elements, called vertices, and a list of unordered pairs of elements, called edges. The set of vertices of the graph G is called the vertex set of G, denoted by V(G), and the list of edges is called the edge list of G, denoted E(G). If v and w are vertices of G, then an edge of the form vw is said to connect v and w. The number of vertices in V(G) is termed the order of the graph (n), and the number of edges in E(G) is termed its size (M). INTERDEPENDENT NETWORKS Interdependent critical infrastructures take the form of large complex technological networks, and for the central and eastern United States, they can have n ~ O(10,000) and M ~ O(100,000). These man-made networked systems typically permit trade of goods and distribution of resources, such as electricity, gas, potable water, or general commodities. Within critical infrastructure networks, the number of vertices, n, represents the inventory of network elements such as generators, substations, and control facilities; while the edges, M, represent the inventory of some predefined coupling between elements, for instance, transmission lines, distribution pipelines, and roads. Modeling these kinds of networks requires knowledge of the nature of their interdependencies. Such interdependencies need to be quantified and stored in inoperability matrices, I, whose (i th, j th ) elements define the probability of inoperability that failure in the j th infrastructure component triggers in the i th infrastructure component. These i th and j th elements also represent the degree of interconnectedness among network components (i.e., define importance of interdependent network edges), and can be used to assemble adjacency matrices, A, which dictate the topologic layout of the network. Interconnectedness indices can be calculated by quantifying each of the following dimensions: (1) type of interdependencies, (2) coupling and response behavior, (3) infrastructure characteristics, (4) infrastructure environment, (5) type of failure, and (6) state of operation (Rinaldi et al., 2001). This quantification requires a Bayesian approach, which is referred to as pre-posterior analysis. With this method, subjective judgments based on experience are incorporated systematically with observed and measurable data to obtain a balanced estimation (Ang and Tang, 1975). The first two dimensions play a major role in defining the interconnectedness amongst network components. The type of interdependency dimension is subdivided into physical, geographical, informational, and logical categories. Physical interdependencies arise when infrastructures depend on tangible linkages among them. This type of interdependency is suitable for quantification with low uncertainty. Geographical interdependencies depend on the spatial proximity to the area of the catastrophic event. Informational and logical interdependencies emerge due to the contemporary need for exchange of data, computerized control, and intervention of human decisions. The coupling and response behavior dimension is subdivided into degree of coupling, coupling order, and complexity. The degree of coupling refers to the tightness or looseness of the interdependency. In highly coupled infrastructures, disturbances tend to propagate rapidly through and across them. Coupling order is concerned with whether two infrastructures are directly connected to one another or indirectly coupled through one or more intervening infrastructures. Complexity of the links represents linearity or nonlinearity of the interactions among infrastructures. The infrastructure characteristics and infrastructure environment dimensions are accounted for in the topology of the network, and in the hazards that threaten network integrity. Lastly, the type of

4 failure and state of operation dimensions largely depend on the nature of the disruption, and the fragility of network components. Once the inoperability matrix, I, is reasonably assembled for a particular geographical area of analysis, there is a need for tractable and reliable models that capture the essential characteristics of critical infrastructure networks. Desirable model characteristics are their ability to reproduce interdependent responses, as well as to replicate observed resilience, failure modes, network growth, and fragmentation evolution. Other characteristics can initially not be considered. For instance, directionality of edges, or refinement of network growth beyond generation and distribution levels (e.g., individual user level) may play unimportant roles in capturing overall network topology. TOPOLOGIC PROPERTIES OF NETWORKS Independent of the models used to represent the shape and evolution of real networked systems, networks exhibit certain properties that are inherent to their topology. Calculation of such properties allows comparison amongst abstract models and calibration of them against real networks. Vertex Degree and Distribution The vertex degree, k v, of undirected graphs (i.e., graphs where edges are bidirectional) is the number of edges connected to a vertex. This parameter computed for all v V(G) permits determination of its cumulative distribution, which for most networks in nature can be described by power laws (Albert et al., 2000). However, some of the man-made technological networks exhibit exponential cumulative distributions. A relevant example is the power grid of the western United States (Watts and Strogatz, 1998). One of the research issues under consideration is determination of the actual vertex degree distribution for an interdependent network, whose components may exhibit either power law or exponential cumulative distributions. So far, it is known that networks with vertices displaying similar degree, which do not depart substantially from the mean degree, show Poisson distributions. When the number of vertices, n, is very large, the vertex degree cumulative distribution tends to be exponential. On the other hand, when the vertex degree displays large variations, i.e., low degree values coexist with extremely large and infrequent degree values, the overall vertex degree distribution follows power laws. Most social, informational and biological networks show power law distributions. Characteristic Path Length The characteristic path length, L, of a graph is the median of the means of the shortest path lengths, d(v i, v j ), connecting each vertex v V(G) to all other vertices. This parameter can be regarded as a global indicator of network connectivity. In large graphs this quantity can be calculated as: L 1 = d(vi, v j ) n(n 1) (1) i j i If L is large, the dynamics within the network are slow due to low connectivity. In the frequent event that any two vertices are not connected at all, or become disconnected due to external

5 disruption, their shortest path length d becomes infinity. One way to handle infinite values is to calculate the inverse characteristic path length, L -1, where 0 L -1 1, and now low L -1 indicates low connectivity. This parameter can be calculated by slightly modifying equation (1): Clustering Coefficient L = n(n 1) d(v, v ) (2) i j i i j An important concept in networks, referred to as neighborhood, is necessary for clustering quantification. The neighborhood, Γ(v) of a vertex v is the subgraph that consists of vertices adjacent to v without including v itself. Then, the clustering coefficient, γ v uses Γ(v) to characterize the extent to which vertices adjacent to any vertex v are adjacent to each other (Watts, 1999). More precisely, E( Γv ) γ v = (3) kv 2 where E(Γ v ) is the number of edges in the neighborhood of v and the denominator represents a binomial coefficient, which amounts for the total number of possible edges in Γ(v). This coefficient can be regarded as a local measure of transitivity, or in more general terms, it measures how connected the network is in local scales (i.e., at the scale of the neighborhood of the vertices). Fraction of Shortcuts The range of an edge R(i,j) is the length of the shortest path between i and j in the absence of that edge. An edge with R(i,j) > 2 is called a shortcut. These types of edges exhibit long-range connectivity; therefore, they are able to connect vertices that are not in the same neighborhood. Given a graph of M edges, the fraction of those edges that are shortcuts is denoted by 0 φ 1. This parameter can be regarded as a global measure of the topologic irregularity or randomness within the network. It is important to highlight the correlation that this parameter has with the clustering coefficient and the inverse characteristic path length. For example, if φ is measured for any particular network, and its value is close to 1, the global inverse characteristic path length L -1 will tend to 1, and the clustering coefficient γ will tend to 0. Most real networks, however, display simultaneously high values of both global connectivity and local clustering. NETWORK MODELS Development of theoretical models that predict responses of real complex networks has become a multidisciplinary effort. Models have moved from static random networks to growing networks. A description of the most significant models and their incremental improvements is presented, including the model proposed by the authors to represent interdependent critical infrastructure network topology, growth, and response to natural or man-made hazards.

6 Random Networks Forty years ago a model to represent complex networks was proposed by Erdös and Rényi, Their model corresponded to a random graph, where links (edges) are placed completely randomly, i.e., all vertices have the same probability of having a link. This model turns out to be structurally unbiased in the sense that for a large graph all nodes have approximately the same number of links. In other words, its vertex degree cumulative distribution turns out to be exponentially distributed. Also, random graphs exhibit large inverse characteristic path length L -1, low clustering coefficient γ, and high shortcut parameter φ. These properties depart from reality in the clustering coefficient, which is in general higher, and in the vertex degree distribution, which for most networks follows a power law. However, technological networks such as the power grid show vertex degree exponential distributions. Small World Networks In order to capture simultaneous high local clustering and high global connectivity, an alternative to the random network model was offered by Watts and Strogatz, Their model starts with the construction of a regular one-dimensional network, referred to as 1-lattice, without weighted or directed edges. The vertex degree k, is assumed to be the same for all vertices. This means that every vertex is connected to its nearest k/2 vertices. Figure 2 shows an example of a particular network with n = 20 and k = 6. Then, each edge is randomly rewired, with probability β, using the following algorithm (Watts, 1999): 1. Each vertex i is chosen along with the edge that connects it to its nearest neighbor in a clockwise sense (i, i+1). 2. A uniform random deviate r is generated. If r β, then the edge (i, i+1) is unaltered. If r < β, then (i, i+1) is deleted and rewired such that i is connected to another vertex j, which is chosen at random from the entire graph (excluding repetitions and self-connections). Isolated Vertex Figure 2. One dimensional lattice of order, n = 20, and vertex degree, k = When all vertices have been considered once, the procedure is repeated for edges that connect each vertex to its next-nearest neighbor (i+2), until k/2 such rounds are completed. When rewiring probability β = 0, the resulting graph remains precisely a 1-lattice, and when β = 1, all edges are rewired randomly, resulting in a close approximation to a random network. In an

7 intermediate range of β [typically for β ~ O(1/n)], the network generated displays both high local clustering and high global connectivity. These are properties of small-worlds, as well as their exponential vertex degree cumulative distribution, which agrees with technological networks. A schematic view of the small world model is presented for the network of order, n = 20, and vertex degree, k = 6 (see Figure 3) (a) (b) (c) Figure 3. Effect of rewiring a one dimensional lattice of order, n = 20, and vertex degree, k = 6. (a) regular network, β = 0, (b) small-world, β = 0.05 ~ O(1/n), and (c) random graph, β = 1. Scale - Free Networks It has been discussed that most social, informational and biological networks are known for displaying power law distributions (Barabási and Albert, 1999). Random and small-world networks reproduce exponential vertex degree distribution in agreement with some technological networks. However, it is still a subject of investigation to determine whether a network of interdependent infrastructures displays power law, exponential, or any other distribution. Scale-free models become important because they reproduce networks with power law distributions. In addition, scale-free models incorporate growth and preferential attachment, both important features of real network evolution. A simple scale-free network is constructed with the following algorithm: 1. Start with two vertices, and at every new time step, t, add a new vertex to the network. 2. Assume that each vertex connects to the existing vertices with two edges. The probability that it will choose a given vertex, v i, is proportional to the number of links (i.e., vertex degree k v ) the chosen vertex has. This step is referred to as preferential attachment. 3. Repeat steps 2 and 3 until the network reaches a desired order, n. This network provides a scale-free topology in the sense that some vertices (in particular the older ones) may accumulate a disproportionate number of edges, resulting in no vertex degree that is typical of the others. In other words, the vertex degree distribution displays a power law. One drawback of this model is that it generates a low clustering coefficient γ, which departs from reality compared with most networks (Barabási, 2002). Therefore, a new step is added: 4. If an edge between v and u was added in the preferential attachment step, then add an edge from v to a randomly chosen neighbor w of u. This forms a triad that increases γ. Interdependent Infrastructure Networks Technological networks, such as the western United States power grid, have shown topological properties that agree with small-world models (Watts, 1999). The network has an order n = 4,941,

8 and an average vertex degree, k = Some statistical properties are: L = 18.7, γ = 0.08, and φ = For an equivalent random graph of this order and degree, the statistics would be: L = 14.89, γ = 0.005, and φ = Whenever network properties show, L L random, γ >> γ random, and φ 1, there is sufficient evidence of small-world behavior. In the present research, similar statistics are being calculated for selected interdependent infrastructure networks in the central region of the United States. However, field data is yet to be completely gathered and analyzed to determine topological properties. Therefore, while data processing is being completed, this study is focusing on development of a theoretical model to reproduce interdependent network response, growth and evolution. This model is based on regional utility network growth as a function of demand trends: 1. Define initial conditions in terms of number and type of vertices (e.g., generators, substations, pumping stations, control stations, distribution nodes), and in terms of number and type of edges (e.g., transmission lines, distribution lines). These conditions depend on the size of the target area for analysis. An initial schematic layout is shown in Figure 4. Transmission Edge Generation Node Control Node Distribution Edge v Distribution Node w Figure 4. Generic components of interdependent infrastructure networks. 2. Divide the chosen geographical region into smaller areas. It is common in the United States to use divisions referred to as census tracks. These units are on average 10 km 2 within the area that contains the city of Memphis. This city is located in the southwest corner of the state of Tennessee on the banks of the Mississippi River. This study uses a group of 216 census tracks contained within Shelby County. An average of 4,200 people live in a census track. The entire County has approximately 890,000 people distributed in an area of 2,000 km 2 (Shinozuka et al., 1998). Figure 5 depicts Shelby County boundaries and includes an independent infrastructure network. Intersections among pipelines are distribution nodes. Pump stations or storage tanks Distribution pipelines Figure 5. Water distribution network within Shelby County, Tennessee, U.S.A. at the pump station and storage tank level of resolution.

9 3. At every time step, t, assign to each existing vertex a parameter of available supply capacity, S w. New distribution vertices appear at the center of subdivisions of census tracks, referred to as census blocks, according to urban development projections. Each new vertex has an associated demand parameter, D v, dependent on the population density. 4. Attachment of new vertices to existing ones occurs in a preferential framework. Each existing node requires calculation of a preference parameter, p w, which is a function of the Euclidian distance between v and w, x vw, and the demand to supply ratio. The vertex with the highest preference parameter is chosen for connection. If demand D v > S w then p w = 0. Otherwise it is calculated as follows: p w 2 xvw Dv = 1 + (4) 3 Max v V(G) (xvw ) 3 Sw 5. In the event that p w = 0 for all vertices in the vertex set, V(G), a new generation node is included if it has been projected in the urban development plan. Otherwise, an intraconnection with a neighbor network of the same type (i.e., power grid with power grid) is included. Such intraconnection supplies the required demand of the new vertex when the existing network works at a demand to supply ratio Distribution nodes evolve into control nodes (e.g., substations, pumping stations, tanks) when their vertex degree exceeds its capacity of branching, which occurs at k w ~ O(10). 7. Each new generation or control node, v, should be linked with interdependent edges to vertices of analogous level in neighbor networks of different type (i.e., power grid with water distribution and gas lines). The type of infrastructures that connect to a particular vertex, v, is defined based on extrapolated trends of the adjacency matrix, A, among infrastructures. The selection of the vertices to connect with is performed based on an interdependent preference parameter, IP w, calculated as a function of the physical proximity to v, and the demand to supply ratio. A sample of the kind of interdependencies that exist among different networks is shown in Figure 6. Steps 3 to 7 are repeated until the interdependent network order reaches an estimated value, n, for a network life cycle of N years. Water network Water for production, cooling, and emissions reduction Power network Power for pump stations, lift stations, and control systems Water for production, cooling, and emissions reduction Gas for heat Gas network Power for compressors, storage, and control systems Gas for generators Figure 6. Illustrative interdependencies generated by growth and evolution.

10 DYNAMIC RESPONSE OF NETWORKS Once a network is developed using any of the models described above, the effect of natural or deliberate disruptions can be investigated. Dynamic responses are captured by two main properties: (1) network resilience, and (2) network fragmentation. Such properties are essential to describe the performance of a network when its components are being removed, and to identify whether or not its basic topological properties influence the response. Network Resilience Network resilience has been investigated as a property that characterized its capacity to remain connected after vertex removals. Such removals are performed incrementally until the fraction of removed vertices within a network of order, n, approaches 1 (Albert et al., 2000). In general, the effect of vertex removals on the inverse characteristic path length, L -1, is recorded to create plots of connectivity versus vertex deletion. Currently, the effect of edge removal is also being investigated (Holme et al., 2002). Different strategies are followed for systematic removal. The simplest is random removal. This amounts to deleting components of the network selected purely at random. A more elaborate removal strategy (e.g., deliberate attacks), selects vertices or edges based on their degree, and preferentially removes the most connected ones. In the context of natural hazards of large extent (e.g., earthquakes), the removal strategy corresponds to neither vertex deletion, nor to edge deletion alone. Instead, it involves simultaneous removal of vertices and edges, selecting components by following a relationship between earthquake hazard intensity and structural response of the physical components of the network. This relationship is referred to as a fragility curve, and represents the conditional probability of exceeding a particular performance level (e.g., full operability, partial functionality) given a particular hazard intensity level (e.g., spectral acceleration at the fundamental period of vibration of the structure, S a, or peak ground acceleration, PGA). Details on generation of fragility curves using approximate models of real structures, referred to as metamodels, are discussed in Dueñas-Osorio et al., In order to visualize the concept of resilience, a small-world network model with n = 100, and k = 10, has been generated (see Figure 7a). Resilience due to vertex removal is shown in Figure 7b for three different removal strategies: (1) random removal, (2) targeted removal based on initial configuration, and (3) targeted removal based on recalculated configuration after each removal. (a) (b) Figure 7. (a) Small-world network model of order n = 100, mean vertex degree, k = 10, and rewiring probability β = (b) Network model resilience to vertex removal.

11 It can be observed that strategic vertex removals are more effective in impairing network connectivity. Edge removal, on the other hand (see Figure 8a), is relatively insensitive to deletion strategies. This can be explained by the assumptions of the small-world model in the sense that the edge degree (i.e., product of the vertex degree of the nodes it connects) is typical for all edges. However, at high removal fractions, the shape of the network has changed and the egalitarian property is lost, therefore, strategic removals become dominant in disruption capacity. Figure 8b presents the effect of simultaneous vertex and edge removal for a random removal strategy. An additional assumed curve is included to represent the expected behavior of earthquake-induced removal of components. Validation of this assumed behavior is part of this ongoing study. The approach being implemented relies on analysis of dynamic response properties of an interdependent infrastructure network that matches the physical network layout in Shelby County, Tennessee, U.S.A. This information is also being used to calibrate and validate expected growth and evolution indicators of the proposed independent infrastructure network model. (a) (b) Figure 8. (a) Network model resilience to edge removal, and (b) Network model resilience to simultaneous vertex and edge removal (e.g., earthquake-induced hazard). Fragmentation and Fragment Size Distribution Knowledge of the effects of component removal on network connectivity is not enough to understand the mechanisms that produce connectivity variations. Therefore, a new property referred to as fragmentation captures the number and size of the portions of the network that become disconnected. There are two extreme failure mechanisms: (1) giant component fragmentation, and (2) total fragmentation. These mechanisms are illustrated in Figure 9a for the small-world model used in resilience analysis. The first mechanism, despite a large amount of removed components, retains a high level of global connectivity (Bollobás, 1985). The second mechanism simply collapses the network into small pieces. Figures 9b and 9c show the physical layout of the network after 80% of vertex removal following random and targeted initial strategies. Random vertex removal fragments the network into 8 clusters, one of them of order 6 (i.e., giant component). Targeted initial removal breaks the network into 10 clusters, one of them of order 4. Finally, targeted recalculated removal (not shown in the figure), explodes the network into 19 clusters, 18 of them of order 1 (e.g., isolated buildings), and one of them of order 2.

12 (a) Failure modes Giant component fragmentation (b) (c) Total fragmentation Figure 9. (a) Association of failure modes with network resilience. (b) and (c) Remaining network components of small-world model after 80% random and targeted initial removals. Keeping track of the fragmentation evolution permits determination of critical fractions of removed components (i.e., fraction of component deletion at which the network becomes disconnected), as well as determination of the effect that each removed component has on network response. In addition, it helps to determine whether a particular failure mode can be associated with a particular network model and a particular removal strategy. Identification of these types of correlations is essential for implementation of mitigation and recovery actions in real interdependent networks. For instance, it was shown that random and strategic edge removals for small deletion fractions induced similar fragmentation in networks in which any edge degree is representative of all others (i.e., networks with exponential vertex degree cumulative distribution). INTERDEPENDENT INFRASTRUCTURE VULNERABILITY In order to illustrate the way in which network component vulnerability determines removal strategies for natural hazards, a small scale scenario in Shelby County is constructed (see Figure 10). It contains four components of different network types: water, gas, power and transportation. Water Station (WS) Gas Supply (GS) Transportation (TR) Abstract Model Electrical Substation (ES) Figure 10. Small-scale scenario for prescribing interdependent infrastructure removal strategies.

13 Following the guidelines to quantify interconnectedness indices presented above, the inoperability matrix, I, for this scenario is assembled: I = WS TR GS ES For example, the fourth column indicates that if the electrical substation fails, there is a 40% probability of inducing inoperability in each of the other network components. The occurrence of a natural hazard is then defined. An earthquake with 2% probability of being exceeded in 50 years is assumed to occur in the New Madrid Seismic Zone. The probability that network component responses exceed the criteria to remain fully operational is obtained from their fragility curves. In order to perform a simplified response comparison, all components are evaluated at the spectral acceleration of the worst earthquake demand. This corresponds to the electrical substation building, which experiences S a = 0.30g at its fundamental period. To propagate the effect of interconnectedness in network fragility, a Leontief-based approach is implemented (Haimes, 2001). This approach essentially states that failure of one component is proportional to the combined damage effect of each of its coupled components. This procedure amounts to solving a system of simultaneous linear equations, whose variables correspond to the interdependent probabilities of failure. If instead of a scenario earthquake, a more systematic approach is followed to investigate the entire range of possible earthquakes, it is feasible to generate interdependent fragility curves. These are basically a collection of scenario-based analyses with monotonically increasing earthquake demands (see Figure 11). Once these types of plots are obtained, performing rigorous searches for optimal mitigation actions becomes feasible, even for large complex networks with hundreds of thousands of vertices. Optimal reduction of vulnerability allows finding retrofit measures to key network components so that the overall inoperability is minimized when subjected to limited resource availability. Additionally, optimal increase of connectivity among network components allows finding network topologies that maximize interconnectedness so that functionality can be restored rapidly and in a geographically representative area. WS TR GS ES (5) 1 Interdependent Fragility g Spectral Acceleration, Sa [%g] Water Transportation Gas Electricity Water Transportation Gas Electricity [ WS, TR, GS, ES] = [ 1.00, 0.72, 0.94, 0.55] [ WS, TR, GS, ES] = [ 0.70, 0.40, 0.50, 0.20] Interdependent Independent Interdependent fragility for scenario earthquake Independent fragility for scenario earthquake Figure 11. Independent and interdependent fragility curves for the illustrative example network.

14 Optimal, suboptimal and original network configurations permit calculation of expected losses before and after interventions with the advantage of explicitly incorporating the effect of network topology and component conditional probabilities of failure. In this way, loss aggregation methods are enhanced because correlation structures are embedded in the network response analyses. CONCLUSIONS Development of network models to represent real networked systems requires capturing not only the underlying topology of the network, but also its growth and evolution processes. This study proposed a model for constructing interdependent infrastructure networks. Ongoing research efforts are focused on verification of model properties against real interdependent networks located in the central United States. Using a simple small-world network model, which displays several properties of real networks (e.g., high clustering coefficient, γ, high inverse characteristic path length, L -1, and exponential vertex degree distribution), the effect of network component removal was investigated. It is shown that post-disaster connectivity depends on the removal strategy. Network responses exhibited different failure modes ranging from clustered failure to fragmented failure. This study speculated on the potential failure mode given earthquake-induced hazards. The possible impact was quite severe due to simultaneous vertex and edge removal, and large geographical extent of the impact. Ongoing research will refine models and evaluate results to support or disprove this hypothesis. The current theorized networks based on small-world models exhibited certain correlation among failure modes, removal strategies, and network topology, so additional efforts will investigate if the phenomenon can be generalized. Acknowledging the potential significance of earthquake-induced removal patterns, this study relates physical structural vulnerability and seismic hazard with prescription of network component failures. It states that the removal strategy for this natural hazard comes directly from the interdependent conditional probability of exceeding an operational performance level. Such values are plotted within interdependent fragility curves, which come from Leontief-based models that couple independent probabilities of failure. Metamodels are used to approximate the dynamic response of different independent network components (e.g., power generation buildings, tanks, pump stations, substations, transmission towers, pipelines, etc.). These approximations allow affordable numerical simulations to obtain independent fragility curves. Dynamic network models and responses to natural or deliberate disruptions can now be seen as potential tools to also estimate monetary losses. This is because dynamic models allow direct mapping of failed components with repair costs by embedding network component correlation in their formulation. In this way, mitigation actions can be more effectively prescribed based on optimal allocation of resources that accounts for the effects of network physical topology and conditional probabilities of network component failures. Further validation of interdependent model growth and evolution as compared with real networks will be the focus of future work. Bayesian updates of adjacency matrices also represent an area for additional research. Finally, improvements are needed in theoretical network development, so that severe modeling assumptions can be removed. For example, all models presented, including that proposed for interdependent networks, used assumed undirected edges. Finally, an incremental step to improve modeling would involve the use of directed graphs to capture interconnectedness asymmetries.

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