# probabilistic response of interdependent infrastructure networks

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Microsoft Word - Duenas-Osorio_Leonardo_MAEC_FINAL.docLeonardo 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 earthquake- induced removal strategy is stressed and illustrated using a theoretical model of a small-scale test- bed. 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 30332-0355, U.S.A. James I. Craig, Mid-America Earthquake Center, School of Aerospace Engineering, Georgia Institute of Technology,

270 Ferst Drive, Atlanta, GA 30332-0150, 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 30332-0355, U.S.A.

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 man- made 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

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

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 (ith, jth) elements define the probability of inoperability that failure in the jth infrastructure component triggers in the ith infrastructure component. These ith and jth 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

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, kv, 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

∑∑ ≠−

1L (1)

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

∑∑ ≠−

Clustering Coefficient

v v k

E γ (3)

where |E(Γv)| 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.

Random Networks

Forty years ago a model to represent complex networks was proposed by Erdös and Rényi, 1960. 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, 1998. 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 = 6.

3. 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

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).

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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, vi, is proportional to the number of links (i.e., vertex degree kv) 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,

(a) (b) (c)

and an average vertex degree, k = 2.67. Some statistical properties are: L = 18.7, γ = 0.08, and φ = 0.79. For an equivalent random graph of this order and degree, the statistics would be: L = 14.89, γ = 0.005, and φ = 0.89. Whenever network properties show, L ≈ Lrandom, γ >> γrandom, 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.

Generation

Node

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 km2 within the area that contains the city of Memphis. This city is located in the southwest corner…

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 earthquake- induced removal strategy is stressed and illustrated using a theoretical model of a small-scale test- bed. 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 30332-0355, U.S.A. James I. Craig, Mid-America Earthquake Center, School of Aerospace Engineering, Georgia Institute of Technology,

270 Ferst Drive, Atlanta, GA 30332-0150, 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 30332-0355, U.S.A.

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 man- made 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

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

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 (ith, jth) elements define the probability of inoperability that failure in the jth infrastructure component triggers in the ith infrastructure component. These ith and jth 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

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, kv, 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

∑∑ ≠−

1L (1)

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

∑∑ ≠−

Clustering Coefficient

v v k

E γ (3)

where |E(Γv)| 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.

Random Networks

Forty years ago a model to represent complex networks was proposed by Erdös and Rényi, 1960. 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, 1998. 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 = 6.

3. 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

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).

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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, vi, is proportional to the number of links (i.e., vertex degree kv) 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,

(a) (b) (c)

and an average vertex degree, k = 2.67. Some statistical properties are: L = 18.7, γ = 0.08, and φ = 0.79. For an equivalent random graph of this order and degree, the statistics would be: L = 14.89, γ = 0.005, and φ = 0.89. Whenever network properties show, L ≈ Lrandom, γ >> γrandom, 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.

Generation

Node

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 km2 within the area that contains the city of Memphis. This city is located in the southwest corner…