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Implementation of multi-objective optimization for vulnerability analysis of complex networks C M Rocco1, J E Ramirez-Marquez2*, D E Salazar3, and I Hernandez1 1 Facultad de Ingeniera, Universidad Central de Venezuela, Caracas, Venezuela 2 Stevens Institute of Technology, Hoboken, New Jersey, USA 3 Ecole Nationale Supe´rieure des Mines de Saint Etienne, France The manuscript was received on 19 August 2009 and was accepted after revision for publication on 12 February 2010. DOI: 10.1243/1748006XJRR274
Abstract: This paper describes the vulnerability analysis of a complex network as the process of identifying the combination of component failures that provide maximum reduction of network performance. By way of a vulnerability analysis, the understanding of these failures can be related to the occurrence of a disruptive event, and also to the fundamental tasks for the protection of critical infrastructures. To describe vulnerability, the paper provides an analytical method to characterize completely the importance of network disruptions and identify a vulnerability set via the solution of a proposed multi-objective network vulnerability problem. This approach makes it possible to recognize that decision-makers (e.g. network managers) could benefit from understanding the relationship between different failure scenarios and network performance, for example, how the increase in protection resources would reduce the vulnerability of the network. Numerical examples, related to a medium-sized network and two complex networks, are solved using the evolutionary algorithm known as the multi-objective probabilistic solution discovery algorithm (MO-PSDA) and illustrate the proposed approach. Keywords: network vulnerability, multi-objective optimization, evolutionary algorithms, sensitivity analysis
1 INTRODUCTION During the last two decades, owing to advances in technology, previously decoupled service systems have now become highly networked and considered as critical national infrastructures. For example, the electric power grid is a critical infrastructure built of different systems – regional power plants, regional transmission substations. and interregional distribution grids – that, if incapacitated, can have significant security, economic, and social consequences on a national scale [1]. Other examples include the transportation infrastructure (where disruptions in the road network may have significant impacts on trade [2] and the port network (where disruptions can create serious economic loss). As the economic vitality and security of a *Corresponding author: System Development & Maturity Laboratory, School of Systems & Enterprises, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030, USA. email:
[email protected] JRR274
nation is more and more reliant on these critical infrastructures, it has also become critical to understand how their service changes as a function of potential unforeseen disruptions. In broad terms, disruptions in these infrastructures can be categorized into (i) design inherent and (ii) design external. The first category considers disruptions due to component/subsystem/system failures because of faulty design, or wear and tear. The impact of disruptions in this category has been analysed via reliability engineering and risk analysis. Research in these areas has provided managing frameworks – see, for example, the use of failure modes and effects criticality analysis (FEMCA), failure reporting and corrective action system (FRACAS), probabilistic risk assessment (PRA), and reliability, availability, and maintainability system (RAMS) management – to prevent and handle undesired failure events that occur in systems at random. The second category considers all potential disruptions that are not design or component influenced, for example natural and man-made disasters. Proc. IMechE Vol. 224 Part O: J. Risk and Reliability
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For the second category, research efforts are focused on creating the necessary methods and tools to analyse the safety and security of critical infrastructures in order to build adequate protection against natural disasters and/or intentional attacks [3–11]. A commonly used approach to model and analyse the performance of services in an infrastructure is to model it as a flow in a network of nodes that are connected by links (refer to reference [5] for illustrative examples). Based on such a model, and on the specific performance/service function of the network (e.g. the amount of energy transmitted between two points of the electric grid), the analysis of potential disruptive events (e.g. the removal of a single link or node, or a group of links or nodes) can be developed with the intent of understanding their effects at the network level. Such an analysis is usually referred to as a network vulnerability analysis. In the present paper, the vulnerability of a service in a network (i.e. the vulnerability of a network) describes the maximum reduction of network performance attained by the occurrence of a disruptive event – a similar rationale has been implemented by several other authors [3, 4, 6–9] to analyse networks with different performance functions. For example, in telecommunications, Crucitti et al. [12] studied the vulnerability of two internet backbone networks (Ca*net3 IS–IS routing network and Infonet 2001 carrying about 10 per cent of the traffic over the United States and Europe) by using the concept of network efficiency, a concept proposed by Latora and Marchiori [10] known as the L–M measure, as a performance function. The L–M measure has been widely applied (e.g. see references [3] and [12]) and used in the context of complex networks (see Newman [5] and the references therein). For transportation applications, Nagurney and Qiang [6] proposed ‘a transportation network efficiency measure that can be used to assess the performance of a transportation network and which differs from other proposed measures, including complex network measures, in that it captures flows, costs, and travel behavior information, along with the topology’. The Nagurney–Qiang measure (N–Q measure) is applicable to urban transportation networks and also the internet [8]. For example, the N–Q measure captures flows, costs, and travel behaviour information, using a single index. However, the L–M measure, unlike the N–Q measure, does not take service demands or network flows into consideration. In this same area, Zio et al. [9] performed a safety analysis of the road transport network in the province of Piacenza in Italy via the implementation of topological measures of interconnection and efficiency. The vulnerability of the network was evaluated with respect to the loss of a road link, for example due to a car accident, roadworks, and other jamming disruptions. Proc. IMechE Vol. 224 Part O: J. Risk and Reliability
This analysis allowed consideration of the implementation of a road development plan that reduced the vulnerability of the network performance function. While these studies have provided the groundwork to analyse vulnerability, the actual assessment for these different implementations has usually been done from a single-perspective scenario. So, for example, based on current analyses it is possible to determine the single most important network component to be reinforced such that network vulnerability is minimized or, from a catastrophic perspective, the most important component to be damaged so as to maximize network vulnerability. In essence, these analyses fail to consider how resources (for example, the cost associated with defending against a potential disruption), disruptive events, and network performance interact with each other. A more holistic vulnerability analysis would allow the network manager (i.e. the decision-maker (DM)) to understand the interaction among failure events, resources, and network performance. Based on this holistic perspective, the DM could, for example in a security sensitive network, understand how the increase in protection resources would reduce its vulnerability. Currently, to solve this challenge, for example in power systems vulnerability analysis, the DM must first identify the most important single component outage, then the most important two component outages, and so on. To address such research gaps, the present paper provides an analytical method to characterize completely the vulnerability of a network via the solution of a proposed multi-objective network vulnerability problem (MO-DNVP). To do so, the evolutionary algorithm known as the multi-objective (MO) probabilistic solution discovery algorithm (MO-PSDA) is used. MO-PSDA allows solutions to be developed to MO decision models where the traditional concept of optimality (a single best optimal solution) cannot be directly applied for comparing the ‘goodness’ of alternative solutions. While different concepts of optimality exist, MO-PSDA considers the Pareto optimality condition in which a number of potentially optimal solutions are compared according to the concept of Pareto dominance [13]. Hence, the final output of MO-PSDA is an approximation to the optimal Pareto frontier (PF). In summary, the key contributions of this paper include: (a) a clearly quantifiable definition related to the concept of network vulnerability under a failure set F ; (b) a novel computational approach to quantify such a vulnerability concept based on multi-objective optimization. The remainder of the paper is organized as follows: section 2 presents the mathematical description of JRR274
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vulnerability in the context of complex networks, while section 3 presents the MO models considered along with a synthesis of MO-PSDA procedure. Section 4 presents illustrative examples, using different performance functions on complex networks. Finally, section 5 presents the conclusions. The assumptions made are listed below. 1. The network and associated parameters are completely identified. 2. The network performance function is known a priori. 3. A failure scenario completely fails components and there is no repair.
As previously mentioned a critical infrastructure can be abstractly modelled as a network G(N, A) where N represents the set of nodes and A represents the set of links (i,j) i,j ¼ 1,. . .,n. Associated with each link (i,j) is an L · 1 attribute vector aij where the lth entry, aiij , describes an attribute of the link (i,j), for example, its length, its reliability, its capacity, its maintenance cost, etc. Moreover, let function Ql(G) > 0 define the performance that allows the assessment of G based on the lth attribute of every link (i,j). Examples of function Ql(G) are ubiquitous: in electric power systems such function is related to its ability to provide an adequate supply of electrical energy [14, 15]. Moreover, in this respect, Rocco et al. [16] analysed the vulnerability of the Italian power system by considering a single function Q(G) as the network flow. Given a coherent performance function Ql(G) > 0 and a set F of possible events (for example the removal of a single link or node, or a group of links or nodes), the importance of each event f 2 F can be obtained as [11] Iðf Þ ¼
Ql ðG ðN; AÞÞ Ql ðG ðN ; Ajf ÞÞ Q l ð G ð N ; AÞ Þ
ð1Þ
Based on equation (1) and assuming set F is completely characterized, the vulnerability of G(N, A) can be defined as the importance of the most important event f* 2 F (i.e. the event that maximizes I(f) 8 f 2 F). To clarify, vulnerability is understood as the most damaging outcome associated with the set of events F. Mathematically, such an event can be defined as f ¼ argmax Iðf Þ
ð2Þ
f 2F
Thus, the vulnerability of the network under the event set F can be obtained as
VF ðG ðN ; AÞÞ¼Iðf Þ JRR274
Conversely, it is also possible to investigate beneficial effects to the network via an improvement analysis. For this type of analysis one would like to determine which among several potential improvement events s, s 2 S, produces the maximum relative increase in the function Ql(G(N, A)) > 0. For this case, the importance of each event s 2 S can be obtained as [11] IðsÞ ¼
Ql ðG ðN ; AjsÞÞ Ql ðG ðN ; AÞÞ Ql ðG ðN; AÞÞ
ð3Þ
ð4Þ
The most important event s* 2 S is the one that maximizes I(s) and the improvement efficiency of the network under the event set S, is given as Ls ðG ðN ; AÞÞ ¼ Iðs Þ
2 MATHEMATICAL MODELLING OF VULNERABILITY FOR COMPLEX NETWORKS
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ð5Þ
It is important to mention that the networks considered in this paper are large and computationally intensive in terms of obtaining exact values for traditional functions Ql(G). However, in networks with thousands of nodes and links, the traditional concepts of two-terminal or k-terminal reliability becomes ambiguous and even irrelevant. For these types of networks, authors have used performance functions adapted or derived from complex network theory [9]. 1. The characteristic path length [17] representing the average of the shortest distances dij between all pairs of nodes in G(N, A) – assuming the cardinality of N equals n – has been mathematically defined as X 1 L¼ dij ð6Þ nðn 1Þ i; j 2 N 2. The network efficiency [17] to measure how efficiently the nodes of the network communicate if they exchange information in parallel, and assuming the cardinality of N equals n, has been mathematically defined as X 1 E¼ 1=dij ð7Þ nðn 1Þ i; j 2 N As described by Zio et al. [9] . . . efficiency plays the role of L in defining the network connection characteristics on a global scale. The fundamental difference is that E is the efficiency of a parallel network of nodes which concurrently exchange packets of information, whereas 1/L measures the efficiency in a sequential system where only one packet of information at the time goes along the network. Thus, 1/L represents well the efficiency of unweighted networks where no distinction is made among the distances in the graph.
Zio [18] introduced the reliability efficiency, an indicator that is intended to assess the reliability of a complex technological network or infrastructure. Proc. IMechE Vol. 224 Part O: J. Risk and Reliability
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This new measure implements the efficiency formulation in point 2, while taking into account the reliability of each link via dij as follows 9 8 > > = < 1 Q ð8Þ dij ¼ min g ij > ð1 qmn Þ> ; : m;n2g ij
Definition 1 A vector x0 dominates x, denoted x0 x, iff fk(x0 ) 6 fk(x), k ¼ 1,. . .,K ^ 9j: fj(x0 ) < fj(x). If no solution dominates x, it is said to be non-dominated. Definition 2 A vector x0 2 X*iff:9 x 2 X: x x. Thus, x0 is a Pareto optimal solution of a K-objective optimization and X* the true Pareto set.
In equation (8) the minimum is obtained with respect to set g ij, describing all possible paths linking node i to j and where qij denotes the probability that the link between nodes i and j is unavailable.
Generally, complex MO optimization problems have large or even infinite solution spaces where the true Pareto set X* can rarely be completely characterized, and thus solution procedures are based on approximating such a set. Moreover, when considering K ¼ 2 and 3, the Pareto set is usually reported graphically as a Pareto front (PF).
3 MULTI-OBJECTIVE OPTIMIZATION FOR VULNERABILITY ANALYSIS
3.2 Vulnerability analysis via MO optimization
3.1 Fundamentals A general MO optimization model is of the form x ¼ arg minðf1 ðxÞ; f2 ðxÞ; . . . ; fk ðxÞÞ x
s.t. x 2 X ¼ fxjHðxÞ60g In this general model, the set X defines the n-dimensional solution space for all feasible decision vectors x satisfying the inequality in function H. The vector F(x), a vector of size K > 1, returns the value of objective function fk : Rn ! R, thereby mapping the decision vectors into a real value of the objectives’ space. Without loss of generality, the general MO model is given in terms of minimization, since, if some of the objectives need to be maximized, one can transform max (fk(x)) ¼ min ( fk(x)). Contrary to traditional single-objective optimization problems, MO problems are characterized by some degree of conflict among the objective functions. Thus, the actual optimization rarely yields a single solution that optimizes all objectives. By contrast when solving the general MO formulation, one obtains a set X* X of optimal decision vectors x*. These vectors provide a description of how the objective functions interact among each other, for example, describing how some objectives cannot be improved without worsening the others. In order to identify the optimal set X* different concepts of optimality exist to determine the MO ‘quality’ of two different vectors, say x and x0 . However, the most commonly used concept, and the one used for this research, is the Pareto optimality condition (see definition 2) in which solutions are compared according to the concept of Pareto dominance as described in definition 1. Proc. IMechE Vol. 224 Part O: J. Risk and Reliability
For a set F of relatively small cardinality, obtaining the vulnerability of a system may be computationally tractable. However, as the cardinality of such a set increases obtaining VF(G(N, A)) can become computationally intractable. To solve this issue, new approaches need to be developed to identify VF(G(N, A)), as proposed in this manuscript by means of MO optimization. While this is one of the first implementations of a MO optimization perspective into the analysis of complex network vulnerability, previous work by the authors suggests this type of approach. For instance, Salazar et al. [19] and Rocco et al. [20] considered different indices of vulnerability that can be related to a system protection strategy. In such approaches, a PF representing the actions that deliver maximal impact to the systems are identified, thereby providing an approach to devise optimal protection strategies. As previously discussed, most current approaches analyse vulnerability based on a single index that is a composite of resources, damages, and performance of the system. It should be noted that when considering service networks, the failure events f 2 F actually generate component failures that translate into a reduction in performance. So in this paper the focus is on identifying the vulnerability of networks when the set F contains all single-component failure events or all two-component failure events, and so on. Notice that such an approach provides a clear description of how vulnerability changes as the complexity of event increases. With this rationale in perspective, the general MO model can be transformed into the MO vulnerability model as follows f ¼ arg minðIðf Þ; Cðf ÞÞ
ð9Þ
f 2F
In equation (9), I(f) is obtained via equation (1) and C(f) is the cost associated with event f. Notice that a Pareto optimal solution provides a description of the impact that an event of set F has on the performance JRR274
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of the network. As such C(f) may relate a failure event to: (a) the number of component failures; (b) the actual monetary cost of a failure. For the first case, the PF describes how network vulnerability changes as a function of the number of failed network components, whereas the second case describes how vulnerability increases with actual failure cost.
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are analysed in each of the ten possible cycles. Moreover, for each of these networks the figure-ofmerit used to assess their performance is the network efficiency described by Zio [8] as in equations (7) and (8). Note that this figure-of-merit describes the efficiency as a function of the reliability between every pair of nodes in the network. Finally, all the examples were solved with an AMD Athon XP at 1.5 GHz and 1 GB of RAM. 4.1 A 52-nodes network
3.3 Solution techniques There are numerous approaches to solve the MO vulnerability problem described in equation (9). In general, these approaches are known as MO evolutionary algorithms (MOEA) and refer to algorithms that use some type of evolutionary approach (i.e. genetic algorithms, ant colony, etc.). MOEA are able to deal with non-continuous, non-convex and/or non-linear objectives/constraints, and objective functions possibly not explicitly known (e.g. the output of Monte Carlo simulation runs). Specific examples of these algorithms include: SPEA2 [21], PESA-II [22], and NSGA-II [23] as well as MO-PSDA [13, 24, 25], among others. MO-PSDA offers a simple, intuitive, and efficient approach to solve the MO vulnerability problem. It is based on three main steps. 1. Simulation of failure events via Monte Carlo simulation (based on the actual probability of failure of each). 2. The assessment of the importance of each failure event (via equation (1) and the respective network efficiency calculation). 3. An evolutionary procedure that selects potential Pareto optimal failure events (via the implementation of definitions 1 and 2). This three-step procedure is repeated according to a predefined number of cycles to obtain an approximation to the optimal PF generated. For a complete description of MO-PSDA, the interested reader is referred to reference [13].
Figure 1 shows the network considered by Manzi et al. [26] comprising 52 nodes and 152 directed links. For the analysis of the vulnerability of this network the reliability efficiency defined by Zio [18] was considered (as described in section 2, equation (8)) assuming a link reliability of 0.95 (i.e. qij ¼ 0.05). Moreover, the vulnerability was computed for two different cases: link disruption and cost of link damage. To clarify, the first example considers obtaining the vulnerability for each and every possible size of link disruption and thus implementing PSDA provides the description of the most important single-component disruption, the most important two-component disruption, and so on. The second case illustrates how the vulnerability changes as a function of link disruption cost (an integer value randomly selected in the interval [1,5] units). For each solution case, a point in the PF represents the outage of a subset of links and thus a particular network configuration. Figures 2 and 3 illustrate a subset of the final PF approximation found using the PSDA. It is very important to stress that both of these fronts are constructed by analysing at most 5000 component failure combinations (i.e. disruptive events) out of the total 2152 potential solutions for this problem. The PF subset described in Fig. 2 was obtained based on the optimization of two criteria, namely minimizing the number of links out while also maximizing the importance of the links out (see equation (1)). Also, as
4 EXPERIMENTAL RESULTS To illustrate the proposed vulnerability analysis this section presents two sets of examples. The first set considers a medium-sized network, while the second set considers two complex networks: the Venezuelan national electric power system and the transportation network of the province of Piacenza, Italy. For each example, the PF has been approximated based on the results of five independent MO-PSDA [13] runs. For each run 500 potential failure events
Fig. 1 Network from Manzi et al. [26]
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Fig. 2 Pareto frontier for example 1: case 1 26 24 22 20 18
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f *describes the event that links between nodes 24–28, 32–36, 28–24, and 36–32 are damaged and, VF(G(N, A)) ¼ 0.1411. Equivalently, when considering the second case, the point (1.13, 4) in Fig. 3 corresponds to the maximum reliability efficiency decrease considering a total cost of four units. In this case, only two links are damaged; those between nodes 22–26 (cost ¼ 2) and 26–22 (cost ¼ 2) with the reliability efficiency decreasing to 0.7525. Note that when considering disruption costs, as would be the case when analysing the actions of potential antagonists, if the interest was to achieve a reliability efficiency reduction near to 0.6537, it is necessary to spend 15 units. Indeed, five links must be damaged: 20–24 (cost ¼ 3), 40–47 (cost ¼ 1), 42–46 (cost ¼ 5), 47–40 (cost ¼ 1), and 46–42 (cost ¼ 5) with an importance equal to 14.07. This example shows that even if the analysis considering links out or cost is similar, its interpretation is quite different. Finally, note that both graphs provide a complete description of how vulnerability changes as a function of cost. 4.2 Complex networks
8 6
4.2.1 The Venezuelan national electric power system
1.13
4 2 0 0
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IMPORTANCE (%)
Fig. 3 Pareto frontier for example 1: case 2
previously discussed, Fig. 3 illustrates the PF subset based on the perspective of minimizing the cost of damages while maximizing the importance. In both figures the first extreme point located at (0,0) – the reporting (VF(G (N, A)) · 100, C(f*)) describes the decrease in base reliability efficiency and the number of links out – represents the solution where no links are damaged with a base reliability efficiency of 0.761 164 069. In Fig. 2, the number of links out can be considered as the total damage cost whenever such a cost is assumed to be the same for every link and equal to one. In this case, Fig. 2 illustrates that the point (14.11, 4) corresponds to the maximum decrease in reliability efficiency when considering the damage of four links. In fact, if links between nodes 24–28, 32–36, 28–24, and 36–32 are damaged, the reliability efficiency decreases from 0.7612 to 0.6537. In order to relate this result to the equations described in section 2, consider that for the set F containing all possible four-component failure events, that is jFj ¼
152! 148!
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Figure 4 illustrates a compact configuration of the Venezuelan national electric power system (VNEPS) as described by Rocco [27]. The VNEPS is a 635-bus meshed network with 175 generating units and 1179 circuits (average circuit reliability of 0.99) with an installed capacity of about 22.54 GW and a peak load of about 15.55 GW. This network was previously analysed by Rocco [27] via a graph-based approach that used advances in mapping the topology of complex networks. Such a network can be considered as a complex network of the single-scale type. In fact, its cumulative degree distribution follows an exponential distribution as P(k)e0.42k. Additionally, in reference [27] it was found that the cumulative distribution of the betweenness [17] (i.e. the number of best paths that pass through the node) can be fitted by a power law Pcum(L)(2774 þ L)1.22. As mentioned in reference [12] ‘this means that, although very homogeneous in the node degree, the network shows a high heterogeneity in the node load: most of the nodes handle a small load, but there are a few nodes that have to carry an extremely high load’. For this network the measure of the network efficiency, as described in equation (8), is used as a performance function; the network efficiency of the network is 0.104 468 26. The network vulnerability is assessed based on solving a MO model that considers maximizing the network efficiency reduction and minimizing the number of links out. JRR274
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Fig. 6 Pareto frontier for Piacenza network [9]
Fig. 4 Venezuelan national electric power system network by Rocco [27]
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0.0078), based on its length and accident failure rate. To address the vulnerability of the network the perspective is on identifying ‘the degradation in the global safety efficiency of the network due to the disconnection, i.e. the interruption, of a set of its road links’. Figure 6 shows the PF approximation found using PSDA considering up to ten links out. For this network, the algorithm approximates the PF by analysing only 5000 solutions out of a solution set with a cardinality of 2789. As in previous cases, each point in the PF represents the outage of a subset of links, and thus a particular network configuration. The efficiency is in the range [5.34 · 102, 6.62 · 102] while the vulnerability is in the range [8.20, 19.68]. The value of 5.34 · 102 is similar to the value found by Zio et al. [9] (5.68 · 102) ‘taking the whole path of 10 most critical links out of the network’.
Fig. 5 Pareto frontier for VNEPS
In this respect, Fig. 5 shows the PF approximation found using PSDA. For this network, the algorithm approximates the PF by analysing only 5000 solutions out of a solution set with a cardinality of 21179. In practical terms, each point in the PF represents the outage of a subset of links, and thus a particular network configuration. For example, the point located at (3.35, 1) corresponds to the maximum percentage decrease in the network efficiency when considering the damage of a single link. Figure 5 also reveals that the maximum efficiency, under a double contingency scenario, is 5.55 per cent. 4.2.2 Transportation network of the province of Piacenza, Italy The last example considers the road transportation system of the province of Piacenza studied by Zio et al. [9]. In reference [9] the system was modelled as a weighted, undirected graph with 687 nodes connected by 789 edges. Each edge is characterized by the probability of an accident (average value of JRR274
5 CONCLUSIONS In this paper a MO optimization approach is used to analyse the vulnerability of complex networks to external sources of failures (i.e. natural or manmade disasters). The MO-DNVP proposed allows the interaction among failure events, resources, and the network performance to be understood (independent of how the network performance function is characterized). Such an analysis can provide the network manager with valuable information for devising defensive policies, as the contingencies that affect the network the most, or the more economic set of failures that, once induced, reduced the network performance down to a predefined level. Likewise, this type of analysis can be used to identify the optimal actions to improve the network performance. The problem described was solved via the MO-PSDA, although the methodology can equally be applied using any other efficient algorithm. Finally, it is important to mention that the MOPSDA as an evolutionary algorithm can only claim to Proc. IMechE Vol. 224 Part O: J. Risk and Reliability
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provide an approximation to the true vulnerability, unless a complete enumeration and evaluation of set F can be performed. Unfortunately, such enumeration increases exponentially as a function of network elements and failure events considered (i.e. for the networks considered it is highly unlikely – owing to the extensive computational effort required – that vulnerability can be obtained for every set F considered). Thus, at this point MOPSDA provides an attractive alternative. Authors 2010 REFERENCES 1 Prezant, D., Clair, J., Belyaev, S., Alleyne, D., Banauch, G., Davitt, M., Vandervoorts, K., Kelly, K., Currie, B., and Kalkut, G. Effects of the August 2003 blackout on the New York City healthcare delivery system: A lesson for disaster preparedness. Critical Care Medicine, 2005, 33(1)(suppl.), S96–S101. 2 Berdica, K. An introduction to road vulnerability: what has been done, is done and should be done. Transp. Policy, 2002, 9(2), 117–127. 3 Crucitti, P., Latora, V., and Marchiori, M. Locating critical lines in high-voltage electric power grids. Fluctuation and Noise Lett., 2005, 5(2), L201–L208. 4 Rosato, V., Issacharoff, L., and Bologna, S. Influence of the topology on the power flux of the Italian high-voltage electrical network. Europhysics Lett., 2009 (in press). 5 Newman, M. E. J. and Girvan, M. Finding and evaluating community structure in networks, 2003, available from http://arxiv.org/abs/cond-mat/0308217. 6 Nagurney, A. and Qiang, Q. A. Transportation network efficiency measure that captures flows, behavior, and costs with applications to network component importance identification and vulnerability. In Proceedings of the POMS 18th Annual Conference, 4–7 May 2007. 7 Nagurney, A. and Qiang, Q. A. A network efficiency measure for congested networks. EPL, 2007, 79(38005). DOI: 10.1209/0295-5075/79/38005. 8 Nagurney, A. and Qiang, Q. A. Robustness of transportation networks subject to degradable links. Europhysics Lett., 2007, 80, 1–6. 9 Zio, E., Sansavini, G., Maja, R., and Marchionni, G. An analytical approach to the safety of road networks. Int. J. Reliability, Qual. Saf. Engng, 2008, 15(1), 67–76. 10 Latora, V. and Marchiori, M. Efficient behavior of small-world networks. Phys. Rev. Lett., 2001, 87, article no. 198701. 11 Latora, V. and Marchiori, M. Vulnerability and protection of infrastructure networks. Phys. Rev. E, 2005, 015103(R), pp 1–4. 12 Crucitti, P., Latora, V., and Marchiori, M. A topological analysis of the Italian electric power grid. Physica A, 2004, 338, 92–97. 13 Ramirez-Marquez, J. E. and Rocco, C. All-terminal network reliability optimization via probabilistic soluProc. IMechE Vol. 224 Part O: J. Risk and Reliability
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APPENDIX
Notation A C(f) dij f f*
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set of links of G cost of event f defines the shortest distance path between nodes i and j in G(N, A) single event optimal event with respect to a vector of objectives
F G (N, A) I(f) K N VF(·) _ gij Ql(G)
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set of events network G importance function of f number of objective functions set of nodes on G vulnerability function with respect to set F OR operator set containing all paths between nodes i and j in G (N, A) performance function on G
Proc. IMechE Vol. 224 Part O: J. Risk and Reliability
