Original Article

Proc IMechE Part O:J Risk and Reliability227(6) 576–585� IMechE 2013Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1748006X13489490pio.sagepub.com

Network vulnerability assessment viabi-objective optimization with afragmentation approach as proxy

Cesar Yajure1, Darihelen Montilla2, Jose Emmanuel Ramirez-Marquez3

and Claudio M Rocco S4

AbstractThe fragmentation of a network is used to understand the effects of element removals on its cohesion. Minimum infor-mation is required to fragment a network, namely: the topology of the network. Continuous fragmentation of a networkcan be used to uncover important/critical elements in the network. This article proposes a bi-objective optimizationmodel that, when solved, provides the most economical network fragmentation strategies for increasing element frag-mentation cost. After description and solution of the model, the manuscript describes, via experimentation, how theresults of the model can be used as a surrogate metric for understanding element importance performance in real ser-vice networks. The experimentation is complemented with a classical example of social network analysis. The resultsshow that the proposed fragmentation models can be used as a guide to identify sets of elements that contribute to thesuccessful performance of a system.

KeywordsBi-objective optimization, network fragmentation, power systems, social networks, vulnerability

Date received: 6 November 2012; accepted: 17 April 2013

Introduction

For a given network represented by nodes and theirinterconnecting arcs, the network fragmentation prob-lem consists on determining a set of nodes that must beremoved in order to reduce the ability of nodes to com-municate with other nodes—that is, reduce the networkcohesion. The application of the network fragmenta-tion problem (NFP) naturally arises in a number of dif-ferent contexts:

Borgatti1 described the NFP in the public health contextas the immunization/quarantine problem: ‘‘which subsetof members of a population should be immunized/quaran-tined so as to maximally hinder the spread of theinfection?’’

Equivalently, but in a telecommunications context, andalso as a way to identify the critical nodes of the network,Arulselvan et al.2 proposed a version of NFP as a poten-tial approach to stop the spreading of a virus over a tele-communication network.

In the military context the NFP can be used for targetselection: ‘‘Given a network of terrorists, which onesshould be chosen in order to maximally disrupt the

network?’’ Finally, Boginski and Commander3 implemen-ted NFP as a means to detect critical nodes in protein–protein interaction networks.

These applications highlight that the NFP can be usedalso as a tool to assess the vulnerability of a specificnetwork.

Similar to reliability or availability, network frag-mentation can be considered as network performanceattribute. By considering it as such, one could for

1Instituto Universitario de Tecnologıa ‘‘Dr. Federico Rivero Palacio’’,

Caracas, Venezuela2Power Engineer at CORPOELEC, Caracas, Venezuela3Development and Maturity Laboratory, Stevens Institute of Technology,

Hoboken, NJ, USA4Facultad de Ingenierıa, Universidad Central de Venezuela, Caracas,

Venezuela

Corresponding author:

Jose Emmanuel Ramirez-Marquez, System Development and Maturity

Laboratory, School of Systems and Enterprises, Stevens Institute of

Technology, Castle Point on Hudson, Hoboken, NJ 07030, USA.

Email: jmarquez@stevens.edu

example quantify the fragmentation of a network basedon different catastrophic scenarios (terrorist attack ornatural disasters) and propose effective protective orcountermeasures to reduce fragmentation.

To illustrate the NFP, consider the Map of the 9/11terrorist network as presented by Krebs4 shown inFigure 1. This network consists of 62 nodes (represent-ing terrorist actors) and 153 links (representing the exis-tence of direct communication between two actors).For this original network, Figure 2 illustrates the statusof the network after the removal of 10 nodes (identifiedby the thicker font). Evidently, such removal has frag-mented the network (i.e. many nodes cannot communi-cate with any other nodes).

It is relevant to mention that fragmentation assess-ment is based only on the topological description of the

network and that no additional component level char-acteristics are required (for example in reliability analy-sis5 or capacitated networks) and so there is no need fordata collection.

In essence, the NFP is an optimization problem ofthe form maximize (or minimize) network fragmenta-tion given some constraints. For example, minimize thepair-wise connectivity among network nodes under theconstraint that the total number of nodes removed doesnot exceed some given threshold. Note that thissingle objective (SO) perspective does not allow thenetwork manager to understand the impact thatchanges (increase or decrease) in the number of nodesremoved have on network fragmentation. That is, SOapproaches cannot concurrently describe the tradeoffsof having two or more competing objectives and

Figure 1. Map of the 9/11 terrorist network; data as obtained from http://vlado.fmf.uni-lj.si/pub/networks/data/ucinet/ucidata.htm#zachary.

Figure 2. Map of the 9/11 terrorist network with 10 nodes removed.

Yajure et al. 577

multiple prospective solutions that may change basedon the preference of the decision maker (DM)—thislimitation has been recognized by several authors.

In the last decade researchers have proposed multi-objective approaches to cope with these limitations.For example, Rocco et al.6 proposed a multi-objectiveoptimization problem to derive the group of compo-nents that, if eliminated from the system, cause theworsening of the global efficiency. Zio and Golea7 andZio et al.8 propose two approaches based on the use oftopologic metrics derived from the complex system the-ory, such as clustering coefficients, to define groups ofimportant nodes or edges, respectively. Each group ischaracterized by an increasing cardinality. In this arti-cle, bi-objective models for the fragmentation problemsare proposed.

The two main contributions of this article are: (1) todefine a bi-objective optimization (BO) model for theNFP that consider the problem of maximizing networkfragmentation concurrently with the minimization of thenumber of network elements removed (or equivalentlythe minimization element removal cost) and, (2) to relatethe fragmentation actions obtained from the NFP-BOmodels with the real performance network analyzed. Todo so, two networks are analyzed in two different con-texts: social networking and electric power system. Tothe best of the author’s knowledge, neither bi-objectivemodels for fragmentation nor applications to electricpower systems have been presented elsewhere.

The remainder of the article is organized as follows.‘‘Mathematical modeling of the NFP’’ presents a litera-ture overview of NFP and the proposed bi objectivemodels. ‘‘Experimentation’’ describes the fragmentationapproaches and different metrics to assess its effects.Finally, ‘‘Conclusions’’ are presented.

Mathematical modeling of the NFP

NFP in literature

Mathematically, several authors have considered theeffects of removing nodes or arcs. Ball and Golden9

analyzed the most vital arcs problem (MVAP), whichcan be described as the problem of finding a subset ofsize k of arcs for which removal results in the greatestincrease in the shortest distance between two specifiednetwork nodes. Myung and Kim10 studied the k-edgesurvivability problem, defined as the percentage of thetotal traffic surviving the failure of k edges in the worstcase. Borgatti1 defines the ‘‘key player problem/nega-tive’’ (KPP-Neg): by evaluating ‘‘the node importanceas the amount of reduction in cohesiveness of the net-work that would occur if the nodes were not present’’.Arulselvan et al.2 defined the critical node problem(CNP): Select a subset of nodes B4V such as |B|4 k,whose deletion results in the minimization of the pair-wise connectivity between the nodes. This problem issimilar to the minimum k-vertex sharing ‘‘where the

objective is to minimize the number of nodes deleted toachieve a k-way partition’’.2 Note that both definitionsare based on determining the subset of nodes that mustbe removed.

Other authors (see for example Hanneman andRiddle11 and references therein) have assessed theimportance of network components using a graph cen-trality approach, consisting in measuring the centralityof each component in the network, then selecting the kmost central. For example, Freeman’ betweenness mea-sure11 has been used to classify node importance: anode with high betweenness is considered importantsince it allows the connection of many pairs of nodes.So if that node is removed the rest of nodes should tobe more distant.1 However, as shown in Borgatti,1 evenif centrality measures could be used to assess the frag-mentation, they are not optimal. Finally, Boginski andCommander3 present a related problem, the cardinalityconstrained critical node detection problem (CC-CNDP). The idea is to determine a minimum cardinal-ity subset of nodes ‘‘whose deletion ensures that thenumber of nodes reachable from any other node in thenetwork does not exceed some threshold value.’’

Since both KPP-Neg and CNPs are NP-hard prob-lems, special heuristics have been proposed: Arulselvanet al.2 proposed a heuristic for detecting critical nodesbased on the determination of the maximal independentset (MIS) for a given specified cardinality k. To enhancethe heuristic, the authors proposed a local search proce-dure. Borgatti1 proposes the use of a genetic algorithmto solve the problem.

It is important to realize that classical fragmentationapproaches have been defined to assess node elimina-tion. However, in real life situations, the assessment oflink removals could be more important. For example,consider an electric power system, where nodes are sub-stations and links represent transmission lines. Theaction required to remove a transmission line couldconsist only on damaging a single tower, while theremoval of a substation involves additional tasks.

Finally, all of these references on fragmentationhave formulated the NFP based on the optimization ofa SO function, for example, by maximizing the networkfragmentation and ensuring that the total number ofdeleted nodes is less than or equal to a particular k.

NFP mathematical background

Consider a network defined as a graph G(N, A) whereN represents the set of nodes, n=|N| is the cardinalityof N and A represents the set of undirected links (i,j)i,j= 1,.,l.

The fragmentation of a network is based on theselection of nodes that is able to reduce the cohesive-ness of a network. A general definition of cohesivenessis ‘‘the resultant of all the forces acting on members toremain in the group’’.12

So the NFP is defined as the selection of a subset ofnodes B with cardinality |B|4K, whose deletion

578 Proc IMechE Part O: J Risk and Reliability 227(6)

results in the maximum fragmentation of the network.So, the key word is fragmentation and the main issue isquantification.

Quantifying fragmentation/connectivity of a network

Table 1 shows five different metrics to quantify networkfragmentation/connectivity. Each of the metrics can bedescribed as follows.

Connectivity C: This index considers if a specificnode i is able to reach any other node j: rij = 1 if nodei can reach node j and rij = 0. The factor 2/(n(n–1)) isused as a normalization factor. Note that if all pair-wise nodes are connected, then C=1. If all pair-wisenodes are disconnected then C=0.

Fragmentation F: Quantifies the degree of fragmen-tation and is defined as 1–C. From here the maximumfragmentation is F=1 if C=0.

Connectivity Cc: This index is equivalent toConnectivity C but determined taking into accountthat, since the graph under study is undirected andnodes are mutually reachable, the metric is expressed asa function of the sizes (sh) of each component (groupsof nodes that are connected but isolated), where M isthe set of all maximal connected components and sh isthe size of the hth component. Cc is the measure usedin Arulselvan et al.2

Fragmentation DF: Even if F is a satisfactory metricfor assessing the fragmentation ‘‘it does not take intoaccount the shape—the internal structure—of compo-nents.’’1 For this reason the author suggests ‘‘to mea-sure the total distance between all pairs of nodes in thenetwork,’’ or ‘‘the sum of the reciprocals of distances,observing the convention that the reciprocal of infinityis zero,’’ where dij is the minimum distance (in numberof nodes) between node i and node j. This measure isable to assess ‘‘the relative cohesion of thecomponents.’’1

Fragmentation E: E measures how efficiently thenodes of the network communicate if they exchangeinformation in parallel. So 1–E could be also consid-ered as a measure of fragmentation.

To illustrate the different metrics let us consider thenetwork shown in Figure 3(a) containing nine nodes.Initially, since no fragmentation actions are performed:C=Cc=1 or F=DF=1� E=0. Figure 3(b) shows thestatus of the network when links connected to node 5are removed. Removal of such links produces threeclusters, described as components in Figure 3(b).Evidently, the removal of such links has significantlydecreased the ability of communication among nodes.For example, node 5 has been isolated and in addition,nodes in each component cannot communicate withnodes in any other component.

For the case considered

C=2(6+0+6)

9 � (9� 1)= 0:33 (orF=0:67),

Cc=0:33 andDF=1� 2 � (5:5+5:0)

10(9)=0:76

Bi-objective fragmentation models

Contrary to traditional SO optimization problems, BOproblems are characterized by some degree of conflictbetween the two objective functions. This article consid-ers a Pareto analysis of potential optimal solutions imply-ing that the actual optimization rarely yields a singleoptimal solution for both objectives. In fact, when sol-ving BO problem from a Pareto perspective, a set X*4 Xof optimal decision vectors x* is obtained. These vectorsprovide the values of the decision variables that, whenimplemented in the objective functions, describe the inter-action between the objective functions from which aPareto front (PF) can be constructed.

To define the BO-NFP let the vector of decisionvariables be represented by the fragmentation profilevector x, where its kth entry describes the status of thebinary decision variable xk - with xk =1 if network ele-ment k is fragmented (removed) and xk =0 otherwise.Based on the fragmentation metrics described in ‘‘NFPmathematical background,’’ let f1(x) define the networkfragmentation (connectivity) a function of the

Table 1. Network fragmentation or connectivity Metrics.

Index Type Formulae Reference Equation

C Connectivity C =

2P

i

Pj \ i

rij

n(n�1) Borgatti1 (1)

F Fragmentation F = 1�2P

i

Pj \ i

rij

n(n�1) =1–C Borgatti1 (2)

Cc Connectivity Cc =

Ph2M

sh(sh�1)

n(n�1) Arulselvan et al.2 (3)

DF Fragmentation DF = 1�2Pi . j

1dij

n(n�1) Borgatti1 (4)

E Fragmentation E = 1n n�1ð Þ

Pi, j2N

1�dij

Crucitti et al.13 (5)

Yajure et al. 579

fragmentation profile. Finally, define the cost associ-ated to fragmentation profile x by function f2(x)=Pk

ckxk. The BO-NFP is given as

Minx�f1 xð Þ, f2 xð Þf g

s.t.

xk 2 Bin (0, 1)

The output of the BO-FP can be described for twocases.

1. When the element fragmentation cost is unitaryfor every element, the Pareto set provides the bestfragmentation considering unitary increases of costin the bound (1, n+ l). That is, these results wouldprovide the highest network fragmentation whenconsidering 1, 2, . n+ l fragmented elements.Note that in this case no additional information isrequired.

2. When the element fragmentation cost is heteroge-neous, that is the process for isolating a noderequires different expenses, the Pareto set wouldprovide the optimal value of network fragmenta-tion for the first cheapest cost, second cheapestcost,., (n+ l)th cheapest cost. In this case, addi-tional information is required.

To obtain the Pareto set from model BO-FP, an evolu-tionary algorithm has been implemented. The algo-rithm known as BO probabilistic solution discoveryalgorithm (BO-PSDA) has been modified from Roccoand Ramirez-Marquez.14 For a complete description ofthe algorithm the reader is referred to this article.

It is important to clarify that algorithm developmentis not a contribution of this manuscript. In fact, as pre-viously mentioned, the contributions are to provide abi-objective description of the NFP and use the resultsas a surrogate of the effect that these types of

disruptions have in the actual performance of networksanalyzed. The following section addresses this secondcontribution.

Experimentation

This section presents results and discussion regardingvery different types of applications: social networksand electric power systems. In the former case, the ele-ments of the network represent actors and their interre-lations with no specific ‘‘performance’’ description ofthe network (i.e. the network itself is not providing anytangible service that can be measured). The latter casecontains elements representing buses, transmissioncomponents (e.g. lines or transformers) and intermedi-ate substations of a power system. Conversely to thefirst case, the network provides a service—the powerflow—for which a performance can be measured. Thus,a comparison is made between the results yielded byconsidering BO-NFP and other approaches that haveconsidered exact power flow models.

All examples consider unitary element fragmentationcost and the solution of BO-NFP considering equation(2) in Table 1. Finally BO-PSDA has been implementedfor every example with running time being negligible.

Social networks—Zachary’s karate club network

The first network considered is classical in social net-working analysis showing the social network developedby Zachary after studying the social behavior betweenthe members of a karate club (http://vlado.fmf.uni-lj.si/pub/networks/data/Ucinet/UciData.htm#zachary). Thenetwork describes a karate club that, owing to a con-flict has been divided into two factions (squared androunded actors). Figure 4 illustrates the interactions(155 links) among the members (34 nodes) of the net-work. Figure 5 show the PF approximation consideringnode fragmentation only, while Table 2 lists the nodes

Figure 3. (a) Original network; (b) network after removal of links from node 5.

580 Proc IMechE Part O: J Risk and Reliability 227(6)

fragmented as a function of unitary increases in the costalong with the optimal network fragmentation value.

As per Table 2, for a single fragmented actor, thebest network fragmentation is obtained by removingnode 1. Note that by visual inspection, node 1 seemscritical to support the interaction among actors in bothgroups, not surprisingly node 1 represents the karateteacher. From the detailed output of the BO-PSDA,the second best network fragmentation, value whenfragmenting a single node, equals 0.08734 and consid-ers removing node 34—the club’s director. It is interest-ing that as the unitary cost increases by one unit, thebest network fragmentation scheme considers removingnodes 1 and 2 with a fragmentation value of0.4902, followed by the simultaneous elimination ofnodes 1 and 34 that produces a fragmentation valueof 0.40285. The extra increase of one unit provides theset of nodes {1,33,34} defining the ‘‘key actors’’ of thenetwork.

While the application of BO-NFP in the socialnetwork context allows for the identification of keyactors and understanding the why of the importance ofinteractions, this social network does not produce atangible/material service. The following experimentintends to illustrate the effects that network fragmenta-tion schemes have on a service-oriented network and itsrelation to the actual decrease in their performance.

Power systems—Venezuelan transmission networks

Currently, to understand the effect of element fragmen-tation (or deletion) in power systems it is necessary thata detailed assessment of the electric phenomena be per-formed; using for example a power flow model. Forsuch an assessment a significant amount of data collec-tion is needed, including, but not limited to: transmis-sion line impedances and capacity, transformerscharacteristics (impedance and taps settings), requiredloads, minimum and maximum generation. A signifi-cant issue related to these data is that it is not readilyavailable and in many cases can be considered as ‘‘sen-sible information’’. To clarify, consider Figure 6, a rep-resentation of the US electric power grid, obtaininginformation for all the sub-networks comprising thesystem is a formidable task (currently there is no singlerepository), and thus providing a single power flowmodel is infeasible at present. Furthermore, even if amodel were accessible, the computational task for iden-tifying key elements via a flow model would also beprohibitive.

Fortunately, the solution of the BO-NFP can beused as a surrogate, which provides guidance into themost important elements of the system. And, that whena detailed model is available it can be used to providean initial subset of important elements to be considered.

Figure 4. The Zachary’s karate network.

0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Network Fragmentation

Num

ber o

f nod

es e

limin

ated

Figure 5. BO-NFP PF for Zachary’s karate network.

Table 2. BO-NFP Pareto set for Zachary’s karate network.

Nodesfragmented

Optimalfragmentation

Optimalsolution

0 0 —1 0.35651 12 0.49021 1-23 0.64349 1-33-344 0.85205 1-3-33-345 0.91979 1-2-3-33-346 0.95009 1-2-3-24-33-347 0.96257 1-2-3-24-32-33-34

Figure 6. Representation of US power grid based on data fromhttp://vlado.fmf.uni-lj.si/pub/networks/data/map/USpowerGrid.net.

Yajure et al. 581

To illustrate the previous discussion consider theVenezuela power system (known as SEN in Spanish).The associated graph, shown in Figure 7, represents thehigh voltage system, from 138 to 765 kV.15 The graphcontains 139 nodes and 270 links. The peak system loadis 22100MW (as per year 2010).

Figure 8 shows the PF approximation consideringnode and link fragmentation separately. As expected,node fragmentation produces higher values of networkfragmentation than link fragmentation. Figures 9 and10 show the effect on SEN of removing four nodes andlinks, splitting SEN in two and three independent areas,respectively.

It must be highlighted that the PF is obtained withthe network topology as the only necessary informa-tion. To assess the effects of the fragmentation schemes,a detailed analysis, based on the electric phenomena,has been developed. Specifically, SEN has been evalu-ated with an alternating current (AC) power flowmodel.

Table 3 shows the results of analyzing four differentfragmentation schemes (each for nodes and links) inthe AC power flow model. Each case shows the loadthat the system is unable to feed after each fragmenta-tion scheme. For example, the worst outage of a singleline causes a fragmentation of 0.096, but no load shed-ding action is performed (indeed the power system isdesigned to withstand a single line contingency). Notethat one could conclude that by analyzing the worstcase single element fragmentation there is no actualload shedding.

When two or three lines are selected, the worst frag-mentations are 0.175 and 0.214. Under this fragmenta-tion scheme there are load-shedding effects near 2% ofthe total system load. The case of a four-line outagesproduces a network fragmentation value of 0.257 andthe power flow does not converge. While cases of noflow convergence do not necessarily represent a systemcollapse, the results suggest that the system could expe-rience important problems and urgent operational deci-sions are required.

When the fragmentation actions are based on nodeoutages, the results show even more critical effects.Indeed, the outage of node 125 is enough to produce anon-convergent power flow, thus reflecting possibleoperational issues. In this case, the SEN is split in twoislands (see Figure 11), where one of the islands corre-sponds to an important generation center.

The simultaneous outage of node 35 and 110 pro-duces a fragmentation of 0.240: the SEN is now split intwo islands, one of them with a high load demand (seeFigure 12).

From Table 3 it is possible to note that for SEN, afragmentation index .0.240, owing to links or nodesremoved, causes the non-convergence of the powerflow.

Figure 7. The Venezuela SEN network.

0

10

20

30

40

50

60

70

80

90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Network Fragmentation

Num

ber o

f Nod

es/L

inks

Rem

oved

NodesLinks

Figure 8. Approximation of PFs for SEN.

582 Proc IMechE Part O: J Risk and Reliability 227(6)

Figure 9. Venezuela SEN four nodes fragmentation result.

Figure 10. Venezuela SEN four links fragmentation result.

Figure 11. Venezuela SEN one node fragmentation result.

Yajure et al. 583

Conclusions

The rationale of network fragmentation is to under-stand which element fragmentation actions decrease thecohesion of a network. This article has proposed theNFP as an alternate/surrogate approach to identify theimportance/criticality of network elements. An advan-tage of evaluating network fragmentation (when com-pared with other data intensive approaches) is that theonly necessary input is the network topology.

The development of the bi-objective NFP model canhelp in identifying increasing cardinality sets of criticalnetwork elements. That is, the solution approach isable to determine fragmentation strategies that yieldthe highest network fragmentation values.

While the conceptual meaning of fragmentation val-ues are difficult to provide, the examples presentedallow for a possible interpretation of the fragmentationindex as it relates to the performance of the networkbeing considered. In particular, the last exampleshowed a fragmentation ‘‘rule of thumb’’ with the realbehavior of the power system: in the Venezuelan SENa fragmentation greater than 0.240 reveals that the sys-tem could experience significant operational problems.

A note of caution is that while, the fragmentationmodel is able to determine, with minimum information,the set of elements that worsens the cohesion, the setdoes not necessarily imply the worst real performancedegradation. Finally, the results provided show thatthere is the potential for fragmentation to be used as aguide in identifying sets of important components inother applications, such as all-terminal networks andtraffic networks.

Declaration of conflicting interests

The author declares that there is no conflict of interest.

Funding

This research received no specific grant from any fund-ing agency in the public, commercial, or not-for-profitsectors.

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Figure 12. Venezuela SEN two nodes fragmentation result.

Table 3. Effects of nodes/links fragmentation: power flow results.

Node outage Node numbers Fragmentation Power flow converges? Load shed (MW)

1 125 0.135 No —2 35-110 0.240 No —3 3-79-83 0.498 No —4 3-4-79-83 0.586 No —Line outage Line numbers Fragmentation Power flow converges? Load shed (MW)1 220 0.096 Yes 02 17-220 0.175 Yes 2933 17-220-247 0.214 Yes 3054 17-188-189-199 0.257 No —

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Appendix

Notation

A set of arcsB subset of NC network connectivity indexCc network connectivity index based on

componentsDF enhanced network fragmentation indexdij minimum distance (in number of nodes)

between nodes i and jE network efficiency indexF network fragmentation indexG stands for graph: no definition is requiredh a generic componentK number of nodes producing the maximum

fragmentation of the networkl=|A| number of linksM set of all maximally connected

componentsn=|N| number of nodesN set of nodesrij Boolean decision variable representing if

node i can reach node j (rij = 1) or not(rij = 0) in the network

sh size of the hth componentx vector of element status xxk binary decision variable representing

element w is removed (xk= 0) or not(xk= 1) in the network

X set of optimal decision vectors x*X* subset of X

Yajure et al. 585