Improving the Survivability of Clustered Interdependent Networks by Restructuring Dependencies
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Improving the Survivability of ClusteredInterdependent Networks by RestructuringDependencies
Genya Ishigaki,
Student Member, IEEE,
Riti Gour,
Student Member, IEEE, and Jason P. Jue,
Senior Member, IEEE ©2019 IEEE. Personal use is permitted. This is the author’s version of an article that has been published in this journal. Changes were made to this versionby the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TCOMM.2018.2889983.
Abstract —The interdependency between different networklayers is commonly observed in Cyber Physical Systems andcommunication networks adopting the dissociation of logic andhardware implementation, such as Software Defined Networkingand Network Function Virtualization. This paper formulates anoptimization problem to improve the survivability of interde-pendent networks by restructuring the provisioning relations. Acharacteristic of the proposed algorithm is that the continuousavailability of the entire system is guaranteed during the restruc-turing of dependencies by the preservation of certain structuresin the original networks. Our simulation results demonstrate thatthe proposed restructuring algorithm can substantially enhancethe survivability of interdependent networks, and provide insightsinto the ideal allocation of dependencies.
Keywords —interdependent networks; network survivability;cascading failure; network function virtualization; cyber physicalsystems.
I. I
NTRODUCTION M ANY network systems encompass layering and integra-tion of the layers in both explicit and implicit manners.For example, Software Defined Networking (SDN) decouplesthe control logic from forwarding functions to realize the flex-ibility and agility of communication networks. Also, NetworkFunction Virtualization (NFV) involves separation of networkfunction logic from hardware. The concept of separating logicfrom hardware implementations is also commonly adopted inCyber Physical Systems (CPS), such as smart grids, in whichcomputing capability manages physical entities.The dissociation of logic and functions, which is effectivefor system flexibility, has accelerated the amount of layeringand obscure dependencies in network systems. The work [1]on software defined optical networks points out the depen-dency of logical nodes on physical nodes that provide physicalpaths for connections among logical nodes, as well as thedependency of physical nodes on the logical nodes throughSDN control messages, which define the operations of thephysical nodes. Similarly, it is revealed that NFV embraces theinterdependency between Virtual Network Functions (VNF)and physical servers hosting the VNFs, when a virtualization
Manuscript submitted March 6, 2019.Genya Ishigaki, Riti Gour, and Jason P. Jue are with the Department ofComputer Science at The University of Texas at Dallas, Richardson Texas75080, USA (Email: { gishigaki, rgour, jjue } @utdallas.edu).An earlier version of this paper has been presented at IEEE InternationalConference on Communications (ICC) 2018. v v ′ v ′ G G Orchestrator v (a) An interdependent network withtwo constituent graphs representingphysical and logical network. v ′ v ′ G G Orchestrator v (b) Initial failure at a physical server v . v ′ v ′ G G (c) Cascading failure affecting a log-ical node v . v ′ G G (d) Cascading failure affecting aphysical server v (cid:48) . The entire net-work becomes nonfunctional.Fig. 1. An example of cascading failure in an interdependent networkrepresenting the dependency between physical servers and NFVs. orchestrator is recognized as one of the VNFs [2]. Further-more, the integration of a control information network and anelectricity network seen in smart grids is a typical example ofthe interdependency of two different layers in CPSs [3]. Thistendency of layering and collaborative functionality of layerednetworks is likely to be more evident for next-generationnetwork systems.However, it has been revealed that certain types of depen-dencies between different layers of networks can deterioratethe robustness of the entire interdependent system [4]. Con-secutive multiple failure phenomena called cascading failures exemplify the unique fragility of such network systems. Innetworks without interdependencies, a failure would influencea certain part of a network. Nonetheless, in networks withinterdependencies, some nodes that are not directly connectedto the failed portion can become nonfunctional due to the lossof service provisioning from nodes in other layers, which aredirectly influenced by the initial failure.Fig. 1 shows an example of such a cascading failure,which starts as a single node failure of v and results in theentire network failure. Suppose that a network G consistsof physical servers v and v (cid:48) , and G represents logicalcomputing nodes v and v (cid:48) hosting VNFs. The orchestrator,which coordinates the mapping between physical and logical a r X i v : . [ c s . N I] M a r EEE TRANSACTIONS ON COMMUNICATIONS 2 layer, is realized as one of the VNFs on v . The arcs from G to G ( ( v , v ) , ( v (cid:48) , v (cid:48) ) ) illustrate the dependency of NFVsor computing nodes on the physical servers, while the arcsfrom G to G ( ( v , v ) , ( v , v (cid:48) ) ) indicate the dependency ofphysical servers on a logical node in terms of the flow ofcoordination messages from the orchestrator to the physicalservers. When the physical server v fails, the logical nodehosting the orchestrator v loses its dependent physical node v , and becomes nonfunctional. This induces another loss ofthe dependent node of v (cid:48) , and eventually the single nodefailure causes a failure of the whole network.Cascading failures can also lead to the malfunctioning ofCPSs. In fact, it has been reported that some major electricityoutages in smart grids, such as the 2003 nation-wide blackoutin Italy [5], and the 2004 blackout over 8 states in US and2 provinces in Canada [6], were due to cascading failuresinduced from poorly designed dependencies between the elec-tricity network and control information network.Many contributions have been made since the first theoreti-cal proposal on the cascading failure model by Buldyrev et al.in 2010 [7]. The pioneering works [7], [8] focus on analyzingthe behavior of cascading failures rather than proposing designstrategies. In contrast, some following works identify vulnera-ble topologies in interdependent networks to avoid such fragilestructures in the design phase by investigating the relationbetween node degree and failure impacts [9], or evaluatingthe importance of nodes exploiting the algebraic expression ofdependencies [10]. Furthermore, other works propose designstrategies in more realistic models to consider the impact offailures caused by a single component [11], integrated factorswithin and between layers [12], or the heterogeneity of nodesin each layer [13].This paper discusses a design problem for interdependentnetworks to improve their survivability, which is a measure ofthe robustness against a whole network failure, by modifyingan existing network topology. The contribution that contrastsour work with other related works is the consideration ofexisting network facilities. Our method is aimed at redesigninga relatively small part of the existing network to enhance thesurvivability so that the entire network remains operationaleven during the restructuring process. In order to realize thiscontinuous availability, a special type of dependency, whoseremoval does not influence the functionality of the entiresystem, is identified in the first step of our restructuringmethod. Our heuristic algorithm increases the survivability ofentire systems by the relocations of these dependencies. Whileour previous work [14] allows a node to have dependencieswith any nodes in the other layer, this paper extends the modelby considering geographical, economic, or logical accessibilityof provisioning by nodes. These constraints are represented asclusters of nodes, and an interdependent network is modeled asa directed graph consisting of multiple clusters. The member-ship of a node in a specific cluster imposes restrictions on thenodes to which the node can provide support, and the nodesfrom which the node can receive support. Hence, possiblemodifications to the dependencies between nodes would vary,depending on the cluster to which a node belongs. Finally,our method is evaluated by simulations in different pseudo interdependent networks.II. R ELATED W ORKS
Most of the preceding works on interdependent networksattempt to analyze the behavior of cascading failures in well-known random graphs, which have certain characteristics indegree distributions and underlying topology [7], [8]. Thoseworks analyze the propagation of failures based on percolationtheory developed in the field of random networks. Followingthe directions shown by a seminal work by Buldyrev et al. in[7], more general models are discussed in [8].The works [9]–[13] focus on the design aspect of interde-pendent networks. The relation between the impact of failuresand interdependencies is empirically demonstrated to decideappropriate dependency allocations in [9]. A method to eval-uate the importance of nodes in terms of network robustnessis proposed in [10] by introducing a novel representation ofinterdependencies based on boolean algebra. This evaluationenables network operators to prioritize the protection of thenodes that contribute more to the robustness of the network.In [12], the authors consider dependency relations not onlybetween layers but also within a single-layer. Combiningmultiple factors that make a node nonfunctional, their methodadjusts the dependency of a node on the other nodes. Thework in [13] also considers the influence within a single-layer, supposing the heterogeneity of nodes. In this model, anetwork can have different types of nodes such as generatingand relay nodes. Zhao et al. [11] formulate an optimizationproblem enhancing the system robustness, defining SharedFailure Group (SFG), a group of nodes that can simultaneouslyfail due to a cascading failure initiated by the same component.Another branch of interdependent network research is re-covery after failures [15]–[20]. The works in [15]–[17] an-alyze the behaviors of failure propagations when each nodeperforms local healing, where a functioning node substitutesfor the failed node by establishing new connections with itsneighbors. The speed of further cascades and resulting networkstates are revealed by percolation theory [15], [16] or steadystate analysis in the belief propagation algorithm [17]. Also,resource allocation problems, which consider the differentroles of network nodes are discussed in [18]–[20]. The orderof assigning repairing resources is a critical problem duringthe recovery phase when the amount of available resourcesis limited. The works in [18], [19] propose node evaluationmeasurements to decide the allocation, while an equivalentproblem in the phase diagram is discussed in [20].Our work proposes a method to improve the survivability ofinterdependent networks, following the survivability definitionin [21]. Our work would be classified into the categoryof protection design methods before failures. Specifically,the proposed method is exploited in a redesign process ofan existing network to enhance the survivability, while theexisting works [9]–[13] discuss the initial design of an entirenetwork. Our protection method, considering the functionalityduring the redesign, would reduce the cost of survivabilityimprovement in contrast to the entire reconstruction of thesystems.
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III. M
ODELING AND M OTIVATING E XAMPLE
In this section, we present a mathematical model for de-scribing interdependent networks, and we present a motivatingexample of our method. Section III-B summarizes related work[21] defining the survivability for interdependent networks,which we adopt to evaluate the networks.
A. Network Model
An interdependent network consists of k constituent graphs G i = ( V i , E ii ) ( ≤ i ≤ k ) and their interdependencyrelationships, which are defined by sets of (directed) arcs A ij ( ≤ i , j ≤ k , i (cid:44) j ) representing the provisioning betweena pair of nodes in different graphs. Edges in E ii ⊆ V i × V i are called intra- edges because they connect pairs of nodes inthe same network. In contrast, arcs in A ij ⊆ V i × V j ( i (cid:44) j ) are called inter- or dependency arcs. If there exists an arc ( v i , v j ) ∈ A ij ( v i ∈ V i , v j ∈ V j ) , it means that a node v j has dependency on a node v i . The node v i is called the supporting node, and v j is a supported node. A node v is saidto be functional if and only if it has at least one functionalsupporting node.When an interdependent network is logically partitioned,each constituent graph G i has a clustering function κ i : V i −→{ , , ..., γ i } , where γ i ∈ N is the number of clusters in G i = ( V i , E ii ) . Then, a graph I xi = ( W xi ⊆ V i , E ii ( W xi )) induced by anode set W xi = { v | κ i ( v ) = x ( ≤ x ≤ γ i )} is called a cluster .Note that this definition insists that a node is in exactly onecluster.In order to emphasize the dependency between constituentgraphs, an interdependent network can be represented as asingle-layer directed graph G = ( V , A ) , where V : = (cid:208) i V i ,and A : = (cid:208) {( i , j )| i (cid:44) j } A ij by abbreviating intra-edges. Withthis notation, a node v is said to be functional if and onlyif deg in ( v ) ≥ . Note that all the discussions in the rest of thispaper follow this single-layer graph representation.Additionally, we introduce a different notation of arcs withrespect to their source nodes. Let A ( v ) ⊆ A represent a set ofarcs whose source node is v ∈ V . To identify each arc duringthe restructuring process, where some arc temporarily loses itsdestination, each arc is denoted as ( v , ·) m ( m = , ..., deg out ( v )) .The index m is a given fixed identification number for each arcin A ( v ) . Hence, every arc in A can be specified by providingsource node v and its identification number m .A set of constituent graphs is totally ordered by the numberof nodes that are the source of at least one intra-arc: | V out i | ,where V out i : = { v ∈ V i | A ( v ) > } . A constituent graph that hasthe least number of nodes with outgoing arcs is named the min-imum supporting constituent graph G i : | V out i | ≤ min j | V out j | . B. Survivability of Interdependent Networks
Parandehgheibi et al. [21] propose an index that quantifiesthe survivability of interdependent networks against cascadingfailures exploiting the cycle hitting set , and they prove that thecomputation of the survivability is NP-complete. They showthat a graph needs to have at least one directed cycle in order tomaintain some functional nodes; in other words, the existence C C v v v v v v v v v Fig. 2. Graph G with ( v , v ) . C C v v v v v v v v v C Fig. 3. Graph G (cid:48) with ( v , v ) . of one cycle prevents an interdependent network from its entirefailure. Thus, the survivability of interdependent networks isdefined as the cardinality of the minimum cycle hitting setwhose removal brings non-functionality for the entire network.Note that a cycle hitting set S is a set of nodes such that anycycle C = ( V ( C ) , E ( C )) in a given graph G = ( V , A ) has at leastone node in the hitting set: S ∩ V ( C ) (cid:44) ∅ , ∀ C ∈ C( G ) , where C( G ) is the set of all cycles in the given graph. This definitionimplies that the entire failure of an interdependent networkoccurs when the corresponding graph becomes acyclic. Let H ( G ) denote a cycle hitting set with the minimum cardinality: | H ( G )| : = min S ∈S | S | , where S is the set of all the cycle hittingsets in G . Formally, the survivability of an interdependentnetwork G is the cardinality of the minimum cycle hittingset, | H ( G )| . C. Motivating Example
Adopting the survivability definition shown above, improv-ing survivability would be equivalent to increasing the numberof disjoint cycles in a graph. Figs. 2 and 3 show an examplecomparing two similar interdependent networks.In graph G in Fig. 2, there exists two cycles: C and C . If v , which is in both V ( C ) and V ( C ) , becomes nonfunctionalbecause of a failure, all the nodes in G eventually lose theirsupporting nodes and become nonfunctional: H ( G ) = { v } .On the other hand, no single node failure can destroy all thethree cycles in G (cid:48) in Fig. 3, while a two-node failure canmake it acyclic (e.g. H ( G (cid:48) ) = { v , v } ). Therefore, the graph G (cid:48) is more survivable than G , since = | H ( G )| < | H ( G (cid:48) )| = , although they differ only in the destination node of onedependency arc ( ( v , v ) in G or ( v , v ) in G (cid:48) ). Supposingthat G is an existing topology of a network, a method thatrelocates ( v , v ) to ( v , v ) can achieve an enhancement of thesurvivability. IV. P ROBLEM F ORMULATION
A. Assumptions
This paper deals with the case in which interdependentnetworks have two types of homogeneous constituent networkswith identical dependencies ( k = ). However, our discussionwith the restriction on k can be easily extended to more generalcases. In more advanced network models, each constituentnetwork can have different types of nodes, such as indepen-dently functional generating nodes and relay nodes, whichneed provisioning from a generating node via paths of intra-edges [13]. Nevertheless, for simplicity, this work follows the EEE TRANSACTIONS ON COMMUNICATIONS 4 assumption in [21] that each node in a constituent network isdirectly connected to a reliable conceptual generating node bya reliable edge (homogeneous constituent graphs). Moreover,it is assumed that each supporting node provides a unit amountof support that is enough for a supported node to be operational(identical dependencies), following the same model in [21].Additionally, this paper presumes that each cluster x re-ceives some support from at least one of the clusters thatare supported by cluster x . In other words, this presumptionexcludes the case that a cluster does not receive provisionsfrom any of the clusters that the cluster is supporting. B. Requirement Specification
One aspect contrasting our scheme to other works is theconsideration to improve the survivability of existing inter-dependent networks by changing some topological structures.Because all the nodes need to remain functional even duringthe relocations of dependency relations, it is necessary toavoid the loss of all supporting nodes for any node at anystage of the restructuring. In other words, each node needsto be survivable from a cascading failure, which requires thedirect or indirect support by the nodes in directed cycles. Thisconstraint is formally represented as the following rule for thelive restructuring.1) Every node remains reachable from a node in a directedcycle via at least one directed path at any stage of therestructuring.In addition to guaranteeing the continuous availability, theamount of provisioning provided by each supporting nodeshould remain the same after the restructuring in order toconsider the capability of each node. The capability could be,for example, the limit on electricity generation, computationperformance, or the number of ports available.2) The number of supports that a node provides mustremain less than or equal to its original provisioningcapability.Furthermore, depending on which cluster a node in graph G i belongs to, the node has a constraint on clusters in G j that it can support. The constraint is given by a supportabilityfunction σ ij : V i −→ γ j , where γ j is the power set ofthe cluster indices in a constituent network G j . This meansthat a node v (∈ V i ) can provide its support to the nodesin the clusters of G j given by the supportability function.This specification corresponds the geographical, economic, orlogical constraints on the accessibility of supports from a nodeto specific groups of nodes. For example, it is impossible forinformation control node v to have electricity supply fromnode u if v and u are geographically far apart or managedby different administrative institutions. The geographical oradministrative domain is shown as a cluster in each constituentgraph, and dependency relations of the nodes should be closedwithin a set of permitted nodes, which are geographicallyclose, or managed by the same company or allied companies,since each cluster should be independent from the outsiders.This constraint relating to network clustering is simply ex-pressed as follows. 3) All the provisionings from a node u are directed towardsthe nodes in the clusters that u can support, as designatedby the supportability function σ ij . C. Clustered ∆ H Problem
This section formulates the clustered ∆ H problem, whichis aimed at enhancing the survivability of a given interdepen-dent network with clusters by restructuring dependency rela-tionships, considering the continuous availability, supportingcapability, and clustering constraint of each node.Considering the continuous availability of an existing net-work during restructuring leads to the formulation of a gradualreconstruction problem, where no relocation of two or moredifferent arcs is conducted at a time. Each phase relocatingone arc is named a step . Let G s = ( V , A s ) denote thegraph representing the interdependent network topology at step s . The improved interdependent network G s + after step s consists of a node set V , which is the same node set as ingraph G s , and an arc set A s + amended by the relocation ofan arc ( u , v ) ∈ A s to ( u , v (cid:48) ) , where v (cid:48) ∈ V is a new destinationfor the arc ( u , v ) .The clustered ∆ H problem is to maximize the difference insurvivability between a given interdependent network, whichis recognized as G , and the resulting network after a sequenceof consecutive improvements. The resulting network is repre-sented as G f , where f denotes the step at which the last arcrelocation is completed. Formally, the objective is to maximizethe difference between | H ( G )| and | H ( G f )| , which is definedas ∆ H . Problem (Clustered ∆ H Problem) . For a given G = ( V = (cid:208) i V i , A ) , the number of clusters γ i ∈ N in each constituentgraph G i , a clustering function κ i : V i −→ { , , ..., γ i } foreach constituent graph G i , and supportability functions σ ij : V i −→ γ j , maximize ∆ H : = | H ( G f )| − | H ( G )| , where G s + = ( V , A s + ) ( ≤ s ≤ f − ) is obtained by the relocation of thedestination of a single arc in A s : A s + = A s \ ( u , v ) ∪ ( u , v (cid:48) ) ,satisfying1) deg in ( v ) G s ≥ ∀ v ∈ V ,2) deg out ( v ) G s + = deg out ( v ) G s ∀ v ∈ V ,3) κ j ( v ∈ V j ) ∈ σ ij ( u ∈ V i ) ∀ ( u , v ) ∈ A s .These three conditions correspond to the three rules de-scribed in Section IV-B. The second and third conditions areeasily derived from the corresponding rules. Lemma 1 showsthe equivalence of the condition 1 and Rule 1. Lemma 1.
When deg in ( v ) G ≥ ( ∀ v ∈ V ) in a connecteddirected graph G = ( V , A ) , (a) G has at least one directedcycle, and (b) any node v ∈ V is reachable from a node u ∈ V that is contained in a directed cycle. Proof. deg in ( v ) G ≥ ( ∀ v ∈ V ) insists that any node v hasat least one parent v (cid:48) . The path v ← v (cid:48) ← ... composed byrepeating the trace of parents can be acyclic until the lengthof the path is | V − | . However, the | V | th node must have atleast one parent from the assumption. Thus, the pigeonholeprinciple indicates that it is necessary that the path forms adirected cycle. (cid:3) EEE TRANSACTIONS ON COMMUNICATIONS 5 G i G j v ′ u ′ u ′′ u ′′′ Fig. 4. Original Dependencies, where ( v (cid:48) , u (cid:48) ) is missing. Note that this fig-ure only shows A ji . The symmetricdiscussion can be done for A i j . (1) G i G j v ′ u ′ u ′′ u ′′′ (2) Fig. 5. Relocation Steps (1) to main-tain the functionality of u (cid:48)(cid:48)(cid:48) , and (2)to form a length-2 cycle with v (cid:48) and u (cid:48) . D. Problem Analysis
This section provides the analysis on the trivial optimalcase of the clustered ∆ H problem with a special setting,where each of constituent graph consists only of one clus-ter. Let ρ (( u , ·) m ) denote the number of relocations that arc ( u , ·) m ∈ A experienced during the restructuring process. Notethat (cid:205) u ∈ V (cid:205) deg out ( u ) m = ρ (( u , ·) m ) = f . From the definition, the optimum survivability cannot ex-ceed the number of supporting nodes, which each have atleast one outgoing arc, in the minimum supporting constituentgraph G i . This is because a set of such nodes covers allthe directed cycles in an interdependent network G . Thisobservation implies that the optimum survivability is achievedwhen every node v i ∈ V i of G i has an injective mapping to anode in V j ( j (cid:44) i ) . In other words, for each node v i in G i , thereexists at least one unique disjoint cycle whose length is 2 with v j in G j . The following lemma gives a sufficient condition toreach the ideal state by repeated relocations while preservingthe problem constraints. Lemma 2.
When the number of relocations for each arc ρ (( u , ·) m ) is not upper bounded, in order to have the opti-mum restructuring, it is sufficient that the minimum support-ing constituent graph G i satisfies | V j | < (cid:205) u ∈ V i | A ( u )| and (cid:205) v ∈ V j | A ( v )| > | V i | ( j (cid:44) i ) . Then, the optimum survivabilitybecomes | V out i | . Proof.
The maximum survivability achievable by restructuringis equal to the number of nodes that have at least one outgo-ing arc | V out i | in the minimum supporting constituent graph G i = ( V i , E ii ) , because the removal of such nodes from G i must destroy all the cycles between G i and another constituentgraph. In order to achieve the maximum survivability via therestructuring process, it is necessary that each node u ∈ V out i belongs to a cycle whose length is 2. Otherwise, the cyclecontains another node w ∈ V out i , and the removals of such w ’smake u lose all incoming arcs. Note that a node in V i \ V out i isnever a part of directed cycles, since it has no outgoing arc.Suppose that we have the minimum supporting constituentgraph G i and another constituent graph G j that satisfy the twoconditions in the lemma. From the definition of the minimumsupporting constituent graph, we can make | V out i | pairs ofnodes (cid:104) u ∈ V out i , v ∈ V out j (cid:105) , which are expected to form alength-2 cycle together after restructuring, so that no two nodesin V i are paired with the same node in V out j .Figs. 4 and 5 illustrate a general example of a restructuringprocess to form such a length-2 cycle by dependency arc relo-cations. Note that the figures only show A ji , but the symmetric argument can be done for A ij . Let (cid:104) u (cid:48) ∈ V out i , v (cid:48) ∈ V out j (cid:105) bea pair such that ( v (cid:48) , u (cid:48) ) (cid:60) A ji . In order to make a length-2cycle between v (cid:48) and u (cid:48) , the arc ( v (cid:48) , u (cid:48)(cid:48)(cid:48) ) should be relocatedto ( v (cid:48) , u (cid:48) ) . However, the relocation makes u (cid:48)(cid:48)(cid:48) lose all of itsincoming arc. The loss of incoming arc of u (cid:48)(cid:48)(cid:48) is alwaysavoided by relocating one of the arcs incoming to u (cid:48)(cid:48) to u (cid:48)(cid:48)(cid:48) (See Figs. 4 and 5 (1)). The supposition in the lemma andthe pigeonhole principle suggest the existence of at least onenode u (cid:48)(cid:48) ∈ V i that has two incoming arcs. After the adjustmentof the provisioning for u (cid:48)(cid:48)(cid:48) by this relocation, the arc ( v (cid:48) , u (cid:48)(cid:48)(cid:48) ) can be relocated to ( v (cid:48) , u (cid:48) ) (See 4 and 5 (2)).For a pair (cid:104) u (cid:48) ∈ V out i , v (cid:48) ∈ V out j (cid:105) such that ( u (cid:48) , v (cid:48) ) ∈ A ij , similar relocations are always possible, because | V j | < (cid:205) u ∈ V i | A ( u )| . Thus, these relocations eventually achieve themaximum survivability by forming | V out i | length-2 cycles thateach consist of a pair (cid:104) u ∈ V out i , v ∈ V out j (cid:105) . (cid:3) Some propositions similar to Lemma 2 appear in related lit-erature [11], [22]. The sufficient condition provided in Lemma2 allows the entire restructuring of inter-arcs by repeatedrelocations of each arc. Therefore, the ∆ H problem is recog-nized as a design problem of an entire interdependent networkdiscussed in [11] under these assumptions. Also, the work [22]claims that such a one-to-one provisioning relation realizes therobustness, while assuming certain structural characteristics ofrandom graphs.However, it is unrealistic to relocate a dependency arcmany times, when considering the overhead of the changesof provisioning relations in network systems. Therefore, thefollowing part of our paper discusses the case where thenumber of relocations are strictly restricted: ρ (( u , ·) m ) ≤ ( ≤ m ≤ deg out ( u ) , ∀ u ∈ V ) . Under this condition, it cannot beguaranteed to obtain the optimum survivability even when thesufficient condition above holds.V. HEURISTIC ALGORITHM FOR ∆ H P ROBLEM
This section proposes a heuristic algorithm for the clustered ∆ H problem. Before providing the details of our heuristicalgorithm, we first define special types of arcs named MarginalArcs (MAs), which are candidates for the relocations inSection V-A. Then, the heuristic algorithm, which consists oftwo algorithms: Find-MAs and ∆ H , is described. The Find-MAs algorithm enumerates all the arcs that match the defini-tion of MAs. With the set of MAs found by the Find-MAsalgorithm, the ∆ H algorithm decides appropriate relocationsof the dependency arcs in the set, considering disjointness ofnewly formed cycles, so that it can improve the survivabilityof a given network.After the discussion for a simple case with only one clusterin each constituent graph in Sections V-B to V-C, Section V-Dexplains how the other cases with multiple clusters are brokendown into the simple case. A. Restructuring of Dependencies
In order to guarantee continuous availability, it is necessaryto classify the dependency arcs into either changeable or fixedarcs. However, it is computationally difficult to know the
EEE TRANSACTIONS ON COMMUNICATIONS 6 v v v v v v v C C Fig. 6. Original graph G with Marginal Arcs ( v , v ) and ( v , v ) . v v v v v v v C C C Fig. 7. Modified graph G (cid:48) with a new arc ( v , v ) . v v v v v v v C C C ′ Fig. 8. Modified graph G (cid:48)(cid:48) with a new arc ( v , v ) . classification beforehand under the condition of ρ (( u , ·) m ) ≤ ( ∀ u ∈ V ) , because this process involves enumeration of allthe permutations of arc relocations and their combinations ofdestinations. Thus, in this paper, the classification is simplifiedby using a sufficient condition, while this enumeration islikely to become another optimization problem for a furtherinvestigation.As observed in Section III-C, increasing disjoint cycles in agiven network could be an important factor to enhance overallsurvivability. Hence, our method maintains all existing cycles,which is sufficient to avoid cascading failures, and tries toreallocate the destinations of the arcs that do not belong todirected cycles and that do not make their descendant nodesnonfunctional. Let the arcs that are not in any cycles in a givendirected graph G = ( V , A ) be called Marginal Arcs (MAs).Formally, the set M (cid:40) A of MAs is defined as M : = {( u , v ) | ( u , v ) (cid:60) A ( C ) ∀ C ∈ C( G )} . (1) Lemma 3.
A removal of any marginal arc never decreases thesurvivability of an interdependent network: | H ( G )| ≤ | H ( G )| ,where G is a given graph, and G is the graph obtained by theremoval. Proof.
Let M be a set of marginal arcs. From the definitionof MAs (Eq. (1)), the removal of MAs does not destroy orconnect any existing cycles in G = ( V , A ) . Therefore, | H ( G )| = | H ( G )| , where G = ( V , A \ M ) . (cid:3) Moreover, appropriate relocations of the removed MAscould improve the survivability of interdependent networks,assuring operability during the relocation process and main-taining the provisioning capability of each node. Let usanalyze the effect of dependency relocations using simpleexamples in Figs. 6-8. The given graph G in Fig. 6 has twomarginal arcs: M = {( v , v ) , ( v , v )} . In order to maintainat least one supporting node for v , one of the MAs hasto remain the same, and the other can be relocated. Fig. 7shows the case of relocating ( v , v ) to ( v , v ) ; on the otherhand, Fig. 8 indicates the case of relocation of ( v , v ) to ( v , v ) . Even though one new cycle ( C and C (cid:48) respectively)is formed by each relocation, the modified graphs G (cid:48) and G (cid:48)(cid:48) have different survivability: | H ( G (cid:48) )| = ( = H ( G )) , and | H ( G (cid:48)(cid:48) )| = . This is because the cycles in G (cid:48) are not disjointwith each other: V ( C ) ∩ V ( C ) ∩ V ( C (cid:48) ) (cid:44) ∅ ; in contrast, V ( C ) ∩ V ( C ) ∩ V ( C (cid:48)(cid:48) ) = ∅ in G (cid:48)(cid:48) . Therefore, it could be saidthat the appropriate relocation for improving survivability isto form disjoint cycles. Algorithm 1 ∆ H -algorithm ( G , l ) Input: subgraph (directed graph) G = ( V , A ) , maximum hop l ∈ N (odd) M ← find-MAs( G ) M ⊂ A for each ( v , w ) ∈ M do if deg in ( w ) ≥ after A \ {( v , w )} then while True do pick C ∈ C( v ) (randomly) for i ← l ; i > ; i ← i − do pick u ∈ V ( C ) : d C ( v , u ) = i if u (cid:60) U then A ← A \ ( v , w ) ∪ ( v , u ) U ← U ∪ { n | d C ( v , n ) ≤ i } break to next arc in M (line 2) end if end for end while pick ( u , v ) ∈ A in ( v ) (randomly) A ← A \ ( v , w ) ∪ ( v , u ) end if end for B. Find-MAs Algorithm
The Find-MAs algorithm first distinguishes MAs M , whichare candidate arcs for relocations, from the arcs in directedcycles in a given graph G = ( V , A ) , by employing Johnson’salgorithm [23]. Johnson’s algorithm enumerates all elementarycycles in a directed graph within O ((| V | + | E |)(|C( G )| + )) . It isenough for distinguishing MAs to obtain elementary directedcycles because any non-elementary cycle can be dividedinto multiple elementary cycles within which dependencyrelationship are closed. After the enumeration of cycles in G by Johnson’s algorithm, the set of MAs is obtained by M ← A \ (cid:208) C ∈C( G ) A ( C ) . C. ∆ H Algorithm
With the set of MAs obtained by Johnson’s algorithm,the ∆ H algorithm (shown as pseudo code in Algorithm 1)relocates the destinations of MAs, considering disjointness ofnewly created cycles. (See the discussion in Section V-A.)For each MA ( v , w ) , our algorithm first checks whether or notthe relocation of this MA causes the loss of supports for thecurrent destination w : deg in ( w ) G = ( V , A \{( v , w )}) ≥ (line 3). EEE TRANSACTIONS ON COMMUNICATIONS 7 C v v v v v v v Fig. 9. A given graph G with M = {( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v )} . C v v v v v v v C C C C Minimal-addRegular relocation
Fig. 10. A modified graph G (cid:48) with new arcs: ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) . If w still has some supporting node after the removal of ( v , w ) , the next step is determining a new destination for ( v , · ) .Our algorithm randomly selects one of the cycles that containsthe source v denoted by C ∈ C( v ) (line 5). There may bemultiple possible candidate nodes for a new destination in thecycle C . Thus, the new destination is decided by the size of thenewly formed cycle, which is a result of the relocation (line6, 7). To represent the size of the newly formed cycle, thedistance from a node v to a node u in an (existing) cycle C inthe counter direction is denoted as d C ( v , u ) in our pseudo code.When the maximum hop is designated by l , the algorithm triesto make a new cycle with size l + using a node u , such that d C ( v , u ) = l , as the destination of the MA. If it fails to formthe cycle, it attempts to compose a smaller cycle using a node u (cid:48) such that d C ( v , u (cid:48) ) = l − . Because of the definition of thedependency, an arc must span between two different layers orconstituent networks. Since the node at d C ( v , u ) = l − in C is in the same constituent network as the source node v , itcannot be a new destination.Consider an example using a given graph G shown in Fig.2 and the restructured graph in Fig. 3. Since the removal of ( v , v ) does not make v lose all its incoming dependency arcs,our algorithm tries to relocate the destination of this arc to oneof the nodes in the cycle C , which are v , v , v . For instance,in the case l = , a new cycle C is formed as depicted in Fig.3 by choosing v , that satisfies d C ( v , v ) = l ( = ) . Similarly,if l is initialized to , a new cycle C is formed using { v , v } .After selecting a destination candidate u in line 7, ouralgorithm checks if u is already used to create a new cycle(line 8). This is confirmed by a set of nodes U storing all thenodes that are in newly formed cycles: { n | d C ( v , n ) ≤ i } (line10). For instance in Fig. 3, U ← U ∪ { v , v , v , v } . As will beunderstood, when another MA tries to form a new cycle usingone of these nodes in U , the new cycle and C share somenodes, which means that those cycles are not disjoint. Also,the arc set A is updated when the new destination is finallyfixed (line 9).If there exists no possible destination for an MA ( v , w ) that satisfies all the conditions, the relocation of the MA is conducted by randomly selecting an incoming arc of v , ( u , v ) and relocating ( v , w ) to ( v , u ) , so that it composes a cycleof length 2 (line 15, 16). This random selection is named Minimal-add process.The MAs relocated by the Minimal-add process satisfyeither of the following cases: 1) The node v does not be-long to any cycles: C( v ) = ∅ , or 2) all the nodes in thecycles of C( v ) are already used to compose new cycles byother MAs. Figs. 9 and 10 show examples of these twoconditions (dashed arcs). A given graph G has the MA set M = {( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v )} .Eventually, the ∆ H algorithm respectively relocates ( v , v ) and ( v , v ) to ( v , v ) and ( v , v ) . Because v is not in anycycles in G (reason 1), the Minimal-add process picks thesource of one of the current incoming arcs in A in ( v ) , v asthe new destination. Also, ( v , v ) does not have any possibledestinations that are not in the set U (reason 2), and it isrelocated to ( v , v ) by the Minimal-add process. D. Application to Clustered Networks
Our heuristic algorithm employs another algorithm named
Decompose-cluster to form subgraphs, which indicate candi-date destinations for the MAs in each cluster, from a giveninterdependent network. When interdependent networks areclustered, the modification of the destinations of MAs needs tobe conducted under more constraints given by supportabilityfunctions σ ij : κ j ( v ∈ V j ) ∈ σ ij ( u ∈ V i ) ∀ ( u , v ) ∈ A . TheDecompose-cluster algorithm selects each cluster (node set W xi ( ≤ i ≤ k , ≤ x ≤ γ i ) ) and collects MAs ( u , v ) whosesources are in the cluster ( u ∈ W xi ), or whose destinations andsources are respectively in the cluster W xi and in a cluster in σ ij ( v ) ( v ∈ W xi & κ j ( u ∈ V j ) ∈ σ ij ( v )) ). Using the collectedMAs and their endpoints, a subgraph Y for reallocations ofMAs in W xi is composed. Each subgraph for each clusteris given to the ∆ H -algorithm so that it can improve thesurvivability by restructuring dependencies in the subgraph.As will be understood, no directed cycles exist if no MAmatches the condition of v ∈ W xi & κ j ( u ∈ V j ) ∈ σ ij ( v ) .However, this is not going to happen in our work due to theassumption mentioned in Section IV-A. Note that the absenceof such MAs means that nodes in a cluster x are not providedany support by the nodes that receive some supports from thenodes in the cluster x . E. Complexity Analysis
The Decompose-cluster algorithm extracts (cid:205) ki = γ i sub-graphs from a given graph G = ( V , A ) . The number of clusters γ i in each constituent graph tends to be much smaller thanthe number of nodes; thus, (cid:205) ki = γ i can be considered as aconstant. In order to compose each subgraph, the algorithmrequires to check the source and destination of each arc in A .However, each edge appears in exactly one subgraph becauseof the used edge set D . Therefore, the total complexity of theDecompose-cluster algorithm is O (| V | + | A |) .The complexity of the ∆ H -algorithm is sensitive to thenumber of cycles in the interdependent network. It is knownthat Johnson’s algorithm finds all elementary cycles within EEE TRANSACTIONS ON COMMUNICATIONS 8
Algorithm 2
Decompose-cluster( G ) Input: interdependent network (directed graph) G = ( V = (cid:208) ki = V i , A ) , clustering functions κ i D ← ∅ for a node set W xi ( ≤ i ≤ k , ≤ x ≤ γ i ) do P ← ∅ , R ← ∅ for each ( u , v ) ∈ A \ D do if u ∈ W xi or ( v ∈ W xi & κ j ( u ∈ V j ) ∈ σ ij ( v )) then P ← P ∪ { u , v } R ← R ∪ ( u , v ) D ← D ∪ ( u , v ) end if end for compose graph Y = ( P , R ) ∆ H -algorithm( Y , l ) end for O ((| V | + | E |)(|C( G )| + )) . The ∆ H -algorithm determinesa new destination after l × C( G ) searches for each MA,in the worst case. When only one cycle whose size is 2exists in the input, and the other nodes are supported bythe cycle, the size of the set M becomes | E | − . It isobvious that the complexity of the Minimal-add process is O ( ) , so the worst case analysis takes the case where all MAsare reallocated by the ∆ H -algorithm. Thus, its complexity is O ((| V | + | E |)(|C( G )| + )) + O ((| E |− )((cid:100) l (cid:101)×|C( G )|)) . Assumingthe maximum hop l is small enough to be considered as aconstant, the overall complexity of our heuristic algorithmbecomes O ((| V | + | E |)|C( G )|) . Note that the assumption on l is valid with our strategy, which tries to increase disjointdirected cycles in a given graph. F. Optimality in Special Graphs
In order to analyze the performance of our heuristic al-gorithm, we consider the survivability improvement in specialgraphs where either an exhaustive search gives us the optimumsurvivability, or some special properties allow us to computethe optimum.In the analysis, the upper bound of the survivability im-provement, which is used as a benchmark for the rest of thispaper, is calculated based on the number of the MAs thatsatisfy the following two conditions. First, let V s be a set ofnodes that hold more than one MA, and M s be a set of MAswhose source nodes are in V s . Even when the MAs from v ∈ V s form more than one new cycles, the removal of such a sourcenode v can destroy all the newly formed cycles. This indicatesthat restructuring increases the survivability by at most | V s | ,when relocating MAs in M s . Second, let V d be a set of nodeswhose incoming arcs are all MAs, and M d be a set of MAswhose destination nodes are in V d . If all the MAs incidentto v ∈ V d are relocated, v loses its functionality during thisrestructuring. Therefore, at least one MA should remain asan incoming arc to v . This implies that the number of cyclesnewly formed by the MAs in M d is at most | M d | − | V d | . Thus,the upper bound U is obtained by | M | − | M s | + | V s | − | V d | .Fig. 11 illustrates a comparison of our algorithm with theoptimum solution in a small interdependent network such that Su r v i v a b ili t y | H | ( a pp r o x i m a t e d ) Method Name
Fig. 11. Numerical comparison with the optimum solution in a smallinterdependent network. Su r v i v a b ili t y | H | ( a pp r o x i m a t e d ) The Size of Cycle in Path-Sunlet Graphs | C | Delta HOPTUpper Bound
Fig. 12. Survivability of MA-saturated Path-Sunlet graphs ζ ( G ∈ L) withtwo length-3 paths: |P | = , k i = ( ∀ P i ∈ P) . each constituent graph has 15 nodes, and the number of depen-dency arcs is 84, including 5 MAs. The optimum solution isobtained by an exhaustive search of 759,375 combinations ofreallocations. This numerical example shows that the solutiongiven by the ∆ H algorithm would not provide solutions thatare exceptionally divergent from the optimum solution. It alsoinfers that the upper bound is not tight in general.Fig. 12 indicates that the survivability obtained by ourrestructuring heuristic algorithm matches the optimum in aspecial class of graphs, which are named MA-saturated Path-Sunlet Graphs ζ ( G ) , G ∈ S . The optimum value of surviv-ability for these graphs is always computable based on thefollowing discussion. Definition 1.
Path-Sunlet Graphs L : A set of graphs satisfyingthe following conditions are named Path-Sunlet graphs. Let L denote the set of Path-Sunlet graphs. • G ∈ L only has one cycle C . • The arcs that are not in the cycle C form a set ofdisjoint paths whose initial nodes are in C : P = { P i = ( v i , v i , ..., v ik i ) | v i ∈ C and P i ∩ P j = ∅ ( ∀ P j (cid:44) i ∈ P)} . Definition 2.
MA-saturation ζ δ ( G ) of a graph G : The MA-saturation is an operation of adding additional arcs to a givengraph until any addition of an arc makes the graph non-simple,maintaining the out-degree constraint that the out-degree ofany node does not exceed a given constant δ ∈ N . Remark.
The optimal restructuring of MAs in MA-saturatedPath-Sunlet graphs ζ ( G ) , G ∈ L consists of forming length-2cycles using an MA and an edge in either P i ∈ P or C .We consider the cases where |P | ≥ , because the surviv-ability in the case of |P | = is obviously (cid:108) | V ( C )| (cid:109) . Lemma 4.
By removing arcs that are not in any cycle, theoptimally restructured MA-saturated Path-Sunlet graph ζ ( G ) is decomposed into some sequence of cycles. EEE TRANSACTIONS ON COMMUNICATIONS 9
Proof.
Three or more cycles do not meet at the same node,since δ = . Therefore, the only possible topology withmultiple length-2 cycles is a chain of cycles, in which twocycles share exactly one node. (cid:3) Lemma 5.
The survivability of the optimally restructuredMA-saturated Path-Sunlet graphs ζ ( G ) , G ∈ L is (cid:205) q ∈ Q (cid:6) q (cid:7) ,where Q is the set of all the sequences of cycles obtained byremoving the arcs that are not in any cycles. Proof.
A removal of one node that is shared by two cyclesbreaks the two cycles. When q is even, the process gives usthe survivability of q . If q is odd, one additional removal isneeded to destroy the remaining cycle. Thus, the survivabilityof a sequence of q cycles is (cid:6) q (cid:7) .Since each sequence in Q is disjoint with the other, thesurvivability of the entire graph is obtained by summing upthe survivability of each sequence. (cid:3) VI. S
IMULATION
In order to understand the performance of the proposedalgorithm, our simulations are conducted in both non-clusteredand clustered interdependent network models of differentsizes. The results from the simplest cases where each con-stituent network only consists of one cluster (non-clustered)are first described, and the clustered cases follow.
A. Network Topology
The performance of the proposed algorithm is analyzedin random directed bipartite graphs that contain at least onedirected cycle. Assuming the situation in which a currentinterdependent network is working normally, each node iseither a member of some cycle or reachable from a node in acycle through some directed path in the input graph. Becauseour algorithm only concerns the dependency arcs between 2constituent graphs ( k = ), any interdependent network isrepresented as a directed bipartite graph whose arcs connecta pair of different types of nodes.Each random bipartite graph is generated by specifying thefollowing parameter: V i , max v ∈ V deg in ( v ) and min v ∈ V deg in ( v ) .In order to observe the performance in different conditions,experiments are conducted in symmetric and asymmetric in-terdependent networks. A symmetric interdependent networkhas constituent networks which each have identical numberof nodes: | V | = | V | , while constituent networks of anasymmetric interdependent network have different number ofnodes: | V | = | V | q ( q ∈ N ) . The degree of each node isdetermined based on the uniform distribution between thegiven maximum and minimum incoming degree. B. Clustering Settings
As the non-clustered cases have symmetric and asymmetricconstituent graphs, clustered interdependent networks are alsoexamined in three patterns of topology configurations. In oursimulations, each constituent graph has three clusters: W i , W i and W i ( i = , ) (See Fig. 13). In symmetric cases, a pair ofcorresponding clusters in different constituent graphs have the W W W W W W NW1NW2 soliddasheddotted
Fig. 13. Dependency models of clustered interdependent networks. Arrowsshow the dependency relationships between clusters. Model 1: solid. Model2: solid and dashed. Model 3: solid, dashed, and dotted. same number of nodes: W x = W x , while a cluster is half-sizedto the corresponding cluster in the other constituent graph inasymmetric models: W x = W x .Also, Fig. 13 illustrates the three models that have differentdependency relationships indicated as arrows. Note that whenan arrow is drawn from W xi to W x (cid:48) j , it means that the nodes incluster W x (cid:48) j can have supports from the nodes in W xi . Model1 consists only of the solid arrows, which means that eachpair of corresponding clusters has dependency relationships.Model 2 has the dependencies illustrated by the solid anddashed arrows, while Model 3 has all the arrows (solid, dashedand dotted). A major difference between these models is thepossibility for a network to have some directed cycles overthree or more clusters. In Model 1 and 2, directed cycles areable to exist only in a subgraph consisting of W and W , W and W , or W and W , while a directed cycle can lie overthe entire graph containing all the clusters in Model 3. C. Metrics
The survivability of the given graphs, restructured graphs,randomly reassigned graphs, and the upper bound of theimprovement are illustrated in our results. The random reas-signments of MAs are conducted with a uniform distributionover all the nodes in the other constituent graph from theconstituent graph that includes the source of an MA.Computing the size of the cycle hitting set is knownto be NP-complete even in bipartite graphs, so the exactvalue cannot be obtained in larger graphs. Our evaluation isconducted using a well-known approximation algorithm whoseapproximation factor is ln | V | + [24].Furthermore, the density of a given graph G = ( V , A ) definedby | A | (cid:206) i | V i | is used to examine the relationship between thesurvivability improvement, and the maximum and minimumdegrees. D. Results1) Non-clustered Cases:
Figs. 14 and 15 illustrate thesurvivability of the given and restructured graphs with identicaland halved size constituent graphs, respectively. In both cases,our method demonstrates more improvement of the survivabil-ity compared to the random reassignment. The survivability ofthe original graphs | H ( G )| maintains a similar value regardlessof the size of graphs, though the survivability of the graphsrestructured by our method | H ( G (cid:48) )| steeply increases alongwith the size of the graph. Since, in the original graph G , arcsare randomly added, it could be difficult to form larger directedcycles. Therefore, it is reasonable that the number of disjoint EEE TRANSACTIONS ON COMMUNICATIONS 10 Su r v i v a b ili t y | H | ( a pp r o x i m a t e d ) The number of vertices | V | and | V | OriginalRestructured ( l = 1)Restructured ( l = 3)RandomUpper Bound Fig. 14. Survivability of interdependent net-works before and after the improvement un-der | V | = | V | , max v ∈ V deg in ( v ) = , and min v ∈ V deg in ( v ) = , and l = , . Su r v i v a b ili t y | H | ( a pp r o x i m a t e d ) The number of vertices | V | and | V | OriginalRestructured ( l = 1)Restructured ( l = 3)RandomUpper Bound Fig. 15. Survivability of interdependent networksbefore and after the improvement under | V | = | V | , max v ∈ V deg in ( v ) = , min v ∈ V deg in ( v ) = , and l = , . ✵✺✶(cid:0)✁✂✷✄☎✆ ✝ ✞✳✟✠ ✡☛☞ ✌✍✎✏ ✑✒✓ ✔✕✖✗✦❍✭❛✘♣r♦①✐♠✙✚❡❞✮ ❚❤✛ ❉✜♥s✢t② ✣✤ ✥ ●✧★✩✪❘✫✬✯✰✱❝✲✉✴✸✹✻✼✽✾✿❀ Fig. 16. The relationship between graph densityand ∆ H . Su r v i v a b ili t y | H | ( a pp r o x i m a t e d ) The number of vertices | W i | (= | W i | ), and | W i | OriginalRestructured ( l = 1)RandomUpper Bound Fig. 17. Survivability of clustered interdependentnetworks (Model 2) before/after the improvementunder | W | = | W | = | W | = | W | , | W | = | W | , max v ∈ V deg in ( v ) = , min v ∈ V deg in ( v ) = , and l = . Su r v i v a b ili t y | H | ( a pp r o x i m a t e d ) The number of vertices | W i | (= | W i | ), and | W i | OriginalRestructured ( l = 1)RandomUpper Bound Fig. 18. Survivability of clustered interdependentnetworks (Model 2) before/after the improvementunder | W | = | W | = | W | = | W | , | W | = | W | , max v ∈ V deg in ( v ) = , min v ∈ V deg in ( v ) = , and l = . Su r v i v a b ili t y | H | ( a pp r o x i m a t e d ) The number of vertices | W i | (= | W i | ), and | W i | Model 1Model 2Model 3Additive
Fig. 19. Comparison of survivability amongdifferent dependency models. cycles indicates the tendency to stay within a similar rangeof values. On the other hand, there would exist more MAs inlarger graphs, because these graphs have more arcs that arenot in directed cycles. This results in dramatic enhancementof the survivability in larger graphs. The difference caused bythe given maximum hop l for our algorithm remains smallover all sizes of a graph.Fig. 16 indicates the relationship between the density ofgraphs and ∆ H , the amount of survivability improvement. Wecompare our method to the random reassignment. The resultshows that, in graphs with lower density, our method hasgreater success in increasing the survivability. An observedgeneral trend of our method is the gradual decrease in ∆ H inaccordance with the density. This trend seems to be inducedby the fact that the graphs with more arcs have a higherpossibility of composing cycles even in the original topology.This implies that graphs with higher density have fewer MAsthat can form new disjoint cycles. On the other hand, therandom reassignment does not demonstrate its effectivenessfor the improvement in graphs with any density, which isthe same result from Figs. 14 and 15. Moreover, the randomreassignment sometimes decreases the survivability ( ∆ H < ).It is conceivable that the reassignment connects two (or more)cycles and make it possible to decompose all these cyclesby the removal of a node. This result implies that imprudentrestructuring of the dependency may cause more fragility ofthe interdependent networks.
2) Clustered Cases:
The results in clustered interdependentnetworks whose dependency relationships follow Model 2 areshown in Figs. 17 and 18. Similar trends to non-clustered cases are observed for both symmetric and asymmetric cases. Theproposed method succeeds in increasing the survivability fordifferent sizes of interdependent networks.Fig. 19 illustrates the difference in survivability after re-structuring among the three types of dependency models ofsymmetric networks. The value of “Additive” is obtained bythe simple addition of non-clustered cases that jointly composea clustered case. For instance, the case of clustered networksconsisting of 20, 40, and 20 nodes clusters is compared withthe sum of the survivability of the cases of non-clusterednetworks of 20, 40, and 20 nodes shown in Fig. 14. Thedependency relations among clusters increase from Model 1to Model 3 (See Fig 13).Model 1 gives similar survivability to the simple addition ofnon-clustered cases, since a pair of corresponding clusters intwo constituent graphs is independent from the other pairs inthis model. In Model 2, the survivability of the entire networkincreases, because the nodes in cluster W i can have moresupports from the clusters whose cycles are disjoint fromthe cycles in W i . Although more supports exist among theclusters in Model 3, its survivability is less than the othermodels. In Model 3, a cycle can lie on more clusters becauseof the bidirectional dependencies among all the clusters. Thistopological characteristic is likely to increase the overlappingof multiple cycles and results in the decline of survivability inthis model. These results cast a doubt on a naive statementclaiming that the increase of dependencies induces morefragility in general interdependent networks. EEE TRANSACTIONS ON COMMUNICATIONS 11 ✷(cid:0)✳✺✸✁✂✄✹☎✆✝✞✟✠✡ ☛☞✌✍✎ ✏✑✒✓✔ ✕✖✗✘✙ ✚✛✜✢✣ ✻✤✥✦✧ ✼★✩✪✫ ✽✬✭✮✯ ✾✰✱✲✴✶✵✿❀❁❂❃ ❄❅❆❇❈❉❊❋●❍■❏❑▲▼◆❖P❚◗❡♥✉♠❜❘❙♦❯❢❱❲❳❨❩❬❭❞❪❫❴❵❛❝❣ ❤✐❥❦❧♣qrst✈✇①②③④⑤⑥⑦⑧⑨⑩❶❷❸❹❺❻❼❽ ❾❿➀➁➂➃ ➄➅ ➆➇➈➉➊➋➌➍ ➎➏➐➑ ➒➓➔ →➣↔↕➙➛➜➝➞➟➠➡➢➤➥➦➧➨➩➫➭➯➲➳➵➸➺➻➼➽➾➚➪➶➹➘➴➷➬➮➱✃❐❒❮❰ÏÐÑÒÓ ÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõö÷øùúûü
Fig. 20. The number of failed nodes (Worst case and Average) after a singlenode failure under | V | = | V | , max v ∈ V deg in ( v ) = , min v ∈ V deg in ( v ) = ,and l = . VII. D
ISCUSSION : I
MPACT A LLEVIATION VS S URVIVABILITY
Although it is not the primary focus of this paper, in thissection, we evaluate the behavior of the proposed algorithmsin terms of its effect on the size or impact of a cascadingfailure. Fig. 20 illustrates the influence of our dependencymodifications on the size of cascading failures induced by asingle node. In this experiment, the impact of a single nodefailure at a node v is defined as the number of nodes θ v thatbecome nonfunctional after a cascading failure initiated by thefailure of v . The results are analyzed in terms of the followingtwo metrics: • Worst (non-filled points): the size of the largest cascadingfailure: max v ∈ V θ v , • Average (filled points): the average size of all possiblecascading failures: (cid:205) v ∈ V θ v | V | .The robustness of restructured networks against a singlenode failure always declines in comparison with the originaltopology. The decline in the size of the largest cascadingfailure is most remarkable in the case of | V | = | V | = in our simulation. In this case, the size of a cascading failureincreases by 1 node after the restructuring.In general, the concentrations of provisioning on a certainportion of a network can improve the survivability, thoughit can make the other portions more fragile. In contrast,appropriate distributions of provisioning are necessary in orderto alleviate the impact of any possible single node failure.This difference in robustness against single node failures andsystem survivability could be a reason for the decline.However, when examining the average size of cascadingfailures, it is observed that the increase in the average numberof failed nodes is suppressed within . nodes over all net-work sizes. Thus, it could be said that our method does notdeteriorate the robustness against single node failures.VIII. C ONCLUSION
This paper addresses the design problem of survivable clus-tered interdependent networks under some constraints relatingto the existence of legacy systems during restructuring. Basedon the definition of the survivability proposed in a relatedwork, it is claimed that the increase of disjoint cycles couldenhance the survivability. The proposed heuristic algorithmtries to compose new disjoint cycles by gradual relocationsof certain dependencies (Marginal Arcs) in order to guarantee the functionality of existing systems. Our simulations indicatethat the algorithm succeeds in increasing the survivability,especially in networks with fewer dependencies. Moreover, theempirical result implies that the number of dependencies, ingeneral, is not the root cause of the vulnerability to cascadingfailures. Rather, the appropriate additions of dependenciescan improve the overall survivability, while poorly designeddependencies make networks more fragile. When redesigningthe interdependency between control and functional entitiesin SDN, NFV, or CPSs based on the proposed algorithm, thepossibility to experience catastrophic cascading failures woulddecrease. R
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Genya Ishigaki (GS’14) received the B.S. andM.S. degrees in engineering from Soka University,Tokyo, Japan, in 2014 and 2016, respectively. Heis currently pursuing the Ph.D. degree in computerscience at Advanced Networks Research Laboratory,The University of Texas at Dallas, Richardson, TX,USA. His current research interests include designand recovery problems of interdependent networks,and software defined networking.
Riti Gour received her BE degree in Electronics andTelecommunication Engineering from S.S.C.E.T,Bhilai, India, in 2012, and her MS degree inTelecommunications Engineering from the Univer-sity of Texas at Dallas, Texas, in 2015. Since 2015,she has been working towards her Ph.D. degreeat UT Dallas, majoring in telecommunications. Herresearch is focused towards survivability of opticalnetworks against correlated failures and disastersusing graph optimization and machine learning tech-niques.