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Designing Optimal Interlink Patterns to MaximizeRobustness of Interdependent Networks against

Cascading FailuresSrinjoy Chattopadhyay∗, Huaiyu Dai†, Do Young Eun‡ and Seyyedali Hosseinalipour§

Dept. of Electrical and Computer Engineering, North Carolina State University, USAEmail: ∗schatto4@ncsu.edu, †hdai@ncsu.edu, ‡dyeun@ncsu.edu, §shossei3@ncsu.edu

Abstract—In this work, we consider the optimal design ofinterlinks for an interdependent system of networks. In contrastto existing literature, we explicitly exploit the information ofintra-layer node degrees to design interdependent structures suchthat their robustness against cascading failures, triggered byrandomized attacks, is maximized. Utilizing percolation theorybased system equations relating the robustness of the networkto its degree sequence, we characterize the optimal design forthe one-to-one structure, with complete interdependence andpartial interdependence, under randomized attack. We alsoextend our study to the one-to-many interdependence structureand the targeted attack model. The theoretically derived optimalinterdependence structures have been verified using simulationson Scale-Free networks.

Keywords—Interdependent networks, optimal interlinks, targetedattack, network robustness.

I. INTRODUCTION

Technological progress over the past decade is making itincreasingly difficult for modern networks to be isolated fromeach other in nature. We find the emergence of interdependentsystems in a wide variety of scenarios in the cyber andphysical worlds. Smart grid is one prominent example, wherethe power network comprising distribution stations dependson the communication network for monitoring and control,while the communication network routers rely on the powernetwork for supply of electricity [1]. Interdependence can alsobe observed in biological processes, where some proteins areinvolved in several interacting processes [2]; and in transporta-tion networks, comprising interconnected networks of railwaysand roadways [3]. Although inter-connectivity of networksincreases the functionality as compared to their isolated parts,such systems are significantly more prone to node failures andattacks [1]. The reason behind this is the phenomenon calledcascading failure, where malfunctioning nodes in one layer ofthe network may cause a recursive cascade of failures amongstthe interdependent layers, which might end up disintegratingthe entire network. We have seen several such examples in thereal world, including large-scale electrical blackouts in the US,Canada, and Italy [4]. Due to the ubiquity of interdependentnetworks and their increased sensitivity to failure of nodes,robustness of interdependent networks is becoming one of theprime research foci for complex networks [4, 5, 6, 7].

Most existing works in this area are devoted towards abetter understanding of cascading failure. One widely adoptedmodel in this field is the randomized attack on the one-to-onesystem with complete bidirectional interdependence, whereevery node in a layer is coupled to a single node in the otherlayer and nodes are attacked randomly without any knowledgeof the network topology [1]. Recently, this basic model hasbeen extended to consider targeted attacks, where nodes areattacked on the basis of their intra-layer degree [8]; partialinterdependence, where some nodes are autonomous, meaningthey can survive without being coupled to the other layer[9]; and multiple interdependence, where a node may havemultiple supporting nodes in the other layer [4, 10]. Theseworks have focused on the modeling and analysis of cascadingfailure to predict the network robustness against node failuresunder the assumption that the interlink structure is randomin nature. One exception is [4], which explores the optimalallocation of interlinks, maximizing the network robustnessagainst random attacks, in the absence of intra-layer topologyinformation. However, to the best of our knowledge, utilizationof the topological properties of the constituent layers to designthe structure of interlinks is largely missing from the currentliterature. A few recent works such as [11] have shown thatparticular interlink designs, like positive correlation of degreesof interdependent nodes, perform best under certain circum-stances. However these works have supported their resultsmainly through simulations. In [9], the authors have used theKirchoff’s index (a metric reflecting structural centrality) toshow that partially interdependent networks are more robustwhen nodes of lowest centrality from two layers are coupled.Similar results have been indicated in [12].

In comparison to these works, which analyze various heuris-tically chosen interlink structures, the main focus of our work 1

is to theoretically obtain the optimal interdependence structurebetween two interdependent network layers, utilizing the nodedegree based topological information, so as to maximize thenetwork robustness against failure of nodes. An importantthing to note here is that since we are using a percolationtheory and mean-field based modeling of the cascading failure

1A subset of the work presented here was published in IEEE Globecom 2015[13]. This work was supported in part by the US National Science Foundation(NSF) under Grants ECCS-1307949, EARS-1444009, CNS-1217341, CNS-1423151, and in part by Army Research Office under Grant W911NF-17-1-0087.

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phenomenon, the results of maximization refer to the meannetwork robustness. This is an inherent feature of all worksin this area which consider the degree distribution of nodes toestimate network robustness. The two important assumptionsof our work are: i) the network comprises two interdependentlayers, and ii) degree based topological information is utilizedfor obtaining optimal structures. Both of these assumptionsserve the purpose of simplifying the mathematical analysis andare prevalent in most works in this area. Although a few workslike [14, 15] do take into consideration networks comprisingmultiple layers, to the best of our knowledge these works aretargeted towards modeling of cascading failure rather thanobtaining an optimal network design. It can also be notedthat there exist relevant real world instances, like the smartgrid with interdependent power distribution networks andcommunication networks, where the network comprises twomajor layers [1, 4, 5, 6, 16] or we can single out the two mostimportant layers and focus on design of their interdependencestructures. Thus although the two-layer scenario is a somewhatsimplified, it bears practical importance as well. With regardto the second assumption,since this is one of the first works toconsider topological information in network design, we adhereto the prevalent assumptions in this area [1, 4, 8, 16] so asto obtain insightful results relating degree based topologicalinformation to network robustness at the cost of simplernetwork models. Although we will discuss the applicabilityof our results to other kinds of topological information likebetweenness and closeness, the theoretical results in our workare derived for a given degree sequence of nodes.

Our work advances the current art in the following aspects.Firstly, we explicitly exploit the topological information; inparticular, the intra-layer node degrees, to obtain the optimalinterlink strategies maximizing robustness of interdependentnetworks. Secondly, our mathematical model directly addressesthe target metric - the fractional size of the surviving compo-nent of the interdependent network, rather than indirect metricssuch as structural centrality [9] or algebraic connectivity [17].Finally, we provide rigorous mathematical proofs for all ourresults. To the best of our knowledge, this is one of the firstworks that mathematically obtains the optimal interdependencestructure maximizing the network robustness against node fail-ures. We believe that the design principles put forward in thiswork would enable the design of more robust interdependentnetworks under various settings.

The remainder of this paper is organized as follows. SectionII presents the basic system model and the derivation ofthe system equations, which will be analyzed for differentscenarios in the remainder of the paper. Section III summa-rizes the main contributions of this work. Section IV andSection V contain the analysis for networks with completeand partial interdependence, respectively, under a randomizedattack model. Targeted attack on interdependent networks isconsidered in Section VI. Finally, we present the simulationresults supporting the theoretical analysis in Section VII, andconclude and indicate possible future work in Section VIII.

Symbol Description

A, B Two layers of the interdependent networksN Number of nodes in each interdependent network layer

λA, λB Mean intra-layer degree of layer A and Bρ Structure of interdependence, represented by a permutation of

indices {1, · · · , N}ρi ith element of ρ, index of a ∈ A coupled to bi ∈ BR Set of all possible interdependence structuresρ? Optimal interdependence structure maximizing network robustnessη Fraction of nodes surviving the initial attack

ψA, ψB Node percolation probability for layers A and BpA, pB Edge percolation probability for layers A and BkAi , kBi Intra-layer degree of the ith node in layer A and Bc Ratio of size of larger (A) to smaller (B) layer under

one-to-many interdependenceti Set of nodes in layer A coupled to the ith node in layer B

QA, QB Vector of indicator variables denoting whether ith nodein layer A and B is autonomous

wAi , wBi Probability of the ith node in layer A and Bto survives the initial attack

Table I: Major notations

Figure 1: Interdependent system model with one-to-onestructure. Solid lines give intra-layer topology and dotted

arrows showing inter-layer topology.

II. SYSTEM MODEL

In this section, we will introduce our network model andobtain the system equations modeling the cascading failurephenomenon. A summary of major notations used in this workis presented in Table I.

A. Details of the Network ModelWe consider an interdependent system consisting of two

layers A and B, as shown in Fig. 1. As the initial step, wewill focus on the simplistic network model of complete one-to-one interdependence, where each node in layer A is coupledto one node in layer B and vice versa. In later sections, wewill generalize the network model from different perspectives,such as the type of interdependence (complete or partial), thestructure of interdependence (one-to-one or one-to-many) andthe nature of attack (random and targeted). We assume that thetwo network layers have small average degrees λA, λB � N ,where N is the number of nodes in each layer. The averagedegree constraint is necessary to ensure the locally tree-likeproperty of the network layers, which is required for the

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(a) Stage I (b) Stage II

(c) Stage III (d) Stage IV

Figure 2: Cascading failure in Interdependent Networks: whiteand black nodes represent the failed and surviving nodes,respectively, at each stage of the cascading failure. StageI- Initial failure of nodes; Stage II- Breakdown of layer Ainto components, the largest of which survives; Stage III-Propagation of failures to layer B; Stage IV- Breakdown oflayer B into clusters and propagation to layer A.

formulation of the system equations. This constraint is alsoconsistent with the observation from a wide range of cyber-physical networks [18].

B. Details of Cascading Failure

As discussed earlier, the phenomenon of cascading failureseparates interdependent networks from their isolated counter-parts. An example of this phenomenon can be seen in Fig. 2. Ateach stage of the iterative cascading failure, a node will surviveif [1, 16, 19]: i) the supporting node survives, and ii) it belongsto the largest connected component in its layer. The cascadestops when a double-layer mutually connected component(MCC) is obtained, where every node belongs to the largestconnected component in its own layer and has a survivingsupporting node. In Fig. 2, we illustrate a cascading failuremechanism here based on the algorithm provided in [19]. Theinitial failure in layer A in Stage I leads to the breakdown oflayer A into three components of sizes, 3,1,1, respectively. Inaccordance to our assumption, the largest component survivesin Stage II. In Stage III, these failures propagate to layerB, where all nodes that are not connected to the survivingcomponent in layer A fail. Finally in Stage IV, the largestcomponent of layer B survives and the failure propagatesto layer A. Since the surviving component after Stage IV ismutually connected, the cascading failure reaches the steadystate, where the fractional size of the surviving component is0.33. Thus any failure of nodes initiates a recursive cascade offailures among the two layers, which ultimately gives rise toa mutually connected network component in the steady state.

The fractional size of the MCC with respect to (w.r.t.) thenetwork layers, i.e. the ratio of the number of layer A (orlayer B) nodes belonging to the MCC to the total number ofnodes in layer A (or layer B), is taken to be the metric forrobustness as is common in literature [1, 8, 16].

C. Representation of Interdependence Structure

For ease of discussion, let us index the nodes in both layersin the decreasing order of their intra-layer degrees, i.e. theith node in each layer has the ith highest intra-layer degree.Let sequence ρ = {ρ1, ρ2, · · · , ρN} be a permutation of thenumber set {1, 2, · · · , N} that represents the interdependencestructure, where the ith node in layer B is coupled to theρith node in layer A. These couplings define the interlinksbetween the network layers, i.e. if nodes a ∈ A and b ∈ B arecoupled, failure of a will cause b to fail and vice versa. Herewe have slightly abused the notation A and B to represent thenames of the two network layers and also the set of nodesin the respective layers. The aim of this work is to obtain theoptimal interdependence structure ρ? ∈ R, where R representsthe search space comprising all N ! possible permutations,which maximizes the network robustness. In later sections, wewill generalize our interdependence structure to partial inter-dependence, where only a fraction of nodes from both layersare interdependent, and one-to-many interdependence, wherea node can support multiple nodes. In these generalizations,the search space of feasible interdependence structures will bedefined accordingly.

D. Derivation of System Equations

Next, we focus on the derivation of the system equationsrelating the steady state size of the MCC (metric for robust-ness) to the network topology. We want to study the steadystate effect of the removal of (1 − η) fraction of nodes fromthe network and thereby obtain ρ? maximizing this steadystate size of the MCC. We have considered both cases ofdouble-layer attack, where nodes are attacked in both layers,and single-layer attack, where only one layer is attacked.Throughout this work, we will be analyzing the case of double-layer attack unless mentioned otherwise. Initially, we considerthe randomized attack case, where the attacker (or mothernature) fails nodes randomly, i.e. all nodes of the network havethe same probability of failure. Thereafter in Section VI, weconsider the case of targeted attack, where the attacker failsnodes on the basis of their intra-layer degrees.

Let the node percolation probability for the two layers berepresented by ψA and ψB , denoting the probability of arandomly chosen node (from layer A or B) to belong to theMCC in the steady state. ψA and ψB can also be thoughtof as the fraction of nodes from each layer which survive thecascading failure. Thus ψA and ψB are nothing but the metricsof robustness that we intend to maximize in this work. For thederivation of the system equations, we need to further definethe edge percolation probabilities (pA and pB) to denote theprobability that a randomly selected edge from layer A or B,on traversal in an arbitrary direction, connects to a node in the

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MCC. For the first set of system equations, we will expressrobustness (ψ) as a function of attack strength (η), intra-layer topology (degree sequence of nodes), interdependencestructure (ρ) and edge percolation probability (pA and pB). Werefer the reader to Appendix A for the details of the derivation.Note that similar system equations have been presented inworks such as [16, 20] for explaining other phenomena oninterdependent networks. The system equations for ψA andψB can be written as:

ψA =1

N

N∑i=1

η2[1− (1− pA)kAρi ][1− (1− pB)k

Bi ] = ψB , (1)

where kli denotes the intra-layer degree of the ith node inlayer l, and ρ = {ρ1, · · · , ρN} represents the interdependencestructure as discussed in the previous section. As explained inAppendix A, the terms η[1−(1−pA)k

Aρi ] and η[1−(1−pB)k

Bi ]

can be understood as the probability for survival of the ρithnode in layer A and the ith node in layer B, respectively. Thusthe product of these probability terms denotes the probabilityof survival of the ith interdependent link coupling bi ∈ B toaρi ∈ A, assuming all probability terms to be independent ofeach other. Note that ψA and ψB are equal in this case becauseof the one-to-one nature of the interdependence. This equalitywill not be valid in later sections, where we generalize thenetwork structure to one-to-many interdependence. Our aim isto optimally design ρ such that ψA and ψB are maximized.Note that ψ (ψA = ψB = ψ) is also dependent on pAand pB , which are themselves intertwined with the networkmodel. Thus for the purpose of obtaining the robustness (ψ)for a particular network topology (degree sequence and ρ) andattack strength (η), we need to define another set of equationsdescribing pA and pB . We refer the reader again to AppendixA for the detailed derivation which gives the final expression:

pA =1

N

N∑i=1

η2kAρiλA

[1− (1− pA)kAρi−1][1− (1− pB)k

Bi ], (2)

pB =1

N

N∑i=1

η2[1− (1− pA)kAρi ]kBiλB

[1− (1− pB)kBi −1], (3)

where λl denotes the mean intra-layer degree of layer l. Forthis case, the terms η[1−(1−pA)k

Aρi−1] and η[1−(1−pB)k

Bi −1]

denotes the probability of survival of nodes aρi ∈ A and bi ∈B reached by traversing a randomly selected edge in layerA and B, respectively. Note that the probability terms havebeen scaled up by the node degrees kAρi and kBi , respectively,since the same node can be reached by traversing each of itsedges. Thus for a particular network topology, ψ is obtainedby solving the self-consistent equations (2)-(3) and substitutingthe solution into (1). We are interested in finding the optimalρ? ∈ R that maximizes the robustness ψ. Furthermore, (2)-(3)defines a set of self-consistent equations since the right handsides of the equations are themselves dependent on pA andpB . The method of solving such self-consistent equations isexplained in Section IV.

III. SUMMARY OF RESULTS

The main contributions of this work are summarized asfollows:• For the one-to-one structure with complete interdepen-

dence, we show that the interlinking of nodes by theirintra-layer degrees in the monotonic order (the ith high-est degree nodes in layer A coupled to the ith highestdegree node in layer B, ∀ i ∈ [1, N ]) leads to maximalnetwork robustness against both single-layer and double-layer randomized attack, i.e. ρ? = {1, 2, · · · , N}. Tothe best of our knowledge, this is one of the firstgeneral results in this field that has been theoreticallyverified, although similar experimental results have beenshown in [11] and some other relevant works. Further-more, we have proved that the anti-monotonic matching(ρ = {N,N − 1, N − 2, · · · , 1}: the ith highest degreenode coupled to the ith lowest degree node) minimizesnetwork robustness.

• We have defined two modes of one-to-many interdepen-dence: i) AND interdependence, ii) OR interdependence.For these two structures, we impose the restriction ofregular interlink allocation (all nodes in a layer have thesame number of inter-layer links) and aim to obtain theoptimal interdependence structure. However due to thecomplexity of the system equations for both cases, wecould only obtain partial results: monotonic arrangementof nodes maximizes network robustness for AND inter-dependence, whereas anti-monotonic arrangement min-imizes robustness for OR interdependence; both resultsholding true for single and double-layer randomizedattacks. The arrangement that minimizes (or maximizes)robustness for AND (or OR) interdependence could notbe obtained. The reason behind this peculiar character-istic is discussed in Lemma IV.4.

• For one-to-one structure with partial interdependenceψA and ψB cannot be simultaneously maximized, andthus we focus on the problem of obtaining a pareto-optimal interdependence structure w.r.t. the two networklayers. We show that for both single-layer and double-layer randomized attacks, decoupling the highest degreenodes from both layers, i.e. allocating nodes of highestintra-layer degrees to the set of autonomous nodes, ispareto-optimal. This result conforms to the finding in [9]which explored the problem from the network geometryperspective.

• For the case of preferential targeted attack on lowerdegree nodes, all results described above hold true. Forpreferential targeted attack on higher degree nodes, wehave shown that the monotonic arrangement of nodes isthe optimal strategy for a perfect targeted double-layerattack, where nodes are attacked in the descending orderof their intra-layer degrees. For a perfect single-layerattack, we have shown that the optimal interdependencestructure is dependent on the strength of attack and thusuniversally optimal structures cannot be obtained by thetools used herein.

• We have presented the simulation results corroborating

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these theoretical findings. We have also tested the appli-cability of our results when the inter-links are designedon the basis of other centrality measures like between-ness and closeness. The simulation results show thateven for these cases the monotonic arrangement is betterthan anti-monotonic and random arrangements. Thisgives some indication as to whether relationships be-tween interdependence structure and robustness, whichhave been theoretically obtained in this work, assum-ing knowledge of degree-based topological information,hold true for other measures of centrality.

IV. ANALYSIS: COMPLETE INTERDEPENDENCE

A. One-to-one structureIn this section, we analyze the system equations (1)-(3). It

can be observed that the relationship between network robust-ness (ψ) and interdependence structure (ρ) is complicated dueto the fact that ψ depends on the values of pA and pB by (1)which themselves are dependent on ρ by (2)-(3). Furthermore,(2)-(3) are defined as self-consistent equations whose solutionis not straightforward. It is known from [21, 22] that such self-consistent equations typically have no closed form solutionsand they are solved iteratively as follows:

pAn+1= fA(pAn , pBn), pBn+1

= fB(pAn , pBn), (4)

where pln denotes the value of pl in the nth iteration, andfA and fB are functions given by (2) and (3), respectively.Starting from pA0 = pB0 = 1, the final solution is obtainedwhen the iterations converge, i.e. pA = limn→∞ pAn , pB =limn→∞ pBn . It is also known from [21] that such iterationsalways converge to fixed points between 0 and 1.

LEMMA IV.1 If there exists an interdependence structureρ? ∈ R, where R is the set of all possible permutations of{1, 2, · · · , N}, such that ρ? individually maximizes all of thefollowing:• ψ for fixed pA and pB ,• pAn+1 for fixed pAn and pBn , ∀ n,• pBn+1

for fixed pAn and pBn , ∀ n,such a ρ? maximizes the network robustness ψ.

Proof: We will prove the above Lemma in two steps.Firstly, we will show that simultaneous satisfaction of thelast two constraints implies maximization of both pA andpB . Thereafter, we will couple the first constraint with the(verifiable) fact that ψ is increasing w.r.t. pA and pB toprove the Lemma. Firstly, denote δAn , pAn − pAn+1

andδBn , pBn − pBn+1 as the difference between the values ofpA and pB at the nth and (n+ 1)th iterations. By the secondcondition stated in the Lemma, ρ? maximizes pAn+1

for a fixedpAn for all n. Thus we can conclude that ρ? minimizes δAn(= pAn−pAn+1

) in each iteration step (for all n). Furthermore,pA0

= pB0= 1. Thus we can write:

ρ? = arg minρ∈R

(∑n

δAn )

= arg maxρ∈R

(1−∑n

δAn ) = arg maxρ∈R

pA, (5)

where the last step follows from the fact that pA0= 1 and the

changes in pA for each iteration n is denoted by δAn . Similarargument holds for δBn as well where it can be shown (bythe third condition) that ρ? minimizes δBn in each iterationstep, using which we can conclude ρ? = arg maxρ∈R pB .Thus we can observe that ρ? maximizes both pA and pB .This indicates that if we can obtain a ρ satisfying the lasttwo conditions of the Lemma, we can prove that such a ρresults in the maximization of both pA and pB . Secondly,it can be easily verified from (1) that ψ is increasing w.r.t.pA and pB . In addition to this, the first constraint states ρ?maximizes ψ for any fixed pA and pB . Combining these, wehave that ρ? maximizes ψ for any fixed pA and pB (by the firstconstraint) and also maximizes both pA and pB (by the lasttwo constraints). Thus it can be concluded that ρ? maximizesψ.

Remark: The above discussion indicates that the maximiza-tion of robustness of a network can be achieved theoreticallyif we can find an interdependence structure ρ? satisfying the(strong) conditions in Lemma IV.1. For networks where suchan optimal structure cannot be found, the tools described in thiswork are insufficient for obtaining optimal interdependence.However, it will be shown below that for several cases,including the complete one-to-one interdependence, such anoptimal ρ? can be found.

To prove the main result in this section, we need thefollowing technical result:

LEMMA IV.2 For any two sequences of real numbers,{x1, x2, · · · , xn} and {y1, y2, · · · , yn} which are monotonicin the same sense and any permutation ρ = {ρ1, · · · , ρn} ofthe set {1, 2, · · · , n}

xny1+ · · ·+x1yn ≤ xρ1y1+ · · ·+xρnyn ≤ x1y1+ · · ·+xnyn.(6)

Proof: This is referred to in literature as the Rearrange-ment Inequality [23].

THEOREM IV.3 For the one-to-one structure with completeinterdependence, the optimal interdependence structure, forboth single-layer and double-layer randomized attack, is thematching of nodes by their intra-layer degrees in the monotonicorder. Furthermore, the anti-monotonic arrangement of nodesis the worst possible interdependence structure.

Proof: We start with the double-layer attack case forwhich the equations (1)-(3) have been derived. By LemmaIV.1, an optimal ρ? exists if we can find a ρ satisfying allconstraints stated therein. We will analyze the three systemequations (1)-(3) to obtain a ρ which satisfies the three con-straints. We present the analysis of (1) and a similar argumentcan be used for (2)-(3). Let us simplify (1) as follows:

ψ =1

N

N∑i=1

aρibi, (7)

where aρi , η[1−(1−pA)kAρi ] and bi , η[1−(1−pB)k

Bi ] can

be understood as the probability of individual nodes in bothlayers of the network to belong to the MCC in steady state, as

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discussed previously in reference to (1). Due to the indexingof nodes in the decreasing order of their intra-layer degree,we can see that {b} = {b1, b2, · · · , bN} is a non-increasingsequence. Thus by Lemma IV.2 we have that the optimalordering of sequence {a} = {aρ1 , · · · , aρN } is the monotonicarrangement, where ρ = {1, 2, · · · , N}. Furthermore, it canalso be concluded from Lemma IV.2 that the anti-monotonicarrangement, i.e. ρ = {N,N − 1, · · · , 1} minimizes the sum.Note that till now we have just obtained a ρ that satisfiesthe first constraint in Lemma IV.1. The completion of theproof requires us to satisfy the remaining constraints usingthe same ρ. It is easy to see that both (2) and (3) can bewritten as (7), where the corresponding sequences {a} and{b} would be different although their monotonicity propertiesare still preserved. Thus following the same argument asdiscussed above we can conclude that monotonic ρ satisfiesall constraints stated in Lemma IV.1. Thus such a ρ is nothingbut the optimal ρ? maximizing network robustness measuredby ψ. The opposite arguments can be applied for the case ofanti-monotonic arrangement to prove that it minimizes ψ.

For the case of single-layer attack, (1) can be written as:

ψA =1

N

N∑i=1

η[1− (1− pA)kAρi ][1− (1− pB)k

Bi ] = ψB , (8)

where the η2 factor is substituted with η since only one ofthe layers is attacked. The same changes apply to the otherequations for pA and pB . It can be noted that a transition toa single-layer attack only changes the system equations by aconstant multiplicative factor without altering the monotonicnature of the defined sequences {a} and {b}. Thus the optimalρ? under a single-layer attack is the same as that for a double-layer attack.

B. One-to-Many StructureIn this section, we extend our study to one-to-many interde-

pendence, where the two layers of the network have differentsizes, |A| = cN and |B| = N , with |l| denoting the numberof nodes in layer l, and c is an integral constant. Due to thedifference in the size of the two layers, our previous assump-tion that every node is coupled to one node in the other layerwill not hold due to the complete interdependence constraint.Let each node i in layer B be coupled to the set of nodesti in layer A, which implies

∑Ni=1 |ti| = cN . In comparison

to the study of one-to-one interdependence, this case opensup many failure propagation models depending on the numberof support links required for a node to survive. In this study,we focus on two extremes for the failure propagation: i) ANDinterdependence, where each node in layer B requires all ofits supporting links for survival; and ii) OR interdependence,where each node requires at least one supporting link.

1) AND Interdependence: We start our study by focusingon the AND structure described above. We stick to the sameindexing scheme as used previously, wherein the nodes of bothlayers are indexed in the decreasing intra-layer degree order.The interdependence structure ρ can be written as a permuta-tion of the set {1, 2, · · · , cN}, where the first t1 elements ofρ represent the indices of the nodes a ∈ A coupled to b1 ∈ B,

the next t2 elements represent the nodes coupled to b2, and soon. Thus the search space of interdependence structure for thiscase is the set of permutations of {1, 2, · · · , cN}. The systemequations can be written as:

ψA =1

cN

N∑i=1

η[1− (1− pB)kBi ]∑j∈ti

η[1− (1− pA)kAρj ], (9)

ψB =1

N

N∑i=1

η[1− (1− pB)kBi ]∏j∈ti

η[1− (1− pA)kAρj ]. (10)

pA=1

cN

N∑i=1

η[1−(1−pB)kBi ]∑j∈ti

ηkAρjλA

[1−(1−pA)kAρj−1

], (11)

pB=1

N

N∑i=1

ηkBiλB

[1−(1−pB)kBi −1]

∏j∈ti

η[1−(1−pA)kAρj ]. (12)

Note that the changes in the system equation arise due to thechange in the interdependence structure and the correspondingfailure propagation model. Since layer A nodes observe a one-to-one interdependence, i.e. each a ∈ A is coupled to a singleb ∈ B, their system equations (9) and (11) are essentially thesame with the difference that multiple nodes a ∈ A are coupledto the same node b ∈ B. For layer B nodes (10) and (12), thesurvival probability is the product of the survival probabilitiesof all its supporting nodes, since we are dealing with the ANDinterdependence structure.

For simplicity of analysis, we impose the additional con-straint of regular allocation, defined as an allocation schemewhere each node i in layer B supports the same numberof nodes |ti| in layer A: |ti| = c for all nodes bi ∈ B.The regular allocation scheme has attracted some researchinterest in works like [4], where it has been shown to be theoptimal link allocation scheme in absence of any topologicalknowledge. In the following analysis, we show how the degreebased topological information can be used to optimize networkrobustness under this regular allocation constraint. Althoughreal world networks might not satisfy this constraint, we feelthat the following study is important as it expands the scopeof our knowledge of design of networks with one-to-manyinterdependence and also provides tools to optimize networkrobustness utilizing topological information. As a part of ourfuture work, we plan to extend our study to the case ofgeneralized allocation schemes. In order to study this problem,we have defined and proved a generalized version of theRearrangement Inequality which will be used in later analysis.

LEMMA IV.4 (Generalized Rearrangement Inequality) Givenany two sequences of positive real numbers, x ={x1, x2, · · · , xcn} and y = {y1, y2, · · · , yn} and any arbitrarypermutation ρ = {ρ1, · · · , ρcn} of the sequence {1, 2, · · · , cn}where ρi denotes the ith element of ρ, let us define six

7

summations:

Θ1 =

n∑i=1

yi

cn∑j=1

xcn−j+1, Φ1 =

n∑i=1

yi

cn∏j=1

xcn−j+1, (13)

Θ2 =

n∑i=1

yi

cn∑j=1

xρj , Φ2 =

n∑i=1

yi

cn∏j=1

xρj , (14)

Θ3 =

n∑i=1

yi

cn∑j=1

xj , Φ3 =

n∑i=1

yi

cn∏j=1

xj . (15)

If both sequences {x} and {y} are decreasing,

Θ1 ≤ Θ2 ≤ Θ3, and (Φ1,Φ2) ≤ Φ3. (16)

Proof: Inequality 1: Θ1 ≤ Θ2 ≤ Θ3

This result directly follows from the Rearrangement inequal-ity (Lemma IV.2) if we introduce a new sequence {y} wherethe each element of {y} is repeated c times.

Inequality 2: (Φ1,Φ2) ≤ Φ3

In this case we start with the expression for Φ3 and showthat any exchange of the xj terms between two groups leadsto a decrement in the value of the summation. Let us definewi ,

∏j∈ti xj , and denote the corresponding xj’s as the ith

group. We exchange one of the terms (xj1) from the kth groupwith another term (xj2) in the lth group, where l > k. Letδ = xj1∈tk/xj2∈tl be the ratio of the two exchanged terms.Note that δ ≥ 1, since {x} is a decreasing sequence and l > k.Also since {y} is decreasing, we have yl ≤ yk. Let Φnew3 bethe value of the summation after the exchange. Thus we have:

Φnew3 − Φ3 =∑i,i6=k,l

yiwi + ylwlδ + ykwkδ−∑i

yiwi (17)

= (δ − 1)

(ylwl −

ykwkδ

)(18)

= (δ − 1)

(yl∏j∈tl

xj − ykxj2∈tlxj1∈tk

∏j∈tk

xj

)(19)

= (δ − 1)

[xj2∈tl

(yl∏j∈tlj 6=j2

xj − yk∏j∈tkj 6=j1

xj

)](20)

≤ 0, (21)

where (21) follows from the fact that yl ≤ yk, δ ≥ 1, and allelements xj , where j ∈ tl, are smaller than all elements in xj ,where j ∈ tk, since l > k and {x} is decreasing. Thus weshow that any change from the ordered arrangement Φ3 yieldsa decrement in the sequence summation. However the oppositeargument for the case of anti-monotonic arrangement does nothold since for that case sequence {y} is still a decreasingsequence but {x} is increasing. Thus (19) comprises theproduct of elements of increasing and decreasing sequencesand there is no general way of characterizing such products.Thus in contrast to the previous case, although Φ3 ≥ Φ2 holdstrue, Φ1 ≤ Φ2 is not always true.

Remark: This peculiar characteristic of the generalized Rear-rangement Inequality will have implications for the analysis ofoptimal interlink structure for one-to-many interdependence. In

particular, in contrast to the results for one-to-one interdepen-dence, we can only obtain the optimally worst or optimallybest structure but not both. In particular, we can obtain theoptimally best structure for AND interdependence and theoptimally worst structure for OR interdependence.

THEOREM IV.5 For an interdependent network with one-to-many AND interdependence and regular link allocation, theoptimal interdependence structure that maximizes networkrobustness, for both single-layer and double-layer attacks, isthe monotonic ordering ρ = {1, 2, · · · , cN}. Thus the optimalinterlink structure is the one where the first c nodes of highestdegree in layer A is coupled to the highest degree nodes inlayer B and so on, where c is the ratio of the sizes of the twonetwork layers.

Proof: Note that a similar argument as Lemma IV.1 can beapplied here. An interdependence structure ρ is optimal onlyif it maximizes all of ψA, ψB , pA, pB given by (9)-(12). Weprovide the analysis of ψA and ψB here and we can extendthe same logic to the case of pA and pB as will be shownlater. We simplify (9) by defining bi , η[1− (1− pB)k

Bi ] and

aρj , η[1− (1− pA)kAρj ], denoting the probability of survival

of the individual nodes:

ψA =1

cN

N∑i=1

bi∑j∈ti

aρj . (22)

Note that due to the indexing of the nodes in the decreasingintra-layer degree order, the elements of the sequence {b} ={b1, b2, · · · , bN} are decreasing in accordance to the constraintin Lemma IV.4. Thus we can see that ψA can be expressed asΘ2 in Lemma IV.4 and using the same Lemma we can showthat the monotonic arrangement (ρ = {1, · · · , cN}) maximizesthe above expression. Similarly ψB in (10) can be written asfollows:

ψA =1

cN

N∑i=1

bi∏j∈ti

aρj . (23)

Note that this expression is similar to Φ2 in Lemma IV.4 andthus we can show the optimality of the monotonic arrangementby utilizing the result in the Lemma. The cases of pA andpB can similarly be written as Θ2 and Φ2, respectively. Thusutilizing Lemma IV.4 we show that the monotonic arrangementmaximizes all the expressions and thus ρ = {1, · · · , cN}is the optimal interdependence structure. Note that the anti-ordered arrangement (Φ1) in Lemma IV.4 does not necessarilyminimize the sum and thus we cannot conclude that the anti-monotonic arrangement minimizes ψ.

The case for the single-layer attack can be solved in a similarway, since the transition to a single-layer attack only leads tochange in the multiplicative factor of the sequence elementswithout altering their monotonicity.

2) OR Interdependence: For an OR interdependence struc-ture, node b ∈ B will survive as long as any one of itssupporting nodes survives. Thus from the perspective of layerA the system equations remain the same as the case of ANDinterdependence. The only difference comes for the case of

8

layer B which can be written as follows:

ψB =1

N

N∑i=1

η[1− (1− pB)kBi ]·[

1−∏j∈ti

(1− η[1− (1− pA)

kAρj ]

)], (24)

pB =1

N

N∑i=1

ηkBiλB

[1− (1− pB)kBi −1]·[

1−∏j∈ti

(1− η[1− (1− pA)

kAρj ]

)]. (25)

The above equations can be explained as follows. Let pi bethe probability of survival of a supporting link i. Thus theprobability of survival of at least one of the ti supporting linksis given by [1−

∏i∈ti(1−pi)], since the survival probabilities

of different links are independent. For this case pi = η[1 −(1− pA)k

Aρi ], thus the above equations follow.

THEOREM IV.6 For an interdependent network with one-to-many OR interdependence and regular link allocation, theoptimal interdependence structure that minimizes network ro-bustness, for both single-layer and double-layer attacks, is theanti-monotonic ordering ρ = {cN, cN − 1, · · · , 1}.

Proof: We start with the analysis of (24) and (25). Similarto the case of AND interdependence, we simplify the aboveexpressions using bi and aρj as defined previously. Thus ψBcan be written as:

ψB =1

N

N∑i=1

bi

[1−

∏j∈ti

(1− aρj )]

(26)

=1

N

N∑i=1

bi −1

N

N∑i=1

bi∏j∈ti

(1− aρj ). (27)

Note that the first term in (27) is independent of ρ. Thus inorder to minimize ψB , we need to maximize the second term.Using Lemma IV.4, we can conclude that ψB is minimizedwhen the sequence {1−aρj} is decreasing, which correspondsto the anti-monotonic arrangement. The same argument canbe applied to the case of pB as well. For the layer A nodes,we can invoke Lemma IV.4 again to conclude that the anti-monotonic arrangement minimizes ψA and pA as they can bewritten as Θ2. Thus the anti-monotonic arrangement can beshown to minimize all of ψA, ψB , pA, pB to prove the theorem.The single layer attack case can be argued by following asimilar methodology as in Theorem IV.5.

Remark: Note that the AND and OR interdependencestructure are conjugate to each other in some sense. ForAND structures, we could obtain an interdependence structuremaximizing network robustness, whereas for OR structureswe could obtain an interdependence structure minimizing thenetwork robustness. We were not able to derive an inter-dependence structure which minimizes (and maximizes) thenetwork robustness for OR (and AND) structures, due to the

Figure 3: Interdependent system model: one-to-one struc-ture with partial interdependence. Hexagonal nodes are au-tonomous, while the rest are interdependent.

shortcoming of Lemma IV.4, where we could not obtain thearrangement that minimizes Φ.

V. ANALYSIS: PARTIAL INTERDEPENDENCE

In this section, we will generalize our system model toanalyze partially interdependent networks where the N nodesin each layer of the network can be divided into two sets:i) autonomous nodes, which do not require support from theother layer, and ii) interdependent nodes, which do requiresupport. The interdependence is still taken to be one-to-one asis shown in Fig. 3. Let us define QA and QB as the allocationstrategy for the two layers, i.e. QA and QB are row vectorsof the form [ql1, q

l2, · · · , qlN ] with:

qli =

{1 if ith node in layer l is autonomous,0 if ith node in layer l is interdependent. (28)

Note that due to the one-to-one interdependence structure, QAand QB will contain the same number of 0’s and 1’s. Thesystem equations for this case can be written as:

ψA =1

N

( N∑i=1

qAi a1i +

N∑i=1

qBi b1iqAρia1ρi

), (29)

ψB =1

N

( N∑i=1

qBi b1i +

N∑i=1

qBi b1iqAρia1ρi

), (30)

pA =1

N

( N∑i=1

qAi a2i +

N∑i=1

qBi b1iqAρia2ρi

), (31)

pB =1

N

( N∑i=1

qBi b2i +

N∑i=1

qBi b2iqAρia1ρi

), (32)

where a1i , η[1 − (1 − pA)kAi ], b1i , η[1 − (1 − pB)k

Bi ],

a2i , ηkAiλA

[1− (1− pA)kAi −1], b2i , η

kBiλB

[1− (1− pB)kBi −1],

and qli denotes the conjugate indicator function to qli andequals 1 if the ith node in layer l is autonomous. Note thatthe physical meaning of the terms a1i, a2i, b1i, and b2i aresame as discussed previously in reference to (1) and (2)-(3).

9

Furthermore, the two components in the system equationscorrespond to the contribution of autonomous and interde-pendent nodes towards network robustness. The problem ofobtaining the optimal interdependence structure in this case isconsiderably more complicated because we have to considerthe joint optimization of the allocation of nodes (QA and QB)and the interdependence structure (ρ). In this case, as will beevidenced below, it is usually not feasible to optimize the MCCfor both layers (ψA and ψB) simultaneously. Instead, we willendeavor to achieve a pareto-optimal robustness which cannotbe strictly dominated by any other interdependence structure.

THEOREM V.1 Given any QA and QB , the optimal interde-pendence structure is ρ? = {1, 2, · · · , N}.

Proof: It can be observed from the system equations (29)-(32), that the interdependence structure ρ only influences thecontribution of the interdependent nodes, which is of the formqBi biq

Aρiaρi . Let {a} and {b} denote the non-zero elements of

the sequences {qAi ai}|Ni=1 and {qBi bi}|Ni=1, respectively. Thus{a} and {b} are decreasing sequences due to the indexingof the nodes in the decreasing order of intra-layer degrees.Thus we can invoke the Rearrangement Inequality (LemmaIV.2) to prove that the monotonic arrangement is optimal, i.e.ρ? = {1, · · · , N}.

Remark: Theorem V.1 essentially allows us to decouple thejoint maximization problem discussed previously. Since theoptimal ordering of interdependent nodes is independent ofthe allocation strategy, we can look at the two optimizationproblems individually. Thus the problem of optimizing inter-dependence structure reduces to the problem of identifying theoptimal allocation strategy Q?A and Q?B .

Imposing the optimal monotonic interdependence structure,we can write the system equations as:

ψA =1

N

( N∑i=1

qAi a1i +

N∑i=1

qAi a1iqBi b1i

), (33)

ψB =1

N

( N∑i=1

qBi b1i +

N∑i=1

qAi a1iqBi b1i

), (34)

pA =1

N

( N∑i=1

qAi a2i +

N∑i=1

qAi a2iqBi b1i

), (35)

pB =1

N

( N∑i=1

qBi b2i +

N∑i=1

qAi a1iqBi b2i

). (36)

Focusing on the objective function for pA in (36) and noticingthat the first term is independent of QB , we observe thatthe allocation strategy maximizing the first component of pAwould be to allocate the highest degree nodes a ∈ A to theautonomous set as a2i > a2i · b1i due to the fact that b1i ≤ 1.For maximization of the interdependent (second) component,the highest degree nodes b ∈ B should be interdependent. Onthe contrary, individual maximization of pB in (36) leads to theopposite allocation strategy: the highest degree nodes a ∈ Aare interdependent, whereas the highest degree nodes b ∈ Bare autonomous. This argument also holds for the case of ψAand ψB given by (34). It can be clearly observed here that

there exists a conflict of interest between the maximizationstrategies for layers A and B, which is why we cannot obtaina joint maximization, in contrast to the previously analyzedcases. Instead we have looked at the problem of obtaining apareto-optimal allocation strategy w.r.t. the two layers, wherewe are interested in obtaining an allocation strategy QA? andQB

? such that no other strategy can lead to an increase in therobustness of both layers. We have used the Nash Bargainingframework to solve this problem. A description of the NashBargaining Problem has been included in Appendix B. It isshown in Lemma B.1 that any solution point which satisfies thesix axioms is the Nash Bargaining Solution. We will use thisidea to obtain the pareto-optimal node allocation for networkswith partial interdependence.

THEOREM V.2 For the one-to-one structure with partial in-terdependence, where the allocation strategies QA and QBhave an arbitrary fixed number of 1’s (= q), the strategyleading to a pareto-optimal robustness of the two networklayers against both single-layer and double-layer randomizedattacks is: Q?A = Q?B = [1, 1, · · · , 1, 0, 0, · · · , 0], where thenumber of non-zero elements in Q?A and Q?B is q.

Proof: We start with the analysis of pA and pB givenby (36). The analysis of ψA and ψB will be shown later.We formulate the optimal allocation problem as a bargainingproblem:• The two players involved in the bargaining game are

layers A and B.• The set of possible agreements or the set of possible

allocation strategies is of the form:

F =

(qA1 qA2 · · · qANqB1 qB2 · · · qBN

),

where N is the number of nodes in each layer of thenetwork. As both layers have q autonomous nodes, theset of possible agreements is the set of all F ’s such thatthe two row-sums are q.

• The utility functions for the two players is given by (36)and denoted by u1 and u2, respectively, which maps anallocation strategy (F ) to the utility for each player:

u1(F ) =1

N

( ∑i∈X1

a2i +∑

i∈Y1,j∈Y2

a2ib1j

), (37)

u2(F ) =1

N

( ∑i∈X2

b2i +∑

i∈Y2,j∈Y1

a1jb2i

), (38)

where X1, X2 denote the set of autonomous nodes inlayers A and B, respectively; and Y1 and Y2 denote theset of interdependent nodes in the two layers. Note thatthe second summation is over all interdependence linksin the network.

• Let us define the disagreement pay-off d = [d1, d2],where d1 and d2 are the pay-offs received by the twoplayers if an agreement is not reached. Both d1 and d2are taken to be the case where nodes are chosen to beautonomous or interdependent randomly, i.e. if the two

10

players fail to arrive at an agreement after the bargainingprocess, a random allocation strategy would be adopted.

• The set of possible pay-offs is denoted by U , which canbe defined as:

U = {(v1, v2)|u1(x) = v1, u2(x) = v2 for some x ∈ F}.(39)

Note here that the utility functions for both players(u1, u2) described by (37)-(38) are bounded above andbelow (ui(x) ∈ [0, 1] ∀ i) and thus have non-empty andclosed support with a convex hull, which is necessaryfor the existence of a Nash Bargaining Solution.

• The Bargaining Problem is thus represented as (U,d)over the space of possible agreements F .

It is known that a solution to such Bargaining Problems has tosatisfy the six axioms described in Lemma B.1 in AppendixB. Let us take a particular allocation strategy for the twolayers, where the nodes of highest degrees in both layers of thenetwork are autonomous. We want to show that this allocationstrategy satisfies all the axioms of Nash Bargaining Solutions(N.B.S.). As we have indexed the nodes in the network in thedecreasing order of their intra-layer degrees, we can write thisallocation strategy as :

F ? =

(1 1 · · · 1 0 0 · · · 01 1 · · · 1 0 0 · · · 0

)where the number of 1’s in each row of the allocation strategyis q. Let us now examine each of the six axioms for thisparticular solution.• Feasibility: It can be clearly seen that the solution

(u1(F ?), u2(F ?)) ∈ U and thus is feasible.• Individual Rationality: At the dis-agreement point, nodes

are allocated randomly. It can be easily examined thatui(F

?) ≥ ui(d) for both players i = 1, 2.• Pareto Efficiency: A closer examination of the utility

functions u1 and u2 defined by (37) and (38) reveals thatit is always better to allocate a node to the autonomousset (X1 and X2) as a2i ≥ a2ib1i and b2i ≥ a1ib2i. How-ever since our particular solution allocates all highestdegree nodes to X1 and X2, the only way of increasingthe utility of a player (say player A) is by modifying thestructure of interdependence, or in particular allocatingnodes of higher degrees from layer B to Y2. However,this will lead to a sub-optimal condition for player Bby the same logic as before that b2i ≥ b2ia1i, i.e. fromplayer B’s perspective highest degree nodes of layer Bshould be allocated to X2. Thus no other solution canstrictly dominate (u1(F ?), u2(F ?)).

• Independence of Irrelevant Alternatives: Let us representthe solution to the Nash Bargaining problem (U ,d) byφ(U,d), i.e. φ(U,d) = (u1(F ?), u2(F ?)). It can betrivially observed that φ(U ′,d) = (u1(F ?), u2(F ?))if U ′ ⊂ U and (u1(F ?), u2(F ?)) ∈ U ′, i.e. if thenew pay-off space (U ′) is a subset of U and containsφ(U,d), φ(U ′, d) = φ(U,d).

• Independence of Linear Transformation: It can be veri-fied from (37) and (38) that for any player i

arg maxx∈F

ui(x) = arg maxx∈F

(αui(x) + β

), (40)

where (α, β) represents a linear transformation τ . As τis independent of the allocation strategy, we can observethat the axiom holds true in this case.

• Symmetry: The symmetry condition occurs in this bar-gaining problem when λA = λB , i.e. two layers areidentically distributed. Under this condition, the twoplayers cannot be differentiated from one another. Asthe allocation strategy F ? does not differentiate betweenthe two layers, we can see that u1(F ?) = u2(F ?) ifλA = λB .

As the NBS is unique and the particular solution chosen by ussatisfies all the six axioms, we can conclude that the particularsolution is the NBS. To complete the proof, we need to verifythat the particular solution leads to pareto-optimal robustnessof ψA and ψB given by (34). Modeling (34) as the utilityfunctions for the two players, we can obtain this result in avery similar way by showing that the same particular solutionF ? satisfies all axioms of NBS for this case and thus thetheorem follows.

Under a single-layer attack (assuming layer A is attackedwithout loss of generality), the system equations can bewritten in the exact same form as (??)-(??) with the dif-ference that the factor η is dropped from the elements of{b1} = {b11, · · · , b1N} and {b2} = {b21, · · · , b2N}. The maindifference in terms of analysis is that the system equationsloose their symmetry even for the case when λA = λB .However, since we intend to achieve a pareto-optimal solutionto the robustness of the two layers, we can proceed as inthe previous section to show that the five axioms (there is nosymmetry in this case) are satisfied by the particular solutionas given in Theorem V.2. Thus the same result holds for thesingle-layer attack as well.

Remark: It is interesting to note here that the particularsolution used in the above proof can be derived by obtainingthe NBS under the symmetry condition of λA = λB , wherethe two layers of the network are identically distributed andthus the two bargaining players cannot be differentiated fromeach other. It can be observed from (37)-(38) that under thesymmetry condition, u1 = u2 ∀ x ∈ F and also d1 = d2,where d1 and d2 are the dis-agreement pay-offs for the twoplayers. Thus we can write the NBS as:

φ(U ,d) = arg maxu∈U ,ui≥di∀i

(u1 − d1)(u2 − d2)

= arg maxu∈U ,ui≥di∀i

(u1 − d1)2, (41)

which is a maximization problem in one variable that caneasily be solved to obtain F ?.

VI. ANALYSIS: TARGETED ATTACK MODEL

In this section, we will examine network robustness undera targeted attack model, where the probability of a node

11

to be attacked is dependent on its intra-layer degree. Asan initial step, we focus on the same model introduced inSection II: completely interdependent network with one-to-one and bidirectional dependency links. Let us define wli asthe probability of the ith node in layer l to survive the initialattack. Note that for a randomized attack structure, wi = η, ∀ i.Using this generalized attack model (w), the system equationsmodeling the cascading failure can be written as:

ψA =1

N

N∑i=1

wAρi [1− (1− pA)kAρi ]wBi [1− (1− pB)k

Bi ] = ψB ,

(42)

pA =1

N

N∑i=1

wAρikAρiλA

[1− (1− pA)kAρi−1]wBi [1− (1− pB)k

Bi ],

(43)

pB =1

N

N∑i=1

wAρi [1− (1− pA)kAρi ]wBi

kBiλB

[1− (1− pB)kBi −1].

(44)All results presented in this paper are dependent on themonotonicity of the functions f1(k, p) = [1− (1− p)k] andf2(k, p) = k[1− (1− p)k−1] with k. It can be observed from(42)-(44) that the survival probabilities of the nodes are theproduct of the attack model w and the functions f1 and f2.Thus if the monotonicity of these products remain unalteredthen all results obtained for the case of randomized attack willbe valid for the targeted attack model as well. Thus if wi is alsoan decreasing function of i all previous results would be valid.A decreasing wi implies nodes of lower degree, i.e. nodes ofhigh indices, have a lower probability of surviving the initialattack. Thus this scenario represents the case of targeted attackon lower degree nodes, where nodes of lower degree are morelikely to be attacked.

However the opposite targeted attack model, which prefer-entially selects nodes of higher degree, is a more practical andinteresting setting. This is because an attack on nodes (hubs)of higher connectivity would cause a much more devastatingeffect on the network than a preferential attack on lower degreenodes or a randomized attack. As a first step towards character-ization of optimal interdependence structures for these targetedattack models, we will analyze the case of perfect targetedattack, where the nodes of the network are attacked in thestrict descending order of their intra-layer degrees. The perfecttargeted attack model can be mathematically represented as:

wi =

{0 for i ∈ {1, 2, · · · , N ′}1 for i ∈ {N ′ + 1, N ′ + 2, · · · , N} (45)

where N is the number of nodes in each layer of the network,η is the fraction of nodes that survive the initial attack, andN ′ = (1− η)N .

THEOREM VI.1 For a double-layer perfect targeted attack oninterdependent networks with complete one-to-one interdepen-dence, the optimal interdependence structure is the monotonicarrangement of nodes.

Proof: Based on Lemma IV.1, note that an optimalstructure is one that maximizes all of the system equations

(42)-(44). We will focus on the case of ψ given by (42) andthe same argument can be applied to prove the other cases aswell. Let us consider the two sequences, {a} and {b}, whereai = wAi [1 − (1 − pA)k

Ai ] and bi = wBi [1 − (1 − pB)k

Bi ].

Thus ai’s and bi’s represent the probability of survival fornodes in layers A and B, respectively. Furthermore, due tothe perfect targeted attack, the top (1 − η)N elements ofboth sequences, where 1− η is the fraction of nodes initiallyattacked and N is the number of nodes in each layer, equal0. In essence, our aim is to find the optimal ordering of thesetwo sequences that maximizes the resulting sum. It is easy toobserve here that the monotonic arrangement of nodes is stilloptimal. Since the first (1−η)N elements of {a} and {b} are 0and the remaining elements ordered non-increasingly, we caninvoke Lemma IV.2 to conclude that the optimal structure isthe monotonic arrangement.

A. Single-Layer Targeted Attack modelA majority of the results derived in this work under the

double-layer attack conform to the case of single-layer attackas well. However, for the case of single-layer targeted attack, aswill be explained in this section, the optimal interdependencestructure is dependent on the attack strength. It is important tonote here that all previous optimal interdependence structures,which have been derived in this work are optimal for anystrength of attack (η). Under a single-layer perfect targetedattack model (assuming layer B is attacked), the systemequations modeling the cascading failure can be written as:

ψA =1

N

N∑i=1

[1−(1−pA)kAρi ]wBi [1−(1−pB)k

Bi ] = ψB , (46)

pA =1

N

N∑i=1

kAρiλA

[1−(1−pA)kAρi−1]wBi [1−(1−pB)k

Bi ], (47)

pB =1

N

N∑i=1

[1−(1−pA)kAρi ]wBi

kBiλB

[1−(1−pB)kBi −1], (48)

where wBi is the indicator function denoting whether theith node in layer B survives the initial attack. As we areconsidering a perfect targeted attack in this case, wBi is 0 forthe first (1− η)N indices and 1 elsewhere.

THEOREM VI.2 For a single-layer perfect targeted attack oninterdependent networks with complete one-to-one interdepen-dence, the optimal interdependence structure is dependentof the strength of attack and is neither monotonic or anti-monotonic.

Proof: Similar to the Theorem VI.1, we focus on theanalysis of ψ given by (46) and the same arguments can beextended to the cases of pA and pB . We again consider two se-quences {a} and {b}, whose elements represent the probabilityof survival for each node in the layers. The difference with thedouble-layer attack case is that here only one of the sequences({b} since layer B is attacked) has the first (1−η)N elementsas 0, while all elements of {a} are non-zero. By Lemma

12

IV.2 we know that the optimal ordering of two sequencesthat maximize their sum of element products is the monotonicarrangement. Now the first (1−η)N elements of {b} are 0 andthe rest comprise a decreasing sequence. However, all elementsof {a} are decreasing. Thus it can be seen that the optimalarrangement would be to couple the smallest (1−η)N elements(last indices) of {a} to the zero elements of {b}, while theremaining nodes are coupled monotonically. Thus the optimalarrangement of layer A nodes (ρ?) is given by:

ρ? = {ηN + 1, ηN + 2, · · · , N, 1, 2, · · · , ηN}. (49)

Thus in ρ?, the first (1 − η)N nodes of layer B (nodes ofthe highest degrees which are subject to the initial attack) arecoupled to the last nodes (having minimum degree) in layerA. Thereafter, nodes of highest degrees in layer A are coupledmonotonically to the nodes in layer B which survive the initialattack. Thus we can observe that the optimal interdependencestructure is dependent on η and is neither monotonic nor anti-monotonic.

Remark: Note that this result is significantly weaker than theother results presented in this work. The reason behind this isthat the network designer needs to have an apriori knowledgeof the fraction of nodes to be attacked to design the optimalinterdependence structure. It is easy to see that such a problemsetting will be rarely be encountered in practice.

VII. SIMULATION RESULTS

For the testing of the theoretical results derived in this work,we have constructed a Python based test bench, using the Net-workX library, to emulate cascading failure in interdependentnetworks, obtaining the largest connected component in eachlayer for the successive stages of the cascading failure. Thealgorithm stops when a mutually connected bi-layer compo-nent is obtained. We have tested the validity of our results onScale-Free networks generated by the Barabasi-Albert model[24] with N =1000. Similar results have been observed forother network sizes as well, and N = 1000 is chosen as anexample without any preference. We present the fractional sizeψ of the surviving component of layer A versus the attackstrength η (note that the smaller the η, the more severe theattack). The curves are average results out of 20 instances ofnetwork generation each with 10 instances of initial attack onthe nodes (thus a total of 200 runs). Next we present a briefdescription of the simulation results.• Fig. 4 presents the simulation results for the comparison

of three different interdependence structures, namely,monotonic, random, and anti-monotonic, for the case ofrandomized attack on networks with complete one-to-one interdependence. It can be observed that the sim-ulation results corroborate with the theoretical findingspresented in Theorem IV.3 that monotonic ordering ofinterdependent nodes on the basis of their intra-layerdegrees is the optimal structure and anti-monotonic isthe worst structure. The left and right figures in Fig.4 depict the values of ψ and p, respectively. It can beobserved that ψ and p are qualitatively similar and thus

Figure 4: Completely interdependent networks with one-to-oneinterdependence. The individual layers have been generatedby the Barabasi-Albert model with m = 3. ‘�’, ‘�’, and ‘4’markers represent the random, monotonic, and anti-monotonicarrangement of interdependent nodes, respectively, on the basisof their intra-layer degrees.

in the interest of space, we omit the figures depicting pfor the remaining cases.

• For the case of randomized attack on networks with one-to-many interdependence structure, we consider c = 3as size ratio of the two interdependent layers in Fig.5. As discussed in Section IV-B, we focus on the caseof regular allocation of interlinks under two differentmodes of interdependence: i) AND interdependence, andii) OR interdependence. It can be observed that thesimulation results support Theorem IV.5 and TheoremIV.6. Particularly, it can be observed that the monotonicand anti-monotonic interdependence structures maxi-mize and minimize the robustness for AND and ORinterdependence, respectively. It can be observed thatthe difference of various interdependence structures forthe OR case is much smaller compared to others. This isbecause the OR interdependence structure is inherentlyvery robust due to the presence of alternative supportinglinks. Thus variations in interdependence structure donot lead to significant variations in ψ.

• The results for the case of partially interdependentnetworks are presented in Fig. 6. As discussed in

13

Figure 5: Completely interdependent networks with one-to-many interdependence under randomized attack. The individ-ual layers have been generated by the Barabasi-Albert modelwith m = 2, and the allocation of interlinks is regular. ‘�’, ‘�’,and ‘4’ markers represent the random, monotonic, and anti-monotonic arrangement of interdependent nodes on the basis oftheir intra-layer degrees, respectively. The left and right figurescorrespond to the cases of AND and OR interdependence,respectively.

Section V, the joint maximization of node allocationand node arrangement can be separated into individualmaximization problems. Here, we have considered thenode arrangement to be monotonic in nature (whichhas been shown to be optimal in Theorem V.1) andsimulated three allocation strategies, namely, high-high,low-low and random, which correspond to the cases ofdecoupling nodes of highest, lowest and random intra-layer degree, respectively. The results, as can be seenfrom Fig. 6, adhere to those presented in Theorem V.2that decoupling nodes of high intra-layer degree is theoptimal strategy.

• In Fig. 7, we present the case of targeted attack networkswith complete one-to-one interdependence. We focus onthe three interdependence arrangements, and show thatthe monotonic arrangement of nodes outperforms theothers for the case of double-layer (left in Fig. 7) perfecttargeted attack. The case of single-layer attack (right inFig. 7) has also been presented, where we can observe

0 0.2 0.4 0.6 0.8 10

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Figure 6: Partially interdependent networks with one-to-oneinterdependence. The individual layers have been generatedby the Barabasi-Albert model with m = 3 and 75% of nodesin the network are autonomous. ‘�’, ‘◦’, and ‘�’ representthe cases of random, low-low, and high-high, respectively,which are strategies of decoupling of nodes. The left and rightfigures correspond to the cases of double-layer and single-layerrandomized attacks, respectively.

that the anti-monotonic arrangement is better for thiscase. Note that neither anti-monotonic nor monotonic isoptimal for the single-layer attack case.

• In Fig. 8, we have considered ordering strategies basedon other measures of centrality, namely, betweennessand closeness. All theoretical results in this work havebeen obtained by utilizing the knowledge of degreecentrality of the nodes. This is a popular approach inresearch on complex networks, since analysis of othermeasures of centrality is very complicated and obtainingsystem equations relating these measures to networkrobustness is a very daunting task. In Fig. 8, we testthe various interdependence structures on other centralitymeasures to observe whether the results obtained hereapply for such cases. To simulate a practical scenario,instead of using random graph generation models, weused the Gnutella peer-to-peer network dataset availablefrom [25] to mimic network layers encountered in thereal world. It can be observed that even in this casethe monotonic arrangement of nodes is the best among

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Figure 7: Completely interdependent networks with one-to-oneinterdependence under perfect targeted attack. The individuallayers have been generated by the Barabasi-Albert model withm = 3. ‘�’, ‘◦’, and ‘�’, represent the random, monotonic,and anti-monotonic arrangement of interdependent nodes, re-spectively, on the basis of their intra-layer degrees. The leftand right figures correspond to the cases of double-layer andsingle-layer attacks, respectively.

the three simulated cases. These results indicate that thetheoretical results derived in this work have a widerapplicability since the suggested design guidelines arevalid for other forms of centrality as well.

VIII. CONCLUSION AND FUTURE WORK

In this work we have exploited the intra-layer node degreeinformation to obtain some quantitative results towards optimalinterlink designs for maximizing robustness of interdependentnetworks. As discussed in Section II, exploitation of intra-layerdegrees can yield the optimal interdependence structure onlyif the strong conditions in Lemma IV.1 are satisfied. We haveshown in this work that these conditions are in fact satisfiedby a wide variety of network and attack models. Howeveras we further generalize our network to include multiplenetwork layers and a combination of the various networkfeatures discussed in this work, like partial and one-to-manyinterdependence, we need to look at other analytical techniqueslike utilization of the adjacency matrix of the coupled net-work, exploiting other types of topological information likebetweenness, closeness etc., which is a part of our on-going

Figure 8: Completely interdependent networks with one-to-oneinterdependence under perfect targeted attack. The individuallayers have been generated by the Barabasi-Albert model withm = 3. ‘◦’, ‘�’, and ‘�’ represent the random, monotonic,and anti-monotonic arrangement of interdependent nodes, re-spectively, on the basis of their intra-layer betweenness (left)and closeness (right).

research work. Another important avenue of future work is theconsideration of the cost and various constraints of interlinkconstruction, such as the geometry of the network, whereinterlinked nodes are preferred to be physically close, or theinherent likelihood of nodes to be interlinked. These extensionswill further enhance the applicability of this work for practicaldesign of interlink structure in real world networks. Althoughthe results presented here represent an initial step in thisburgeoning field, we believe that such rigorously derivedresults regarding the effect of interdependence structure onnetwork robustness enable the scientific community to gaina better understanding of interdependent networks and obtainprinciples to design better networks.

APPENDIX ADERIVATION OF SYSTEM EQUATIONS

The node percolation probability (ψ) depends on threefactors: (i) the node survives the initial attack, (ii) at least oneof the neighbors of the node belongs to the MCC, and (iii)the coupled node in the other layer also satisfies the previoustwo conditions. Note that the second condition is equivalentto stating that the node belongs to the MCC. Utilizing results

REFERENCES 15

from works in literature [1, 16, 21], these conditions can bequantified as:

(i) Since (1− η) fraction of nodes are removed randomly,the probability of any node surviving the initial attackis η.

(ii) Let ka denote the degree of a node a ∈ A. Theprobability of a having at least one neighbor in theMCC is given by the expression: [1−(1−pA)ka ], sincea can connect to the MCC via any of its ka outgoingedges.

(iii) Let b ∈ B with degree kb be the supporting node of a.The probability that b belongs to MCC is nothing butthe combination of the above two conditions for layerB, which is given by the expression: η[1− (1−pB)kb ].

Combining these three conditions, we get:

ψA =1

N

N∑i=1

η2[1−(1−pA)kAρi ][1−(1−pB)k

Bi ] = ψB , (50)

where kli is the intra-layer degree of the ith node in layerl, and ρ = {ρ1, ρ2, · · · , ρN} represents the interdependencestructure where ρi denotes the index of layer A node which iscoupled to the ith node in layer B. Note that the summationof probabilities for the individual nodes has been normalizedby the total number of nodes N to obtain the fractional size ofthe MCC. Furthermore, ψA is equal to ψB in this case due tothe particular construction of the one-to-one interdependencestructure. Since every node is coupled to one node in the otherlayer, the steady state connected component will also containthe same number of nodes on both layers. This equality willnot be valid when we remove the one-to-one restriction on theinterdependence structure.

The ideas used in the derivation of the system equationsfor pA and pB are very similar. The only difference is thatwe are examining the probability of a node, reached bytraversing a randomly selected edge in an arbitrary direction,belonging to the MCC. Thus the first and third conditionsdescribed during the derivation of (50) would remain the same.The key difference lies in the second condition, described asfollows. Let a ∈ A with its intra-layer degree ka be the nodereached on traversing a randomly selected edge in an arbitrarydirection. This node a should connect to the MCC via anyof its remaining ka − 1 edges and the probability of this isgiven by the expression: [1 − (1 − pA)ka−1]. Combining thethree conditions, pA and pB must satisfy the following self-consistent equations:

pA =1

N

N∑i=1

η2kAρiλA

[1− (1−pA)kAρi−1][1− (1−pB)k

Bi ], (51)

pB =1

N

N∑i=1

η2[1− (1−pA)kAρi ]kBiλB

[1− (1−pB)kBi −1]. (52)

Another difference w.r.t. (50) is that in this case the normal-ization factor is Nλ as we are normalizing over all edges inthe network layer. Furthermore, it can be noted that each termhas been scaled up by the intra-layer node degrees (kAi or kBj )

because each node of degree k has to be counted k times asit can be reached by k different edge selections.

APPENDIX BDESCRIPTION OF NASH BARGAINING PROBLEM

We provide a brief description of the Nash BargainingProblem here. Let P = {1, 2, · · · , P} be the set of playersinvolved in the bargaining game. Let S be a closed convexsubset of RP representing the set of all possible pay-offs whichcan be received by the different players participating in thebargaining game. Let d = {d1, d2, · · · , dP } denote the setof dis-agreement pay-offs for the P players, representing thepay-offs that the various players will receive if they do notreach a mutual agreement through the bargaining game. Thepair (S,d) is called a P -person bargaining problem. Multiplekinds of co-operative game solutions to the bargaining problemexist in literature [26]. One of the most popular ones amongthem is the Nash Bargaining Solution (NBS) as it providesa unique and fair pareto-optimal solution to the bargainingproblem. Next, we will formally define the Nash-BargainingSolution.

LEMMA B.1 s? = φ(S,d) is said to be a NBS for thebargaining problem (S,d), if it satisfies the following axioms:

1) Feasibility: s? ∈ S.2) Individual Rationality: s?i ≥ di ∀ i, where s?i is the

pay-off received by the ith player.3) Pareto Optimality: There does not exist any s ∈ S such

that s � s?, i.e. no other solution can strictly dominatethe NBS thus leading to a better pay-off for all players.Therefore the solution s? should be such that s?i cannotbe increased without decreasing s?j for some j 6= i, i.e.there is no alternative allocation which can result in ahigher pay-off for all players.

4) Independence of Irrelevant Alternatives: Let the solu-tion to the bargaining problem (S,d) be denoted byφ(S,d). If s? = φ(S,d) and s? ∈ S′ ⊂ S, thens? = φ(S′,d).

5) Independence of Linear Transformation: For any linearscale transformation τ , τ(φ(S,d)) = φ(τ(S), τ(d)).

6) Symmetry: If the players of the game are indistinguish-able, s?i = s?j ∀ i, j ∈ P .

Furthermore, the solution point satisfying the above axiomsis unique and can be obtained by solving the maximizationproblem:

s? = φ(S,d) = arg maxs∈S,si≥di ∀ i

P∏i=1

(si − di). (53)

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