Maximization of Robustness of Interdependent Networks Under Budget Constraints

1 Maximization of Robustness of Interdependent Networks under Budget Constraints

Srinjoy Chattopadhyay, Student Member, IEEE, Huaiyu Dai, Fellow, IEEE, and Do Young Eun, Senior Member, IEEE Abstract—We consider the problem of interlink optimization in multilayer interdependent networks under cost constraints, with the objective of maximizing the robustness of the network against component (node) failures. Diverting from the popular approaches of branching process based analysis of the failure cascades or using a supra-adjacency matrix representation of the multilayer network and employing classical metrics, in this work, we present a surrogate metric based framework for constructing interlinks to maximize the network robustness. In particular, we focus on three representative mechanisms of failure propagation, namely, connected component based cascading failure [1], load distribution in interdependent networks [2], and connectivity in demand-supply networks [3], and propose metrics to track the network robustness for each of these mechanisms. Owing to their mathematical tractability, these metrics allow us to optimize the interlink structure to enhance robustness. Furthermore, we are able to introduce the cost of construction into the interlink design problem, a practical feature largely ignored in relevant literature. We simulate the failure cascades on real world networks to compare the performance of our interlinking strategies with the state of the art heuristics and demonstrate their effectiveness. Index Terms—Multilayer failure propagation, cost constrained optimization, network robustness, interdependent networks. !

1 INTRODUCTION branching processes [10] and percolation theory [11] to mul- Interdependent networks comprise multiple network layers, tilayer interdependent networks, revealing the relationship where the components of different layers are dependent on between the network robustness and the degree distribution each other for their functioning [4, 5]. Over the past few of the constituent nodes. To the best of our knowledge, years, interdependent networks have emerged as a topic this is the only rigorous model that can track the failure of broad and current interest due to their applications in cascades in multilayer networks. This branching process diverse areas, for instance, smart grids for power distribu- (BP) based framework relies on several assumptions. In tion [1], with interdependent network layers of power sta- particular, the system equations tracking the robustness are tions and communication routers; infrastructure design to exact for infinite trees and only node degree information control traffic congestion [6] in multi-modal transportation is considered, i.e. nodes of the same degree are considered systems, where the different modes of traffic constitute the to be statistically equivalent. Although simple graph gener- network layers; and recommendation systems [7], where ators, like the Erdos-Renyi model, lead to locally tree-like various social networks in which the users are active, like networks, i.e. the local neighborhood of a node has a loop Facebook and Twitter, are considered as the interdependent with vanishing probability, many topologies encountered layers. The fundamental phenomenon distinguishing multi- in the real world can be dramatically different. Complex layer networks is the back-and-forth propagation of failures topological features, like clustering, community structure, across the interlinks between the network layers, popularly etc., have been shown to cause non-trivial effects on the net- named cascading failure. This recursive cascade of failures work dynamics. Modification of the classical BP framework can increase the susceptibility of interdependent networks to account for non tree-like structures is an area of active to node failures compared to their isolated constituents. research [12, 13, 14]. However, extending these works to Our objective in this work is to optimize the design of devise multilayer network design strategies is challenging the interlinking structure between isolated network layers, due to the complexity of the mathematical equations track- maximizing the robustness of the resulting interdependent ing the robustness. It is important to note that even after network against failure cascades. the topological simplifications and degree-based characterization of nodes, the BP approach leads to self-consistent 1.1 Motivation equations of robustness, which typically do not admit closed form solutions and are usually solved numerically. This lack Research interest in interdependent networks is largely at- of amenability to mathematical analysis severely restricts tributed to the seminal paper by Buldyrev et al. [1], where the applicability of the BP approach to practical network the authors study a connected component based mechanism design problems involving large networks and complex of failure propagation in a two-layer interdependent net- topologies. In addition to the connected component based work. Subsequently, the failure cascades corresponding to mechanism of failure propagation, other mechanisms of this mechanism have been studied on more general network failure propagation have also been proposed in literature. structures [8, 9]. These works extend classical results from However, studying these complex mechanisms by the BP approach is a laborious task. Some recent works [2] have • The authors are with North Carolina State University, Raleigh, NC, made considerable progress in this area but the practicality 27695. of such results from the perspective of devising network 2

design strategies is debatable, owing to the computationally us to consider practical design constraints with the cost of expensive iterative computation of the robustness. We argue interlink construction, which has been largely overlooked that while a rigorous modeling of the failure cascades in in multilayer network design. The goal of this work is to multilayer networks is of significant theoretical importance, optimally distribute a total budget for interlink construc- it is unclear whether pursuing this avenue of research will tion, so that the robustness of the resulting interdependent lead to practical network design guidelines. network against different mechanisms of failure propaga- We can conclude from the above discussion that the tion is maximized. For three representative mechanisms, computation of robustness via an exact characterization namely, i) connected component based cascading failure, of the failure cascades is not possible for most practical ii) load distribution in interdependent networks, and iii) applications. To circumvent this problem, researchers have connectivity in demand-supply networks, we define our used various network properties as indirect measures of surrogate metrics and pose the design of interlinks as a robustness, for example, algebraic connectivity [15], number convex optimization problem. Due to the tractable metrics, of spanning trees [16], and total effective resistance [17]. this optimization problem can be readily solved to obtain For the robustness of networks against component failures, the budget allocation maximizing the proxy measures of most works [15, 18, 19] adopt the algebraic connectivity robustness. Through extensive simulation experiments on metric, defined as the second smallest eigenvalue of the real-world networks, we show that the surrogate metric supra-adjacency (SA) matrix representation of the multi- based interlink design outperforms the state of the art layer network, where all links (inter- and intra-layer) in heuristics. the multilayer network are stacked together: diagonal (off- The remainder of this paper is organized as follows. diagonal) blocks indicating interactions inside (across) lay- Section 2 introduces the preliminaries for this study. Section ers. The algebraic connectivity (λ2) of a network is closely 3 contains the main contribution, including the deficiency of related to its partitioning and synchronization properties existing approaches, the surrogate metric based framework [20]. However, in the context of diverse mechanisms of for optimizing interlinks, and an algorithm for learning the failure propagation in multilayer networks, it is not clear optimal allocation of resource distributively. We compare whether λ2 can effectively characterize the robustness. In our interlinking strategies with the existing heuristics in fact, we present simulation results indicating that such Section 4. Finally, we conclude our work in Section 5 and metrics cannot capture the robustness for the mechanisms of indicate future avenues of research. failure propagation considered here. Additionally, the metrics based on the SA matrix are agnostic to the attack models 2 PRELIMINARIES or the propagation mechanisms. One of the most important In this section, we introduce our system model and discuss ideas which we aim to elucidate through this paper is existing approaches for characterizing the robustness of that the mechanism of failure propagation can significantly multilayer networks. Finally, we describe the three repre- affect the robustness and network design guidelines do not sentative mechanisms of failure propagation considered in generalize across these diverse mechanisms. This warrants this work. the development of new metrics of robustness fine-tuned to particular mechanisms. Furthermore, these metrics should 2.1 System Model be amenable to mathematical analysis to allow us to develop Let us represent a network layer by G=(V,E), where V and network design guidelines. E ⊆ V ×V are the set of nodes and intra-layer (undirected) To summarize, the BP approach captures the true nature edges, respectively. We consider a two-layer interdepen- of the failure cascades but is limited in its applicability due dent network G =(GA,GB, E˜) where GA = (VA,EA) and to the complexity of the self-consistent system equations, GB=(VB,EB) are the constituent layers with |VA| = m and whereas the SA matrix approach leads to tractable metrics |VB| = n. Our control variable E˜ ⊆ VA × VB represents the of robustness but is agnostic to the propagation mecha- weighted interdependence structure between GA and GB. nism. This motivates us to develop novel metrics that can We assume that the cost structure, explained next, and the characterize the robustness of multilayer networks against isolated network topologies (GA,GB) are known and our different mechanisms of failure propagation. objective is to design the inter-layer links (E˜) to maximize 1.2 Our Contribution the robustness of G. Most practical network design problems are constrained The main contribution of our work is to introduce math- by the availability of resource, where heterogeneous costs ematically tractable metrics of robustness, which can be are associated with the construction of different links. De- utilized to optimize the interlink structure of multilayer signers strive for the optimal allocation of resources for networks. As these metrics are not based on a rigorous interlink construction under such cost constraints. Resource modeling of the failure cascades, we refer to them as sur- constraints have received limited attention in network de- rogate metrics of robustness. They were designed by study- sign literature, although simplified cost structures [18], lim- ing the dynamics of the failure cascades for the different iting the total number of interlinks, have been studied. We propagation mechanisms. These metrics are shown to vary consider a general cost structure given by: monotonically with the empirical measure of robustness for wl = r(xl), (1) both theoretical and real-world network structures. This monotonic relationship ensures that the maximization of where l ∈ {1, 2, ··· , mn} indexes the interlinks, and r : + + the robustness can be achieved by the maximization of R → R is the mapping from the allocated resource (xl) these metrics. Furthermore, the surrogate metrics enable to the interlink weight (wl). We assume r to be increasing, 3

M1 concave and differentiable with respect to (w.r.t.) xl with network robustness ψ (G(x), η) for the connected com- the additional constraint that r(0) = 0. It is not difficult to ponent based cascading failure mechanism, represented by see that these assumptions occur frequently in practice and M1. Under simplistic assumptions, such as, locally tree-like are also supported by economic theories, like diminishing topology, infinite-sized networks, arbitrary interlinking and returns and marginal utility. Our interlink optimization the randomized attack [1, 23, 24], the robustness is related framework is applicable for any r satisfying these criteria. to the degree distribution by inter-coupled self-consistent We provide results for two specific choices of r to serve as equations of the form: examples. Note that the interlink weights have been defined ψA(G, η) = η · f1 ψA(G, η), ψB(G, η), dA, dB (2) as unbounded positive real numbers. The physical interpretation of the weight can differ for various mechanisms ψB(G, η) = f2 ψA(G, η), ψB(G, η), dA, dB , (3) of failure propagation and for particular cases, bounded weights might be necessary. The cost structure r is defined where ψA and ψB are the fractional size of surviving accordingly for such cases. nodes in the two interdependent layers, dA and dB are Given a total budget b, our goal is to optimally distribute the degree distribution of the two layers, and η denotes it among all possible interlinks, maximizing the robust- the attack strength. f1 and f2 are inter-coupled functions, ness of the constructed interdependent network. The vector whose details can be obtained from the references. The T mn×1 x = [x1 ··· xmn] ∈ R denotes the resource allocation multiplicative factor η appears only in (2) due to the attack strategy, specifying the resource allocated to all mn inter- strategy, which only affects layer A. Although (2)-(3) can links. The cost structure (1) maps the allocated resource x to be solved for simple topologies, an iterative solution is the interlink structure E˜(x). Let G(x) denote the resulting required for most real-world applications. Thus, even under interdependent network, where G(x) , (GA,GB, E˜(x)). the simplified settings, the BP approach involves numerical Robustness can be intuitively understood as the resilience computation of the robustness, requiring global information of the network against node failures, although exact def- about the network topology and the failure spreading dy- initions might vary across applications. Let ψM(G(x), η) namics. When realistic conditions, like weighted interlinks denote the robustness of G(x) against cascading failures, and heterogeneous costs associated with interlink construc- where M specifies the failure propagation mechanism and tion, and more advanced problems, like the optimization η denotes the infection strength, defined as the fractional of the interlink structure, are considered, pursuing the BP size of the nodes comprising the initial infection. The choice approach does not seem promising. of the failure seeds is dependent on the attack model: 2.2.2 Supra-Adjacency Matrix (SA) the randomized attack arbitrarily chooses nodes, whereas the targeted attack preferentially chooses nodes based on The mathematical complexity of the BP approach resulted some topological measure of centrality. Recent works [21] in the development of various metrics, which can ap- have identified the targeted attack based on betweenness proximately characterize the robustness of networks. These as an extremely disruptive strategy. Due to this reason, we metrics of robustness utilize the supra-adjacency matrix [19] representation. The SA matrix for an interdependent consider the betweenness-based targeted attack model for ˜ the results presented here, where an attack strength of η im- network G(x) = (GA,GB, E(x)) is defined as: plies that η fraction of nodes with the highest betweenness (A1)m×m E˜(x) centrality measure comprise the infection seeds. In addition A = T , (4) E˜(x) (A2)n×n to this, we also study the degree-based targeted attack and the randomized attack models. The results corresponding where A1 and A2 are the adjacency matrices of the con- ˜ m×n to these two attack models are presented in the supplemen- stituent layers GA and GB, respectively, and E(x) ∈ R tary material. Adhering to classical models [1], the initial is the interlink structure. In the context of robustness of infection is assumed to occur in layer A and the robustness networks against component failures, the most popular is defined by the surviving fraction of nodes in layer B. metric is the algebraic connectivity, defined as the second Similar to works like [1, 22], we define the robustness of the smallest (smallest non-zero) eigenvalue of the Laplacian L. multilayer network as follows. The Laplacian matrix (L) of A is given by: L = D − A, where D , diag(A1), here 1 is the all-one column vec- Definition 2.1. The robustness of an interdependent network tor, and diag(v) is a diagonal matrix with entries from G(x)=(GA,GB, E˜(x)) against an attack strength η for a failure T v = [v1, v2, ··· , vm+n] . The surrogate measure of robust- propagation mechanism M, is defined as the fraction of nodes ness of the network is expressed as λ2(L(x)). The relevance in layer B which survive the failure cascades triggered by the of λ2 to the network robustness stems from its relationship removal of η fraction of nodes from layer A. We represent this to vertex and edge connectivity, uncovered in the classical robustness by ψM(G(x), η). work [20] and its later extensions. The impact of λ2 on network dynamics is a mature area with decades of research 2.2 Existing Approaches studying it. However in the context of multilayer networks, 2.2.1 Branching Process (BP) we argue that the role of λ2 as a metric of robustness is Extending classical works in branching processes [10] and much less understood. This is because there is a fundamen- percolation theory [11], researchers have been able to track tal difference between robustness problems on single and the cascade of failures among the interdependent layers in multilayer networks: cascading failure. In the former, ro- multilayer networks. These works present equations map- bustness is a single step process, whereas the latter involves ping the degree distribution of the nodes of G(x) to the recursive propagation of failures among the interdependent 4

network layers. Until any relationship between the algebraic load distribution in multilayer networks with weighted in- connectivity of a multilayer network and its robustness is terlinks, we extend the single layer model in [2] to multiple established, the use of λ2 as a universal metric of robustness layers using ideas from [31], which considers a simplified is debatable. model involving uniform distribution of the inter-layer offloaded load. We present the mathematical details of M2 in 2.3 Failure Propagation in Interdependent Networks the supplementary material. We study three representative mechanisms in this work. Although this is not an exhaustive list, to the best of our 2.3.3 Connectivity in Demand-Supply Networks (M3) knowledge, most failure spreading mechanisms considered in literature can be thought of as variations of these. Interdependent demand-supply networks have recently come to the attention of researchers [3]. These networks 2.3.1 Connected Component based cascades (M1) comprise a supply and a demand layer, where supply nodes This is the most popular model for cascading failure in feed the demand nodes to maintain their functionality. Let interdependent networks. Under this mechanism [1], node the supply rate of node a ∈ VA be denoted by sa. Let the failures can occur in two ways: i) by not belonging to the survival threshold of the demand node b ∈ VB, defined largest connected component in its own layer, or ii) by as the minimum supply required for its functioning, be failure propagated from the other layer(s) through the inter- denoted by tb. A demand node b survives if the following links. At each stage of the cascades, the nodes isolated from conditions hold: i) total supply available at b exceeds the P the largest connected component become non-functional threshold, i.e. a wabsa ≥ tb, where wab is the weight of and these failures propagate to the other layer through the interlink between a and b; and ii) b belongs to the largest the interlinks. We consider a generalized version of this connected component in the demand layer GB. [3] studies model involving weighted interlinks, where link weight (wl) a simplified version of this model, where wab is binary and represents the probability of propagation of failure across sa =tb =1, ∀a, b. We model the supply rate of each node (sa) an interlink. Classical models consider wl = 1, implying to be equal to its intra-layer degree. Note that M3 involves perfect propagation of failure among inter-layer neighbors. unidirectional dependence, since the supply nodes are not It is easy to see that M1 involves both local and long- affected by the failed demand nodes. This is in contrast range propagation of failure. The long-range propagation to the previous cases with back-and-forth propagation of is due to the first condition, where only the constituents of failures. This difference in the interlink functionality leads the largest connected component are assumed to survive. to distinct properties, discussed in Section 4, that are not If a node failure partitions the network into disconnected observed for M1 and M2. components, all but the largest component are assumed to It is easy to see from the above discussion that mech- fail according to M1. The local propagation arises from the anisms of failure propagation can be dramatically differ- second condition, where failure of a node can lead to failure ent from each other. Even for multilayer networks with of its inter-layer neighbors conditioned on the interlink identical intra-later topology, interlink structure and initial weight. infection seeds, distinct mechanisms can lead to significant differences in the fraction of nodes surviving in the steady 2.3.2 Load Distribution in Interdependent Networks (M2) state. We have illustrated this for a toy example in the Interdependent load distribution has recently attracted re- supplementary material. Since the illustration requires a search interest due to its diverse applicability. In power detailed description of the propagation mechanisms, we distribution networks [25], the load of a station is offloaded do not include it in the main document for brevity (see to its neighbors upon failure. In multilayer transportation Appendix A). The main idea we want to highlight is that networks [26], a load distribution model is used to emulate mechanisms of failure propagation strongly affect the ro- multi-modal (e.g. airways and railways) traffic to study bustness of multilayer networks and thus, effective met- congestion, where upon failure of an airport, its traffic rics of robustness should be dependent on the particular is redirected to neighboring airports and railway stations. mechanisms. In other words, a universal metric, capable of Although the system models for these different applications characterizing the robustness of networks against different have slight differences, the underlying principle is similar. mechanisms of failure propagation while easy to work with, Upon failure, the load of a node is fractionally offloaded does not exist. to its neighbors, proportional to the link weight. A node fails when its current load, comprising the initial and the offloaded load, exceeds its capacity. Following the model in [2], we restrict M2 to local propagation, where the 3 MECHANISMBASEDINTERLINKOPTIMIZATION load of a node is offloaded only to its intra- and inter- layer neighbors upon failure. Note that this restriction only We start this section by formally defining the cost con- applies to this mechanism. M1 and M3 involve both local strained robustness maximization problem. Next, we dis- and long-range propagation of failures. Depending on the cuss the deficiency of the existing approaches for charac- particular application, the load of a node in a network can terizing the robustness. Finally, we introduce our surrogate be defined in many ways. It is popularly defined in terms metrics and utilize them to solve the maximization problem of some topological information, such as degree [2, 27, 28] to obtain the optimal allocation of resources for the construc- or betweenness [29, 30]. We proceed with the degree-based tion of interlinks. We also present an algorithm for learning definition in this work. Due to the lack of works studying this resource allocation strategy distributively. 5 3.1 Problem Statement 1.0 

 The interlink optimization problem can be written as: 0.8  M 0.6 2 ψ maximize ψ (G(x), η) x (5) 0.4 1 T 0.2 subject to 1 x = b, and x 0, 2 3 0.0 mn×1 − − − − 0.0 0.2 0.4 0.6 0.8 1.0 where x ∈ R denotes a resource allocation strategy ψ and b is the total available budget. ψM(G(x), η) represents (a) (b) the robustness of G(x) = (GA,GB, E˜(x)) against an initial Figure 1: Deficiency of traditional approaches: a) Variation failure of strength η for a failure propagation mechanism of ψ over different degree based interlink structures b) M. The mapping between x and E˜(x) is given by the Variation of λ2 with ψ. cost structure r following (1). In (5), the first constraint verifying their performance through simulations. Mono- specifies the budget and the second constraint restricts the tonic and anti-monotonic interlinking have been identified allocated resource to be non-negative. Note that the budget as favorable strategies under certain conditions, by both the- constraint has been specified as an equality. Let us con- oretical [23] and simulation based studies [32]. The mono- sider a particular example to explain the intuition behind tonic (anti-monotonic) strategy interlinks the ith highest de- this. Interlink weights in M represent the probability of 1 gree node in layer A to the ith highest (lowest) degree node propagation of failure across the interconnected layers. Since in layer B, ties broken randomly. The simulation results are inter-layer failure cascades cannot occur without interlinks, presented in Fig. 1a, where the two constituent layers and ψM1 (G(x), η) is maximized at x = 0. It is easy to see the infection seeds are fixed and the network robustness that x = 0 corresponds to a non-functional interdependent ψM(G(x), η) is plotted for various interlinking strategies network. For instance, in smart grid for power distribution, at η = 0.1. Qualitatively similar results were observed for although the absence of interlinks between the power and other values of η. We re-scale ψ for each mechanism, so communication network layers eliminates the possibility of that the values vary between 0 (minimum robustness) and 1 failure cascades, it compromises the fundamental property (maximum robustness) for all cases. We simplify our system of interdependent networks: the interdependence. In such model to focus on the impact of different interlinking struc- cases, if the budget for the interlink construction is con- tures. We assume that G(x) = (GA,GB, E˜(x)) comprises strained by an inequality (1T x ≤ b), the network design layers of same size (|VA| = |VB| = n), generated by the problem can produce trivial solutions. Due to this reason, Erdos-Renyi model. The interlinking structure is complete we specify the budget constraint as an equality to ensure one-to-one, where each node a ∈ VA is interlinked to a the inter-connectivity among the network layers. For spe- unique node b ∈ VB. We consider the simplest cost structure cific mechanisms of failure propagation, like M3, where (linear and binary), where wl =xl and xl is a binary variable ψM(G(x), η) is increasing in the elements of x, the solution indicating the presence or absence of the lth interlink. These to (5) also solves the case where the budget is specified as assumptions reduce the domain of x from the continuous an inequality. space Rmn×1 to the discrete space comprising all permu- tations of n indices. Note that these assumptions on the 3.2 Deficiency of Existing Approaches network structure are not enforced when we evaluate the The two approaches for estimating the robustness are: i) the performance of our interlinking strategies on real world net- branching process (BP) approach, involving the mathemati- works in Section 4. To explore different degree-based inter- cal modeling of the true nature of the failure cascades; and linking strategies, we consider partially ordered monotonic ii) the supra-adjacency (SA) matrix approach, involving the and anti-monotonic interlinking. In Fig. 1a, an x-coordinate SA matrix representation and employing classical metrics, of +0.6 indicates that 60% of the interlinks in the network are like algebraic connectivity, as the heuristic measures of constructed monotonically, where the positive x-coordinate robustness. In the following, we discuss the shortcomings denotes monotonicity, while the remaining 40% interlinks of these approaches in characterizing ψM(G(x), η). are constructed arbitrarily. The x coordinates denote the interlinking strategies while the y coordinates denote the 3.2.1 Branching Process (BP) average robustness over 500 independent instances of the The development of rigorous mathematical models describ- interlink structure. It can be observed that no interlinking ing ψM(G(x), η) as a function of x for a general mecha- strategy is optimal for all cases, and in fact our simula- nism of failure propagation and a general cost structure tion tests reveal that the maxima varies with η as well. is a daunting task. Although (2)-(3) allow us to compute Thus, degree-based interlinking structures cannot be used the robustness through numerical iterations under certain to characterize the robustness of networks against the three conditions, it is unlikely that such equations can be utilized representative mechanisms. to solve cost constrained robustness optimization problems. Additionally, the BP approach is fundamentally limited by 3.2.2 Supra-Adjacency Matrix (SA) its reliance on the locally tree-like topology and only degree- Under this approach, the robustness of the network is based topological information. measured by applying classical metrics to the SA matrix Due to the difficulties in tracking the failure cascades for representation of the multilayer network. An important the distinct mechanisms, an alternative approach might be drawback of this approach is that the measure of robustness to devise various degree-based interlinking strategies and is only affected by the network topology. As a result, this ap- 6

M proach is agnostic to the mechanisms of failure propagation To summarize, we intend to design f0 (x) for each rep- as well as the attack models. resentative mechanism that are: i) monotone with the empir- Let us evaluate the performance of algebraic connectivity ical values of robustness, to serve as a surrogate measure of (λ2) as a metric of robustness. In Fig. 1b, we compare robustness; ii) concave in x, in order to pose the robustness M λ2 to the empirical robustness ψ (G(x), η) for different maximization as a convex optimization problem; and iii) interlinking strategies under the three mechanisms. Note amenable to mathematical analysis, so that we can solve that the y coordinates represent the re-scaled values of λ2, (6) to obtain interlink design strategies. In the following, we where the absolute values of λ2 corresponding to the three propose a family of surrogate metrics that can be utilized to mechanisms are re-scaled to the range [0,1]. This operation study the robustness against various mechanisms of failure was employed to improve the readability of Fig. 1 as the propagation. Later on, we choose specific metrics from this ranges of ψ and λ2 for the different cases vary significantly. family, corresponding to the particular mechanisms studied The interlinking structures and infections seeds are the same in this work. The family of surrogate metrics can be written as that in Fig. 1a. For a metric to characterize the robustness, as: mn M X M M it should be monotone with ψ for all interlinking structures f0 (x) = F (r(xl),Dl ), (7) x. Note that the simplifications of the cost structure lead l=1 to the equivalence between the resource allocation strategy M where r(xl) is the weight (wl) of the lth interlink, D w = x l and the interlink structures, i.e. l l. A monotonic rela- represents the contribution of interlink l towards enhancing ψ M tionship ensures that the maximization of can be achieved network robustness, and F is some function of these λ by maximizing 2. This requirement is not satisfied in Fig. arguments. Note that a particular metric from this family (7) 1b, implying that λ2 is not an effective metric for mea- M is specified by two factors: i) Dl , capturing the importance suring the robustness of networks against the mechanisms of the lth interlink; and ii) the function F M. In this work, we of failure propagation considered in this work. It can be M M show that even straightforward definitions of Dl and F observed from Fig. 1b that for all three mechanisms, the can produce metrics which can track the robustness for the ψ λ joint maximization of and 2 is never achieved. In fact, the representative mechanisms and lead to better interlinking robustness for each mechanism is maximized at a different strategies than existing heuristics. In this work, we choose value of λ2, which again is dependent on η. This highlights M M different models for F and Dl to illustrate that there a key aspect of our work, that the robustness of networks is no universal format for the surrogate metrics. Although against various mechanisms of failure propagation is maxi- different metrics of robustness might lead to different per- mized under different interlinking strategies and a universal formance gains, we intend to elucidate through this work λ metric of robustness, like 2, is not viable. that even simple choices of these metrics can lead to a significant gain in performance. A comparative study on the 3.3 Surrogate Metrics of Robustness performance of different metrics for the same mechanism of Complexity of the BP approach limited its applicability to failure propagation is an important avenue of research and study the robustness of networks against general mecha- beyond the scope of this work. nisms of failure propagation. Employing universal metrics Next, we define the surrogate metrics corresponding to of robustness, like λ2, is also not a promising direction due the three representative mechanisms and solve (6) to obtain to the non-monotone relationship to ψ. Furthermore, Fig. the resource allocation strategy maximizing them. Interest- 1 demonstrated that the maximization of the robustness ingly, the surrogate metric based framework is amenable occurs at different interlinking strategies for different mech- to distributed optimization and we present an algorithm anisms, discouraging the search for other universal metrics. that can learn the resource allocation strategy through local This motivates us to develop new metrics of robustness exchange of information among the nodes. specific to each mechanism. These metrics, represented by M 3.3.1 Connected Component based cascades (M1) f0 (x), were designed by studying the failure cascades for the representative mechanisms. Simulation experiments The basic idea behind the metric for this case is the observa- i reveal that the proposed metrics vary monotonically with tion that anti-monotonic ordering, where the th highest de- A i the empirical measure of robustness for different values of gree node in layer is coupled to the th lowest degree node B attack strength (η) under all three mechanisms. This allows in layer , outperforms the monotonic and the random us to remove the dependence on η from the surrogate ordering for the case of the degree-based targeted attacks metrics. Using these metrics, our interlink design problem [23, 34]. Although this observation was made under the (5) becomes: degree-based attack model, our simulation studies reveal M that similar results hold for the betweenness-based attacks maximize f0 (x) x (6) as well. Intuitively, since the attacker chooses nodes of subject to 1T x = b, and x 0. high betweenness (more important nodes), coupling them to nodes of low betweenness (less important nodes) checks Our objective in this work is to devise surrogate metrics the spread of failures. Using a multiplicative combination M M f0 (x) that are monotone with the empirical value of ro- (F 1 (x, y) , xy), our metric can be written as: M mn bustness ψ (G(x)) for each mechanism M. Additionally, if X f M1 (x) = r(x )DM1 . the surrogate metrics are concave in the resource allocation 0 l l (8) strategy x, then (6) becomes a convex optimization problem l=1 due to the affine constraints, allowing us to utilize the rich Since we want to preferentially construct interlinks between M1 literature [33] studying it. nodes of high and low betweenness, Dl is modeled to 7

constructed by an investment of xl resource is determined 1.0 US Power Grid and 1.0 US Flight Network and Amtrack Network Internet Routers Erdos Renyi Graph by the quality parameter αl ∈ [0, 1], governing the ease 0.8 Erdos Renyi graph 0.8 Scale Free Graph Scale Free graph of construction of the lth interlink, i.e. interlinks with low 0.6 0.6 2 ( x ) 1  0

values of αl are expensive to construct. Note that the shifted 

0 0.4 0.4

0.2 0.2 sigmoid representation upper bounds the interlink weight

0.0 0.0 to 1. This satisﬁes the physical interpretation of the interlink 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ψ ψ weight as a probability, for the case of M1. Under this cost (a) (b) structure, we can utilize the KKT conditions to solve the convex optimization problem (6) to obtain the following 1.0 US Flight Network and US Power Grid Erdos Renyi Graph closed form solution for the resource allocation strategy: 0.8 Scale Free Graph

0.6

2 q q αlDl αlDl ?  0

+ − ν 0.4 1 2 2 x? = log , (9) 0.2 l q q αl αlDl − αlDl − ν? 0.0 2 2

0.0 0.2 0.4 0.6 0.8 1.0 ψ ? where xl is the optimal resource allocated to the lth inter- (c) ? link, αl is the interlink quality parameter, ν is the optimal Figure 2: Plot of the surrogate metric values with the empir- value of the equality constraint multiplier ν, and Dl, i.e. ical network robustness for different mechanisms. M1 Dl , denotes the absolute modeling of the importance of the lth interlink. The mathematical details are presented in have a high value, when the betweenness difference be- the supplementary material. It can be observed from (9) that M1 ? tween the endpoints of l is large. As f0 (x) in (8) is a a small ν (close to 0) implies that a non-zero resource is M1 weighted sum of the Dl values, maximizing it is equiv- allocated to most interlinks in the optimal condition, thereby alent to allocating more resource on interlinks with higher indicating a strong interaction between the interdependent M1 M1 ? Dl . Note that Dl can be defined in many ways as long layers. ν can be computed through binary search by substi- P ? as its value increases with the betweenness difference, for tuting (9) into the budget constraint l xl =b. A distributed example: algorithm for obtaining this solution is presented in Algo- M1 • Dl = |blA − blB | (absolute), rithm 1. M1 −1 • Dl = exp(−|blA − blB | ) (exponential), 3.3.2 Load Distribution in Interdependent Networks (M2) where b is the intra-layer betweenness of i, and lA, lB i We adopt a logarithmic modeling of F for this case, where are the two end-points of the lth interlink. Note that the F M2 (x, y) , log(x + y). The surrogate metric of robustness proposed network design strategies hold for any arbitrary M for this mechanism of failure propagation can be written as: definition of Dl , as long as it is decoupled from the interlink structure, i.e. DM is not a function of x. We consider mn l M2 X M2 M1 f0 (x) = log(Dl + r(xl)). (10) the absolute modeling of Dl here. The next logical step is to ascertain how well our l=1 proposed metric, given by (8), captures the robustness. This kind of modeling is widely adopted in information The plots in Fig. 2a compare the true network robustness theoretic studies [35], where it is desirable to allocate power M1 M1 ψ (G(x), η) with the metric values f0 (x) for different optimally to different communication channels to maximize M2 interlinking structures, taking η = 0.1. We consider theoreti- the throughput. The Dl term in (10) corresponds to the in- cal graph generation models along with real world networks terference in each channel, while the r(xl) term corresponds as the intra-layer topologies. The real networks are chosen as to the allocated power. For interlink l that enhances the M2 practical candidates for interdependent networks following network robustness, the value of Dl should be low indi- the representative failure propagation mechanisms. This cating a low-noise (high-quality) wireless channel to which is discussed further in Section 4. Each point reflects the more power (resource) should be allocated. On the contrary, average empirical robustness (x co-ordinate) and average high interference channels should receive less power. The M2 metric value (y coordinate) over 500 independent instances logarithmic modeling of F prefers links of low Dl as of interlink structures. It can be observed that the surrogate opposed to the multiplicative case. M1 M2 and the empirical measures of robustness, f0 (x) and In order to define Dl , let us develop a better under- ψM1 (G(x), η), respectively, are correlated monotonically. standing of the failure cascades. We define the free space This behavior allows us to substitute the maximization of a node i as Si , Ci − Li, where Ci and Li denote of the intractable ψM1 (G(x)) with the maximization of the capacity and load of the ith node, respectively. Since M1 f0 (x), which is a much simpler problem. The concavity a node fails when its current load exceeds the capacity, M1 and mathematical tractability of f0 (x) allows us to further the difference between the capacity and load denotes the incorporate the cost of construction of interlinks into the additional space available to handle the load offloaded from interlink design problem. In order to solve the robustness neighbors. Following [36, 37], it can be conjectured that maximization problem (6), we need to specify our cost interlinking nodes of low free space to nodes of high free structure r. We remind the reader that we assume r in space is a good strategy. This is intuitive as nodes of high (1) to be increasing, concave and differentiable. For M1, free space can handle the load offloaded from nodes of low let us choose a shifted sigmoid modeling for r, given by free space upon failure. By coupling the nodes which are r(x ) = 2 − 1 l 1+e−αlxl , where the weight of an interlink most likely to fail to the nodes which are best at handling 8

failure, we check the spread of failures thereby enhancing 3.4 Algorithm M2 robustness. We can deﬁne Dl as: We present the basic framework for obtaining the resource

M2 −1 allocation strategy maximizing the surrogate metrics in Al- • Dl = |SlA − SlB | (absolute) M2 −1 gorithm 1. This algorithm can be theoretically applied for • D = 1 − exp(−|S A − S B | ) (exponential) l l l any separable metric by varying the EB (evaluate budget) M M These definitions ensure that links connecting nodes with block corresponding to the definitions of r, Dl and F . high difference in free space are given preference. We use An important aspect of our work is that the interlinking the exponential modeling for this case. Simulation tests strategy can be obtained distributively, i.e. the constituent presented in Fig. 2b reveal a monotonic relationship be- nodes can locally exchange information to learn the resource M2 M2 tween f0 (x) and ψ (G(x), η), validating the choice of allocations maximizing the surrogate metrics. Distributed the metric. The optimal resource allocation under a shifted implementation of network design algorithms has become sigmoid r is given by: more of a necessity than a feature in recent years, as many q practical applications involve networks of large sizes for ? 2 ?2 ? −Dlν + αl + α + ν − 2Dlν αl which centralized algorithms are not feasible. We distribute ? 1 l xl = log ? . (11) the decision of finding the resource allocation strategy for αl (Dl − 1)ν each interlink to their end-points in layer A or B. Let us The details of the analysis is similar to the previous case define the algorithm w.r.t. layer A. Algorithm 1 runs on all and is omitted for brevity. nodes a ∈ VA in order to compute the resource allocation strategy, i.e. each node a learns the resource to be allocated 3.3.3 Connectivity in Demand-Supply Networks (M3) to all interlinks connected to it. The two main blocks of the Similar to the preceding case, a logarithmic modeling is used Algorithm 1 Optimal Resource Computation for F M3 , under which the metric can be written as: 1: procedure OPT ALLOC(α,GA,GB, b, ∆th) m n M3 X 1 X 2: p1 ← 0, p2 ← max(size(GA), size(GB)), ∆ ← inf f0 (x) = log + sjr(xij ) , (12) b 3: while ∆ > ∆th do i=1 i j=1 4: ν ← (p1 + p2)/2 5: ˜b ← P EB(a , ν) where r(xij ) is the weight of the interlink between nodes l l ˜b > b i ∈ VA and j ∈ VB, bi is the betweenness centrality 6: if then p ← ν of the ith demand node, and sj is the supply rate of j. 7: 2 Here, betweenness is used to measure the importance of 8: else the demand nodes in maintaining the connectivity of the 9: p1 ← ν demand network. As demand nodes of high significance 10: ∆ ← |˜b − b| M3 should receive more supply, the interference term (Di ) 11: xl ← EB(αl, ν) is modeled as 1/bi. Note that there exists a fundamental 12: return xl, ∀l difference between (12) and the load distribution metric (10). Although both are based on a logarithmic modeling, the distributed implementation are: i) broadcast, employed to interlinks for these processes perform different functions. inform every node about the budget b and ∆ , and ii) dis- M th In 2, interlinks offload load to their neighbors indepen- tributed consensus, employed in Step 5 to compute the sum M dently. In 3, the demand node cares about the total supply of the allocated resource. There exists extensive literature it receives and the interlinks cooperate in maintaining the [38, 39] studying the distributed implementation of these supply and are not independent. Due to this reason, the operations. In Algorithm 1, the dual variable ν is computed proposed metric is not separable w.r.t. the interlinks indexed at every node in Step 4 and the resource allocation strategy l i by but is separable w.r.t. the demand nodes indexed by . corresponding to ν is computed by the EB block, based on Simulation studies (Fig. 2c) justify the choice of the metric. the solutions (9), (11), and (13), respectively. The sum (˜b) of We use a logarithmic modeling of the cost structure, given the allocated resources is obtained by distributed consensus by r(x ) = log(α x + 1), where the parameter α is the l l l l algorithms. This sum is compared with b to update ν in quality parameter. We remind the reader that the various Steps 6-9. This descent process is repeated, till a threshold settings of F M, DM and r, are chosen so as to illustrate l budget usage performance (|˜b − b| ≤ ∆ ) is achieved, to the generality of the surrogate metric based framework. The th estimate ν?. Essentially, the algorithm learns ν? via binary solution for this case can be written as: search between pre-defined upper and lower limits. The 1 1 upper limit (p2) is dependent on the objective function. We x? = max 0, − , ij ? ? (13) found that the maximum of the two network sizes works ν (1/bi + wi ) αij for all three cases. The choice has a marginal effect on the ? Pn ? ν where w , sjr(x ) is the total supply received convergence time of . Finally in Step 11, the EB block i j=1 ij ? by the ith demand node. Note that unlike (9) and (11), computes the optimal resource xl for each interlink using (13) is self-consistent, since the right hand side of (13) is the optimal value of ν. a function of x? . This is due to the non-separability of ij 4 SIMULATION RESULTS the objective function w.r.t. the interlinks. Unlike the closed form expressions obtained in the previous cases, (13) is We compare the performance of our interlinking strategies solved by numerical iterations. with the state of the art heuristics for the three mechanisms 9

of failure propagation. We use Python and its associated 0 0 libraries for our simulations. We present simulation results 00 00 in two domains. Firstly, we simulate the failure cascades 0 0

00 00

for different strengths of attack (η) to compare the network 0 0 0 0 M 0 0 robustness ψ (G(x), η) of various interlinking strategies, 00 00 0 0 00 00 under two different regimes of available budget b. Secondly, 00 00 000 0 000 0 we study the variation of performance gain of these inter- 00 00 0 00 0 00 0 00 00 0 00 0 00 0 η η linking strategies with the available budget b. Instead of Figure 3: Connected component based cascading failure un- relying on theoretical graph generation models, we use real- der stringent (left) and sufficient (right) budget constraints. world networks for the layers GA and GB. These real-world networks have been chosen to serve as examples of practical random allocation strategies, the total budget is divided uni- scenarios, where the representative mechanisms of failure formly among all nodes, and then allocated to the highest propagation might occur. ranked inter-layer node. 4.1 Performance of different interlinking strategies 4.1.1 Connected Component based cascades The main roadblock in the way of comparing our strategies Smart power distribution grids, involving interdependent to the existing heuristics is that since we consider a general network layers of power stations and communication system model with weighted interlinks and include the routers, is a classical practical example [1] of M1. Under the effects of interlink construction cost, similar models have degree-based targeted attack, current art [23, 34] suggests not been studied in relevant literature, to the best of our that anti-monotonic interlinking outperforms other degree- knowledge. Most works that study the impact of interlink based (random and monotonic) strategies. Supported by structure, consider unweighted links and are agnostic to simulation experiments as discussed earlier, we choose the interlink construction cost. In order to provide a fair the betweenness-based anti-monotonic interlinking as our comparison, we modify the state of the art heuristics to heuristic design strategy, where the node with the ith high- incorporate cost. This is achieved by considering a linear est betweenness in VA is coupled to the node with the combination of the heuristics with the interlink quality ith lowest betweenness in VB. Starting from the node of parameter αl. Let us assume that for each node a ∈ A, the highest betweenness, for each node a ∈ A, the un- the heuristics rank the nodes b ∈ B in some order. The interlinked neighbors in VB are ranked in increasing order modified heuristics incorporate αl into this ranking scheme on the basis of their betweenness and modified betweenness in order to avoid spending resource on expensive interlinks (additive cost correction), for the heuristic and the modi- with small values of αl. Our experiments indicate that an fied heuristic strategies, respectively. Arbitrary ranking is additive modification, where the modified rank for interlink adopted for the random allocation. l is the sum of the original rank and αl, works best for the In our simulations, we consider an Autonomous Sys- cases considered here. tem AS-733 topology [40], representative of the network In the following set of results, we study the network of routers comprising the Internet, as GA and the West- robustness under four interlink design strategies, or more ern States Power Grid of the United States [41] as GB; specifically, constrained resource allocation strategies: i) the with |VA| = 6474, |VB| = 4941. This corresponds to the random resource allocation; ii) the state of the art heuristics, problem of optimally interlinking the power stations to based on existing works; iii) the modified state of the art the communication routers to maximize the robustness of heuristics, considering interlink construction cost; and iv) the smart grid against failure cascades. The simulation the surrogate metric based optimization, proposed in this results are presented in Fig. 3. Under both stringent and work. Distinct heuristics are used for the three distinct sufficient budget conditions, it can be observed that the mechanisms of failure propagation based on the relevant lit- interdependent network is extremely vulnerable to node erature. We present the results under two different regimes failures. For all interlinking strategies, even η as low as distinguished by the availability of budget: stringent, where 0.05 leads to the failure of more than 95% of the net- the resource availability is low and few nodes can be in- work in the steady state. This establishes the importance of terlinked; and sufficient, where the resource availability is studying the robustness of multilayer networks, which can abundant and most nodes are strongly interlinked. Note catastrophically react to node failures due to the recursive that the numerical values of the budgets is not the same cascade of failures between the interdependent layers. Fig. for all three mechanisms. These values were chosen to give 3 clearly establishes the inferiority of the random resource the reader a visually representative idea of the comparative allocation strategy, as it almost coincides with the x axis. performance of the different resource allocation strategies at This indicates that random allocation can have catastrophic both ends of the spectrum of available budget. The simula- effects on the robustness for M1 and even state of the art tion results for the variation of performance with available heuristics perform much better. It is interesting to note that budget is presented in Section 4.2. As defined in Section 2, this phenomenon does not generalize to other mechanisms, M ψ (G(x), η) is measured by the fraction of nodes in VB as will be evident from Fig. 4, where performance of the that survive in the steady state after the cascade of failures, random and the heuristic strategies are comparable. Since initiated by an infection seed of size η|VA|. Qualitatively the heuristic strategies do not consider cost, we compare similar results is obtained when the layer roles are reversed, our interlink design algorithm w.r.t. the modified heuristics, i.e. layer B is attacked and robustness is measured w.r.t. which outperform the other two strategies in all cases. It layer A. Under the heuristic, the modified heuristic and the can be observed from Fig. 3 that the performance gain of 10

1.0 1.0 Random Random Heuristics Heuristics 0.4 Modified Heuristics 0.8 Modified Heuristics 0.8 0.8 Optimized Optimized

0.3 0.6 0.6 0.6 0.2 0.4 0.4 0.4

Random Random 0.1 0.2 0.2 Heuristics 0.2 Heuristics Modified Heuristics Modified Heuristics Optimized Optimized 0.0 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 η η η η Figure 4: Load distribution in interdependent networks un- Figure 5: Connectivity in demand-supply networks under der stringent (left) and sufﬁcient (right) budget constraints. stringent (left) and sufﬁcient (right) budget constraints.

our framework over the modified heuristics is negligible for disjoint paths between them are identified. Distinct supply stringent budgets but significant under sufficient budgets. nodes are connected to the individual disjoint paths. This The reason behind the absence of gain at stringent budgets strategy is intuitive because the failure of a supply node is discussed in Section 4.2. only affects a unique disjoint path in the demand network and the source and target demand nodes remain connected 4.1.2 Load Distribution in Interdependent Networks through alternative paths. We consider a multi-modal transportation network: Amtrak For designing a resource allocation scheme correspond- G railway routes [42] as A and the US Airport network [43] ing to this strategy, the supply of each node a ∈ VA is as GB, with VA = 11526 and VB = 1574. The problem we distributed uniformly among all constituents of each node are looking at is optimizing the interlinks, which may be disjoint path. Under the modified strategy, this distribution shuttle bus services, between the airports and the railway is modified so that the resource allocated to each interlink stations to maximize the robustness of the airports against is proportional to its quality parameter αl. The simulation the failure of the railway stations. Adhering to the conven- results are presented in Fig. 5. Here we can observe that in tion in related works, the load, representative of the traffic contrast to the previous cases, the surrogate metric based carried by the nodes, is denoted by the intra-layer degree. framework has a significant performance gain even at strin- The capacity of a node is given by: Ci = (1+β)Li, where Li gent budgets. This counter-intuitive result is explained in and Ci denote the load and capacity of node i, respectively, the following. and β = 0.5. Simulation based works [36] from literature suggest that load based anti-monotonic interlinking is a 4.2 Performance gain variation with available budget good strategy. As a result, the heuristic strategy for M is 2 We are also interested in studying the performance of the the degree-based anti-monotonic interlinking, since the load proposed interlinking strategies under different budget con- is measured by the intra-layer degrees. straints b. The performance gain for a strategy is computed The simulation results for the two regimes are presented w.r.t. the random allocation of resource. We define the gain in Fig. 4. An interesting point to note here is that under of an allocation strategy as the total difference between the stringent budgets, the random resource allocation can out- robustness corresponding to the strategy in question and the perform the heuristics. This is counter-intuitive due to the random allocation, for the different values of η. Fig. 6 reveals fact that a random distribution of resources is agnostic the that the performance gain is marginal under stringent bud- network topology as well as the interlink construction cost. get conditions for M1 and M2. This is because the majority The reason behind this observation is that the heuristic of network components are not interlinked owing to the interlinking strategies do not consider cost and the recom- low availability of budget. Due to this weak interlinking of mended interlinks might end up being very expensive ow- the network layers, the cascade of failures between them ing to low α values. This is in contrast to the previous case, l is not pronounced, leading to similar performance for all where even heuristic strategies agnostic to cost performed interlinking strategies. This phenomenon is not observed for much better than the random allocation of budget. This M3, since it does not involve any recursive propagation of establishes the importance of cost constraints in real world failure due to the unidirectional interdependence. For this interlink optimization problems, where for certain mecha- case, interlinking strategies have a pronounced effect on nisms (M ) the heuristic strategies can perform reasonably 1 network performance in lower budgets. It is interesting to well, whereas for others (M ) their performance might be 2 note that although the surrogate metric based framework worse than random. outperforms the other strategies in all cases, the perfor- 4.1.3 Connectivity in Demand-Supply networks mance gain of the surrogate metrics varies considerably Demand-supply interdependencies are common in many with the mechanism of failure propagation. A significant practical instances of multilayer networks. We consider the variation is also observed when other attack models, like US Power Grid network [41] as the supply layer (GA) and the degree-based targeted attack and the randomized attack, the US Airport network [43] as the demand layer (GB), with are considered; the corresponding results are presented in |VA| = 4941 and |VB| = 1574. We study the problem of the supplementary material. These simulation results clearly interlinking the airports to the power distribution stations. show that the surrogate metrics of robustness cognizant of Path-based interlink assignment has been identified as a the mechanism of failure propagation and the attack model good heuristic algorithm in [3] and we use it as the state outperforms the state of the art heuristics. The variation of of the art heuristic. In this strategy, a source and target node the gain obtained by the surrogate metric based framework from the demand layer are picked randomly and all node under different conditions reveals other interesting properties about the robustness of complex networks. REFERENCES 11

whether approximate characterizations, like those presented 70000 Heuristic Modified Heuristic 60000 in this work, can achieve reasonably good performance. Optimized 50000

40000 6 ACKNOWLEDGMENTS 30000 This work was supported in part by the National Science % Gain 20000 Foundation under grants ECCS-1444009 and CNS-1824518, 10000

0 and in part by Army Research Ofﬁce under Grant W911NF- 0 1000000 2000000 3000000 4000000 5000000 Total budget (b) 17-1-0087.

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Parandehgheibi and E. Modiano. “Robustness of Huaiyu Dai (F’17) received the B.E. and M.S. interdependent networks: The case of communication degrees in electrical engineering from Tsinghua networks and the power grid”. In: GLOBECOM. IEEE. University, Beijing, China, in 1996 and 1998, 2013, pp. 2164–2169. respectively, and the Ph.D. degree in electrical [26] P. Zhang et al. “The robustness of interdependent trans- engineering from Princeton University, Prince- portation networks under targeted attack”. In: EPL (Eu- ton, NJ in 2002. He was with Bell Labs, Lucent rophysics Letters) 103.6 (2013), p. 68005. Technologies, Holmdel, NJ, in summer 2000, [27] Z. Wu et al. “Cascading failure spreading on weighted and with AT&T Labs- Research, Middletown, NJ, heterogeneous networks”. In: Journal of Statistical Mechan- in summer 2001. He is currently a Professor of Electrical and Computer Engineering with NC ics: Theory and Experiment 05 (2008), p. 05013. State University, Raleigh. His research interests [28] J. Wang and L. L. Rong. “Cascade-based attack vulner- are in the general areas of communication systems and networks, ability on the US power grid”. In: Safety Science 47.10 advanced signal processing for digital communications, and commu- (2009), pp. 1332–1336. nication theory and information theory. His current research focuses [29] K. I. Goh, B. Kahng, and D. Kim. “Universal behavior on networked information processing and crosslayer design in wireless of load distribution in scale-free networks”. In: Physical networks, cognitive radio networks, network security, and associated Review Letters 87.27 (2001), p. 278701. information-theoretic and computation-theoretic analysis. He has served [30] A. E. Motter and Y. C. Lai. “Cascade-based attacks on as an editor of IEEE Transactions on Communications, IEEE Transac- complex networks”. In: Physical Review E 66.6 (2002), tions on Signal Processing, and IEEE Transactions on Wireless Com- munications. Currently he is an Area Editor in charge of wireless com- p. 065102. munications for IEEE Transactions on Communications. He co-edited [31] Y. Zhang, A. Arenas, and O. Yagan.˘ “Cascading failures two special issues of EURASIP journals on distributed signal processing in interdependent systems under a ﬂow redistribution techniques for wireless sensor networks, and on multiuser information model”. In: Physical Review E 97.2 (2018), p. 022307. theory and related applications, respectively. He co-chaired the Signal [32] J. Wang, C. Jiang, and J. Qian. “Robustness of interde- Processing for Communications Symposium of IEEE Globecom 2013, pendent networks with different link patterns against the Communications Theory Symposium of IEEE ICC 2014, and the cascading failures”. In: Physica A: Statistical Mechanics and Wireless Communications Symposium of IEEE Globecom 2014. He its Applications 393 (2014), pp. 535–541. was a co-recipient of best paper awards at 2010 IEEE International [33] S. Boyd and L. Vandenberghe. Convex optimization. Cam- Conference on Mobile Ad-hoc and Sensor Systems (MASS 2010), 2016 IEEE INFOCOM BIGSECURITY Workshop, and 2017 IEEE Interna- bridge university press, 2004. tional Conference on Communications (ICC 2017). [34] B. Min et al. “Network robustness of multiplex networks Do Young Eun (M’03–SM’15) received the B.S. with interlayer degree correlations”. In: Physical Review E and M.S. degrees in electrical engineering from 89.4 (2014), p. 042811. Korea Advanced Institute of Science and Tech- [35] R. G. Gallager. Information theory and reliable communica- nology (KAIST), Daejeon, Korea, in 1995 and tion. Vol. 588. Springer, 1968. 1997, respectively, and the Ph.D. degree in elec- [36] X. Peng et al. “Load-induced cascading failures in in- trical and computer engineering from Purdue terconnected networks”. In: Nonlinear Dynamics 82.1-2 University, West Lafayette, IN, USA, in 2003. (2015), pp. 97–105. Since 2003, he has been with the Department [37] Z. Chen et al. “Cascading failure of interdependent net- of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC, USA, works with different coupling preference under targeted where he is currently a Professor. His research attack”. In: Chaos, Solitons & Fractals 80 (2015), pp. 7–12. interests include network modeling and performance analysis, mobile [38] D. M. Aoyama and D. Shah. “Fast distributed algorithms ad hoc/sensor networks, mobility modeling, social networks, and graph for computing separable functions”. In: IEEE Transactions sampling. Dr. Eun has been a member of the Technical Program Com- on Information Theory 54.7 (2008), pp. 2997–3007. mittee of various conferences including IEEE INFOCOM, ICC, GLOBE- [39] B. Williams and T. Camp. “Comparison of broadcasting COM, ACM MobiHoc, and ACM Sigmetrics. He served on the Editorial techniques for mobile ad hoc networks”. In: Proceedings Board of the IEEE/ACM Transactions on Networking and Computer of the 3rd ACM international symposium on Mobile ad hoc Communications, and was TPC Co-Chair of WASA 2011. He received networking & computing. ACM. 2002, pp. 194–205. the Best Paper awards in the IEEE ICCCN 2005, IEEE IPCCC 2006, and IEEE NetSciCom 2015, and the National Science Foundation CAREER [40] J. Leskovec et al. “Graph evolution: Densiﬁcation and Award 2006. He supervised and coauthored a paper that received the shrinking diameters”. In: ACM Transactions on Knowledge Best Student Paper Award in ACM MobiCom 2007. Discovery from Data (TKDD) 1.1 (2007), p. 2.