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. F 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 character- ization 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 met- rics 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