Estimation of Robustness of Interdependent Networks against Failure of Nodes Srinjoy Chattopadhyay∗ and Huaiyu Daiy Department of Electrical and Computer Engineering, North Carolina State University. Email: ∗[email protected], [email protected] Abstract—We consider a partially interdependent network and from a percolation theory and branching process point of develop mathematical equations relating the fractional size of the view, based on the classical work [6]. This method of analysis connected component of the network, surviving the cascading is more amenable to analysis and has been used in some failure, to the intra-layer degree distribution of the nodes. We show that these system equations can be mathematically analyzed recent works like [7] to obtain system equations modeling and closed form expressions for the metrics of robustness can the cascading failure. Using this technique of analysis, we be obtained for the Erdos-Renyi (ER) model of random graph have been able to obtain relationships between the robustness generation. We have described the application of our analysis of interdependent networks and its structural parameters for technique to networks with general degree distributions. In our various network models and attack models which, to the analysis, we consider the two extremes of the attack model: randomized attack, where nodes are attacked at random without best of our knowledge, do not exist in literature. The work any knowledge of intra-layer degrees and perfect targeted attack, presented in this paper is in two broad domains. Firstly, we where nodes are attacked based on the strict descending order of have applied our technique of analysis on a general framework their intra-layer degrees. Our results can enable researchers to of partially interdependent network to mathematically obtain gain a better understanding of the robustness of interdependent the robustness against randomized node failure for networks networks. Index Terms—Interdependent networks, Critical fraction, net- generated as ER graphs. Furthermore, we have been able to work robustness. obtain closed form expressions for the critical fraction of nodes I. INTRODUCTION and phase transitions points, which are important properties Over the past decade, inter-connectivity of systems has [3], [7] governing the variation of robustness of networks emerged as a prominent avenue for research in a wide variety for different strengths of attack. We have also discussed the of fields. Throughout the spectrum of cyber and physical application of the analysis techniques developed here to real systems, we can observe a steadfast transition to interde- world networks with known degree distribution. Secondly, we pendent systems: smart power distribution grids which have have considered the case of a targeted node failure model, inter-coupled power distribution and communication networks where the attacker selects nodes in the order of their intra- [1] being the prime example in this case. The increased layer degrees and developed theoretical approximations for the functionality of interdependent systems however comes at corresponding robustness for any general network. the price of enhanced sensitivity to node failures due to the The remainder of this paper is organized as follows. Section phenomenon of cascading failure, where failure of a fraction II presents the system model and system equations which have of nodes initiates a recursive cascade of failures between the been analyzed for randomized attack in Section III and tar- different layers of the network. geted attack in Section IV. Section V presents a comparison of A better understanding of the cascading failure phenomenon the theoretically derived results with the simulations. Finally, has been a prime focus of research work in this area. The tradi- we conclude and indicate possible future work in Section VI. tional technique of analysis in this field utilizes the generating function of the intra-layer node degree distribution. Using II. SYSTEM MODEL this approach, researchers have obtained equations modeling We consider a system of partially interdependent network the cascading failure for various scenarios like completely comprising two layers A and B with N nodes each. The interdependent networks with one-to-one interdependence [1], layers consist of two types of nodes: autonomous nodes (X one-to-many interdependence [2] and partial interdependence in Fig. 1), which do not require support from nodes of the [3], [4]. Some works have also considered the case of targeted other layer; and interdependent nodes (Y in Fig. 1), which attack [5]. Due to the inherent complexity of the generating are dependent on the other layer nodes for their survival. function approach, closed form results relating the robustness Let the fraction of interdependent nodes be denoted by q, to the structural parameters of the network were only obtained where they are chosen randomly so that all nodes have the for a small subset of networks, particularly, completely inter- same probability (q) of being interdependent. The two layers dependent networks with identically distributed layers. are constructed as random graphs with small average degrees In this work1, we have analyzed interdependent networks λA and λB respectively (λA; λB N). This construction 1This work was supported in part by National Science Foundation under ensures the locally tree-like property [6] of the layers, which Grants CNS-1016260, ECCS-1307949 and EARS-1444009. is necessary for the formulation of the system equations networks as will be shown in Section III-C. To formulate the system equations, we need to define node ( A and B) and edge (pA and pB) percolation probabilities denoting the probability that a randomly chosen node (or edge) belongs to the MCC. The survival of any node in the network depends on three conditions [6], [7], [8]: 1) the node survives the initial attack; 2) the node belongs to the largest connected component in its layer; and 3) if the node is interdependent in nature, its support node in the other layer should also satisfy Fig. 1. System model for partially interdependent network. the above two conditions. On the basis of these conditions, we have obtained percolation theory [6], [7] based self-referencing modeling cascading failure. The interdependency between the system equations, the details of which can be obtained from two layers is taken to be one-to-one and bidirectional, which the report [9]: has been widely adopted in relevant literature as a first critical X X X A = (1 − q) f2(kA) + q f2(kA)f2(kB); (1) step towards understanding the operation and key phenomena kA kA kB of interdependent networks. Thus all interdependency links X X X correspond to an interdependent pair of nodes, (a; b) with pA = (1 − q) f1(kA) + q f1(kA)f2(kB); (2) a 2 A and b 2 B, which depend on each other for their kA kA kB kl k −1 survival. The interdependency links are assumed to be inde- where f1(kl) = ηP [kl] [1 − (1 − pl) l ] and f2(kl) = λl k pendent with respect to (w.r.t.) the intra-layer degrees of the ηP [kl][1 − (1 − pl) l ], and kl is the intra-layer degree of a nodes, i.e. inter-links are constructed without the knowledge node in layer l with mean degree λl. Here f1 (or f2) can of the graph structure. be understood as the probability that a randomly chosen edge In this model, we are interested in studying the effect of the (or node) is connected to a node of (is of) degree kl and failure of a fraction (1 − η) of nodes from both layers. For the belongs to the MCC. Thus the summation of these individual first part of our analysis we will assume that the attack strategy contributions of edges and nodes gives the edge (pl) and node is random in nature, i.e. the attacker (or the mother nature) fails percolation ( l) probabilities. Note here that is nothing all nodes with the same probability. Subsequently in Section but the fractional size of the MCC which is our metric IV, we will consider a targeted attack scenario and investigate quantifying the robustness. The system equations for layer B the corresponding robustness. Our main interest is to relate the can be symmetrically obtained. Similar system equations exist robustness of the network to its structural parameters (λ, q). in literature [7] and thus we have omitted the details of the The fractional size of the mutually connected component derivation of these system equations. In this work, we have (MCC) of the network which survives the cascading failure theoretically analyzed these system equations for networks is classically [1]–[5] taken to be the metric for robustness. generated as ER graphs to derive expressions describing the It is known from works like [1], [5] that for completely network robustness ( A and B), the critical node fraction interdependent networks, the phase transition of the network (ηc), and percolation phase transition points (qc) in terms of robustness with varying attack strength (η) is of first order, structural parameters (q; λ). i.e. there exists a minimum η = ηc which would lead to a III. ANALYSIS:RANDOM ATTACK η < η non-zero MCC size in the steady state and for all c We start our analysis considering random node failure, and the average size of the MCC is arbitrarily close to 0. In consider the case when the network layers are generated other words, the variation of robustness with attack strength is according to the ER model. We analyze the system equations η = η discontinuous at c. Furthermore, partially interdependent for layer A (analysis for layer B is symmetrical) in parts by networks [3], [4] undergo a phase transition from first order focusing on the individual terms. The details of the analysis (discontinuous) to second order (continuous) as the fraction of is included in the report [9] which gives us: interdependent nodes (q) decreases below a certain threshold fAg 1 @ X q kA defined as the phase transition point ( c).