Cascading failures in scale-free interdependent networks M. Turalska,1 K. Burghardt,2 Martin Rohden,3 Ananthram Swami,1 and Raissa M. D’Souza3, 4, 5 1)CISD, US Army Research Laboratory, Adelphi, MD 2)ISI, University of Southern California, Marina del Ray, CA 3)Dept. of Computer Science, University of California, Davis, CA 4)Dept. of Mech. and Aerospace Eng., University of California, Davis, CA 5)Santa Fe Institute, Santa Fe, NM

Massive failures, such as blackouts in power grids1 or bustness of interconnected systems. systemic default of financial institutions2, are associated with disproportionately high cost to the systems in which they occur. Thus the ability to predict and control such events is an area of active studies. Over the past decade, the research into networks’ resistance to failure has fo- cused on interdependent systems. In particular, Brum- mitt et al.3 demonstrated that connections between sim- ple regular networks act as a control mechanism regu- lating frequency of catastrophic failures. However the impact of more realistic topologies, such as broad-scale degree distributions or assortativity, on the cascading process between interdependent layers, has yet to be ex- plored. In this paper we present a systematic study filling this gap. We study failure cascades with the BTW sand- pile model, which self-organizes to an apparent critical state, in which cascades sizes are distributed as a power law4,5. The BTW model captures a common feature of many systems in which individual elements carry load, but have a fixed capacity. We explore the propagation of cascades across networks possessing realistic network topologies, such as heterogeneous degree distributions, as well as intra- and interlayer degree correlations. In particular we show that vulnerability of individ- ual nodes to fail correlates with degree of assortativity present in the network. The lack of degree correlations in FIG. 1. The probability of a large cascade occurring in a neutral scale-free network causes the toppling rate to be system of interconnected scale-free networks is a function of almost independent of node degree (Fig.1), while in assor- intra- and inter-layer assortativity. Coupling between neutral tative scale-free networks high degree nodes have much (top) networks leads to a reduction in the number of large larger chance to topple than low degree ones. Thus, the events for both hub-to-hub and random inter-layer connec- inter-layer coupling that preferentially couples rich club tions. Inter-layer connectivity has a strong impact on the nodes in assortative networks reduces the chance of cas- BTW dynamics on assortative networks (middle), with ran- cades propagating across layers. Randomness of neutral dom coupling resulting in an increasing occurrence of cas- networks reduces occurrence of large cascades through a cades. Those differences are a direct result of how the inter- different process, in which heterogeneity of the topology nal network topology affects the chance for a link to transport allows for redistribution of the additional load without sand during a cascade (bottom; neutral network on the left, assortative on the right). Bottom figures correspond to adja- leading to large cascades cency matrices, where nodes are sorted from the lowest (left) We demonstrate that correlations present in the struc- to the highest degree (right). ture of the multilayer network influence the dynamical cascading process and can prevent failures from spreading across connected layers. We find that all three studied 1S. T. P. Hines, J. Apt, Energy Policy 37 (2000). properties: scale-free degree distribution, internal net- 2M. Bardoscia, S. Battiston, F. Caccioli, and G. Caldarelli, Nature work assortativity and cross-network hub-to-hub connec- Communications 8 (2017). 3 tions, are all necessary components to significantly reduce C. D. Brummitt, R. M. D’Souza, and E. A. Leicht, PNAS 109, E680 (2012). size of cascades in the BTW sandpile model. 4P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 These findings highlight the importance of considering (1987). the internal and cross-network topology in optimizing ro- 5P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. A 38, 364 (1988).