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Suppressing Cascades of Load in Interdependent Networks Suppressing cascades of load in PNAS PLUS interdependent networks Charles D. Brummitta,b,1, Raissa M. D’Souzab,c,d,e, and E. A. Leichtf aDepartment of Mathematics, bComplexity Sciences Center, cDepartment of Mechanical and Aerospace Engineering, and dDepartment of Computer Science, University of California, Davis, CA 95616; eSanta Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501; and fComplex Agent-Based Dynamic Networks Complexity Centre, Saïd Business School, University of Oxford, Park End, Oxford OX1 1HP, United Kingdom AUTHOR SUMMARY Modern society depends on characterized by a power law increasingly interdependent distribution: They are often systems that are prone to small but occasionally widespread failure. For enormous. example, transportation, The Bak–Tang–Wiesenfeld communication, power grids, model was originally and other infrastructures formulated on a lattice. support one another and the Given the relevance of world’s interconnected networked systems, the economies. Barrages of dynamics have recently been incidents large and small— studied on isolated networks, downed power lines, but not yet on grounded aircrafts, natural interdependent (or modular) — disasters, and the like cause networks. Here we study it APPLIED avalanches of repercussions on two networks with sparse MATHEMATICS that cascade within and connections between them. among these systems (1). The two networks model a b Although interdependence Fig. P1. The chance that a network coupled to another network either two different types of suffers a cascade spanning more than half the network (gold curve) has a confers benefits, its effect on pà infrastructures or two the risk of failure at the stable minimum at a critical amount of interconnectivity . Networks seeking to mitigate their largest cascades would prefer to build or modules within one individual system level and at demolish interconnections to operate at this critical point pÃ. The blue infrastructure, and the the collective level remains (red) curve is the chance that a cascade that begins in a (b) topples at interconnections between poorly understood. Here we least 1,000 nodes in a. Increasing interconnectivity only exacerbates the them model their analyze how the cascades inflicted from b to a (red), but interestingly, it initially suppresses interdependence. We interconnectivity the local cascades in a. (From simulations on coupled random three- explicitly study networks (interdependence) between regular graphs; the inset depicts a small example with 30 nodes per extracted from two network and p ¼ 0.1). networks affects the sizes of interdependent power grids their cascades of load in the southeastern United shedding. For networks derived from interdependent power States and an idealization of them that is more amenable to grids, we show that interdependence can have a stable mathematical study. In this idealization, each node is equilibrium. An isolated network suppresses its large cascades connected to a node in the other network with probability p by connecting to other networks, but too many (Fig. P1, Inset). interconnections exacerbate its largest cascades—and those of Our main result is that interdependence can have a stable the whole system. We develop techniques to estimate this equilibrium (Fig. P1). Some interconnectivity is beneficial to optimal amount of interconnectivity, and we examine how an individual network because the other network acts as a differences in network capacity and load affect this reservoir for extra load. The gold curve of Fig. P1 shows that equilibrium. Our framework advances the current the chance of a large cascade in a network can be reduced by mathematical tools for analyzing dynamical processes on 70% by increasing the interconnectivity p from 0.0005 to interdependent (or modular) networks, and it improves our 0.075. Too much interdependence, however, becomes understanding of systemic risk in coupled networks. detrimental for two reasons. First, new interconnections open In the basic process that we consider, a system contains pathways for the neighboring network to inflict additional many elements, each of which sheds its load onto neighboring load. Second, each interconnection augments the system’s elements whenever it reaches its capacity. This is captured by capacity, making available more load that fuels even larger – – the classic sandpile model of Bak Tang Wiesenfeld, a cascades in each network. As a result, the chance of a large paradigm for the power law statistics of cascades in many disciplines, from neuronal avalanches to financial instabilities to electrical blackouts (2). In a basic formulation on a graph Author contributions: C.D.B., R.M.D., and E.A.L. designed research; C.D.B., R.M.D., and of nodes and edges, each node has a capacity for holding E.A.L. performed research; C.D.B., R.M.D., and E.A.L. contributed new reagents/analytic tools; C.D.B. and R.M.D. analyzed data; and C.D.B., R.M.D., and E.A.L. wrote the paper. grains of sand (interpreted here as load or stress). Grains of sand are dropped randomly on nodes, and whenever a node The authors declare no conflict of interest. has more sand than its capacity, it topples and sheds all its This article is a PNAS Direct Submission. sand onto its neighbors, which may in turn have too much Freely available online through the PNAS open access option. sand and topple, and so on. Thus, dropping a grain of sand 1To whom correspondence should be addressed. E-mail: [email protected]. can cause an avalanche (cascade) of toppling. These See full research article on page E680 of www.pnas.org. avalanches, like blackouts in power grids (3), occur in sizes Cite this Author Summary as: PNAS 10.1073/pnas.1110586109. www.pnas.org/cgi/doi/10.1073/pnas.1110586109 PNAS ∣ March 20, 2012 ∣ vol. 109 ∣ no. 12 ∣ 4345–4346 Downloaded by guest on October 1, 2021 cascade in an individual network eventually increases with capture these differences with a parameter that controls the à interconnectivity p,sop is a stable minimum. rates at which cascades begin in either network. We show that, This second factor—that new interconnections increase a in any asymmetric situation, the equilibrium will be frustrated, ’ — network s capacity for load has global consequences. With with only one of the grids able to achieve its optimal level of more load available, larger cascades in the system as a whole interconnectivity. become possible. Therefore, networks that interconnect to one Determining how interdependence affects the functioning of another to mitigate their own cascades (Fig. P1) may networks is critical to understanding the infrastructures so vital unwittingly cause larger global cascades in the whole system. This consequence of increased capacity is a warning for the to modern society. Whereas others have recently shown that interconnections under construction among, for example, interdependence can lead to alarmingly catastrophic cascades different power grids to accommodate long-distance trade and of failed connectivity (5), here we show that interdependence renewable sources of energy (4). also provides benefits, and these benefits can balance the The results in Fig. P1 show that networks suppressing their detriments at stable equilibria. We expect that this work will largest cascades would seek the optimal level of stimulate calculations of critical points in interconnectivity à interconnectivity p . However, as shown in the main article, among networks subjected to other dynamics. As critical pà building interconnections to reach increases the occurrence infrastructures such as power grids, transportation, of small cascades. Conversely, networks can suppress their communication, banks, and markets become increasingly smallest cascades the most by seeking isolation, p ¼ 0. But interdependent, resolving the risks of large cascades and the suppressing their smallest cascades exacerbates their largest ones (left side of Fig. P1), just as extinguishing small forest incentives that shape them becomes ever more important. fires can incite large ones, and engineering power grids to suppress small blackouts can increase the risk of large 1. Little RG (2002) Controlling cascading failure: Understanding the vulnerabilities of in- terconnected infrastructures. J Urban Technol 9:109–123. ones (3). 2. Bak P, Tang C, Wiesenfeld K (1988) Self-organized criticality. Phys Rev A 38:364–374. Finally, we determine how asymmetry among networks 3. Dobson I, Carreras BA, Lynch VE, Newman DE (2007) Complex systems analysis of series affects the level of interconnectivity that each network of blackouts: Cascading failure, critical points, and self-organization. Chaos 17:026103. considers optimal. For instance, two interconnected power 4. Joyce C (April 28, 2009) Building power lines creates a web of problems. (National Public Radio), http://www.npr.org/templates/story/story.php?storyId=103537250. grids may differ in capacity, load, redundancies, demand, 5. Buldyrev SV, Parshani R, Paul G, Stanley HE, Havlin S (2010) Catastrophic cascade of fail- susceptibility to line outages, and age of infrastructure. We ures in interdependent networks. Nature 464:1025–1028. 4346 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1110586109 Brummitt et al. Downloaded by guest on October 1, 2021.
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