Suppressing cascades of load in interdependent networks

Abstract

Understanding how interdependence among systems affects cascading behaviors is increasingly important across many fields of science and engineering. Inspired by cascades of load shedding in coupled electric grids and other infrastructure, we study the Bak-Tang-Wiesenfeld sandpile model on modular random graphs and on graphs based on actual, interdependent power grids. Starting from two isolated networks, adding some connectivity between them is beneficial, for it suppresses the largest cascades in each system. Too much interconnectivity, however, becomes detrimental for two reasons. First, interconnections open pathways for neighboring networks to inflict large cascades. Second, as in real infrastructure, new interconnections increase capacity and total possible load, which fuels even larger cascades. Using a multitype branching process and simulations we show these effects and estimate the optimal level of interconnectivity that balances their trade-offs. Such equilibria could allow, for example, power grid owners to minimize the largest cascades in their grid. We also show that asymmetric capacity among interdependent networks affects the optimal connectivity that each prefers and may lead to an arms race for greater capacity. Our multitype branching process framework provides building blocks for better prediction of cascading processes on modular random graphs and on multitype networks in general.

Fig. 1.

The power grid of the continental United States, illustrating the three main regions or “interconnects”—Western, Eastern, and Texas—and new lines (in red) proposed by American Electric Power to transport wind power (11, 13).

Fig. 2.

A random three- and four-regular graph connected by Bernoulli-distributed coupling with interconnectivity parameter p = 0.1 [R(3)-B(0.1)-R(4)]. We also illustrate the shedding branch distribution q_ab(r_ba,bb), the chance that an ab shedding causes r_ba, r_bb many ba, bb sheddings at the next time step. Note these random graphs are small and become tree-like when they are large (⪆1,000 nodes).

Fig. 3.

The multitype branching process (red curves) approximates simulations of sandpile cascades on the power grids c, d (blue curves) surprisingly well, given that power grids are among the most difficult network topologies on which to predict other dynamics (54). We plot the four marginalized avalanche size distributions (using the labels a, b for grids c, d). For example, Fig. 3A, plotting versus t_a, is the chance an avalanche begun in a topples t_a nodes in a. Vertical axes’ labels are the expressions in the plots. Here dissipation f = 0.1.

Fig. 4.

Interconnectivity is locally stabilizing, but only up to a critical point. The main plot, the results of simulations on R(3)-B(p)-R(3) (2 × 10⁶ grains; f = 0.01; 2 × 10³ nodes/network), shows that large local cascades decrease and then increase with p, whereas large inflicted cascades only become more likely. Their average (gold curve) has a stable minimum at p^∗ ≈ 0.075 ± 0.01. This curve and its critical point, p^∗, are stable for cutoffs in the regime . (Insets) Rank-size plots on log–log scales of the largest cascades in network a (Left) for p = 10^-3, 10^-2, 10^-1 and in power grid d connected to c by 0, 8, or 16 edges.

Fig. 5.

Increasing the interconnectivity p between two random three-regular graphs extends the tail of the total avalanche size distribution s(t), which does not distinguish whether toppled nodes are in network a or b. The inset shows a rank-size plot on log–log scales of the number of topplings t in the largest 10⁴ avalanches (with 2 × 10⁶ grains of sand dropped), showing that adding more edges between random three-regular graphs enlarges the largest global cascades by an amount on the order of the additional number of interconnections. As expected theoretically (39), when a and b nodes are viewed as one network, s(t) ∼ t^-3/2 for large t (green line).

Fig. 6.

(A) Networks mitigating the smallest cascades 1 ≤ T_a ≤ 51 seek isolation p = 0, whereas (B) networks suppressing intermediate cascades 100 ≤ T_a ≤ 150 seek isolation p = 0 or strong coupling p = 1, depending on the initial interconnectivity p in relation to the unstable critical point p^∗ ≈ 0.05. But networks like power grids mitigating large cascades (C and D) would seek interconnectivity at the stable equilibrium p^∗ ≈ 0.075 ± 0.01. The bottom figures and the location of p^∗ are robust to changes in the window ℓ ≤ T_a ≤ ℓ + 50 for all 400 ≤ ℓ ≤ 1,500.

Fig. 7.

The critical load disparity at which inflicted cascades in d become equally large as its local cascades is r^∗ ≈ 15 (for 16 interconnections, “bridges”). (For eight interconnections, r^∗ > 20; see Fig. S9). Here we show a rank-size plot in log–log scales of the largest 10⁴ avalanches in power grid d, distinguishing whether they begin in c (inflicted cascades) or in d (local cascades), for 8 and 16 interconnections (solid, dashed curves), for r = 1 (Main), r = 15 (Inset), in simulations with dissipation of sand f = 0.05, 10⁶ grains dropped after 10⁵ transients.

Fig. P1.

The chance that a network a coupled to another network b suffers a cascade spanning more than half the network (gold curve) has a stable minimum at a critical amount of interconnectivity p^∗. Networks seeking to mitigate their largest cascades would prefer to build or demolish interconnections to operate at this critical point p^∗. The blue (red) curve is the chance that a cascade that begins in a (b) topples at least 1,000 nodes in a. Increasing interconnectivity only exacerbates the cascades inflicted from b to a (red), but interestingly, it initially suppresses the local cascades in a. (From simulations on coupled random three-regular graphs; the inset depicts a small example with 30 nodes per network and p = 0.1).