  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 tradeoffs. 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 multi-type networks in general.
