Taken 10 km east of Hvolsvollur Iceland on April 18th, 2010. Lightning flashes and glowing lava illuminate parts of Eyjafjallajokull's massive ash plume in this 30-second exposure.

Way back in 2004, John Robb wrote a piece on scale-free networks:

Scale-free networks are everywhere. The can be seen in airline traffic routes, connections between actors in Hollywood, weblog links, sexual relationships, and terrorist networks. So what exactly is a scale-free network? A scale-free network is one that obeys a power law distribution in the number of connections between nodes on the network [emphasis mine].

Obviously, considering the plight of the airlines right now in the midst of an apocalyptic (yet curiously invisible) ash cloud is particularly fascinating to do in the context of Robb’s networks. In characterizing the nature of scale-free ones, he comes up with a positive and a negative:

Scale-free networks are extremely tolerant of random failures. In a random network, a small number of random failures can collapse the network. A scale-free network can absorb random failures up to 80% of its nodes before it collapses. The reason for this is the inhomogeneity of the nodes on the network — failures are much more likely to occur on relatively small nodes.

Scale-free networks are extremely vulnerable to intentional attacks on their hubs. Attacks that simultaneously eliminate as few as 5-15% of a scale-free network’s hubs can collapse the network. Simultaneity of an attack on hubs is important. Scale-free networks can heal themselves rapidly if an insufficient number of hubs necessary for a systemic collapse are removed.

Examining the fallout from Eyjafjallajokullin in this light does present an interesting dichotomy. If we consider the entire globe as one big air traffic system, then it definitely is showing resilient capabilities. Flights are diverted around the affected nodes and redistributed to areas unaffected by the ash cloud. It’s as if Europe was a tumor that has been surgically removed from the rest of the airborne world.

Thus, of course most everyone can continue to fly whether or not Europe’s airports are open. The global network is continuing to function.

And in fact, it’s hard to conceptualize European airspace as an isolated network. At this point all air traffic to and from the continent is inextricably bound to the rest of world, and so it’s hard to imagine an inverse scenario in which the rest of the world ceases to fly while Europe muddles on.

However, this picture changes slightly if we consider the voluntary closure of most European airspace as an intentional attack. Robb gives the threshold as 5-15% of a system’s capability. Of the 30 busiest airports in the world in 2009, seven are in Europe, those seven with total passenger traffic of 268 million people a year. If 1.5 billion people travel by plane every year, that’s roughly 18% of world capacity (and that’s before taking into account all the other European airports that didn’t crack the top 30). For the rest of the world, it’s a relatively stable – if infuriating – situation. I suppose the real determining factor is that while the initial closures were shocks to the system, they didn’t begin on a Europe-wide scale, and by the time those in the east started closing, it was no longer a surprise.

Either way, the system is voluntarily taking at least a fifth of itself offline, which gives rise to an interesting third possibility that Robb doesn’t mention: how much of a system can turn itself off before collapse?