If you’ve traveled enough, you’ve inevitably been hit by every vacationer's nightmare: a cancelled flight.

It sucks,  but it could be a lot worse. Like, what if I couldn’t be  here in Montana because the plane scheduled to bring me here from Boston got snowed in at an airport in  New Zealand, two days earlier?

Delays and cancellations are always happening somewhere in the world, but  through storms, volcanic eruptions, and outbreaks of war, airlines  are usually able to keep planes in the skies and passengers  relatively on schedule. For that reliability, we can thank  networks. Not computer networks.

I mean the mathematical kind.  And while well-planned networks make modern air travel possible,  it only takes disruptions in a few key places to bring  everything to a grinding halt. [♪INTRO] The simplest way to imagine an airport  network is as a series of points joined by lines. Mathematicians call  this kind of picture a graph, which, in this context, means something  different than “a data visualization”. In math, a graph is a model  of how systems like the air traffic industry are fundamentally structured.

There are nodes, the points  that represent each airport, connected by edges, the lines that  represent flights between airports. And when the graph represents something tangible, like airports, they’re sometimes called networks, and we study them with network theory. Now you could plot the graph  on a map or a grid, however you want really.

Mathematicians are  only interested in the connections. This might feel like an odd  game of connect-the-dots, but network theory lets us do  things like organize flight patterns that withstand interruptions,  determine which layovers are most common, and understand  airline pricing methods. To do that we need to create the graph.

We could try making a complete  graph, meaning that every airport in the whole world has a  direct flight to every other airport. In graph language, that means every node shares an edge with every other node. Never needing a layover sounds pretty sweet.

The other perk to this  setup is that if one airport drops out of the network,  any other airport can pick up the slack and get you to your destination. But since there are about  4,000 airports in the system, the cost of those conveniences would be running more than 16 million flights globally. Which is why real airports, sadly, don’t have direct flights to everywhere.

It doesn’t make economic or  environmental sense to allocate tons of resources to the tiny  percentage of people wanting to fly directly from the Caribbean  island of Saba to, say, Nepal. In reality, there are only about  70,000 different direct flights in the world, less than one percent of  what a complete network would have. So here’s the challenge for network  theorists: how do you assign those routes to get people from  A to B in the best way possible?

We need a network that allows people to get where they’re going, even without a direct flight. Paths are the solution, meaning any  route between airports, including routes that go through additional  nodes. In other words, a layover.

Mathematicians call the idea of being able to get between any two places connectedness. One way to calculate the connectedness  of a graph is to find the largest set of nodes that always has some kind of path between any two of them. Ideally, every airport in the  world would be connected to the global airport network,  so that you have a chance to get there, no matter  how many layovers it takes.

But there are lots of ways  to achieve connectedness. For instance, say you want  a path from Indianapolis, in to Missoula, Montana. You could  fly from Indianapolis to Portland, to Huntsville, to Bismarck and  finally to Missoula.

Easy peasy! Just kidding, we all know that sounds wacky. Because in practice, closeness also matters.

Closeness is what we call the smallest number of individual flights you take to  get from one place to another. If it takes one layover to get from Indianapolis to Missoula, that’s a closeness of two. In our fantasy world of a complete  airport network, every flight is direct, so every path would have a closeness  of one.

But in the real world, airlines have the challenge  of balancing connectedness and closeness with their available  resources and consumer demand. The practical solution is  a handful of big airports whose main job is to connect to  a lot of different locations. Most routes from Indianapolis  to Missoula would pass through Chicago or Salt Lake City  because these big airports have a high degree, meaning they’re  connected to many other nodes.

Especially compared to an airport like Missoula where they have,  like, four flights a day. I kid. I kid.

No shade to  MSO, I actually love it here. I love your short lines, I love  your eclectic decor, the big bear… it’s great! And there are  more than four flights a day, there are four whole gates now!

Network analysis shows that  carefully implementing high degree airports really does a better job  of creating shorter flight paths. In a 2007 paper published in the  journal Chaos, Solitons and Fractals, a pair of Italian researchers found that real life closeness in the Italian  airport network was about two. That means, based on the shape of the network, getting between any two airports in  Italy typically required one layover.

If the airports were randomly connected, the closeness factor would be closer to four. Without high-degree airports,  like Fiumicino in Rome, you’d typically need two or three layovers to travel elsewhere within Italy! Speaking of layovers, we also need to take a quick break for this ad.

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Even balancing connectedness and closeness, we’re still missing some logistics. Like the duration of each flight,  and the demand for a given route. So to make our graph even more  sophisticated, we can attach numbers to each edge in our  network, creating a weighted graph.

Weighted graphs help us determine which airports should be the high degree ones. Taking the weights into account,  you end up with what’s called a hub-and-spoke network that  pretty successfully balances closeness and connectedness with consumer demand. The airports with the highest overall demand are high-degree hubs that connect  to lower-degree airports called spokes, which have lower demand.

When you draw the network,  it sort of resembles a wheel, hence the name. In 2013, Italian and Spanish  researchers found that airlines actually select their network shape  based on their business model. For example, the European airline  Lufthansa is arranged in the hub-and-spoke model, like other major airlines, to balance both closeness and connectedness.

Connecting one additional spoke  to the hub instantly increases the connectedness in the network  without adding more layovers on average, because most routes will  take you through the hub anyway. This lets airlines serve lots of customers coming from a lot of different places. If you’re going to Singapore,  it doesn’t matter much whether you’re coming from Lisbon or  Warsaw, because the airline can bundle you all into the same  flight using a layover at Frankfurt.

Aggregating routes in this way  creates more consistent schedules for anyone at the spokes to  get to their final destination. Meanwhile, the budget airline  Ryanair had a much more random-looking network, driven  by the individual demand between any two airports, rather than  a high degree of connectedness. These flights tend to be direct,  but focus on smaller airports and smaller fleets, which keeps the costs low.

You can’t fly to as many  places on Ryanair’s network, but budget airlines like this  help fill in network gaps where the demand is usually low, meaning  they can offer lower prices. So even though we’ve been talking  about the overall global airport network, it’s actually divided into  subnetworks of different airlines, each optimizing their own networks  for their specific business goals. And for airlines that care  a lot about reliability, they also need to account for robustness.

If you’re a business flyer who  has a very important meeting or video shoot to get to,  you’ll probably book an airline with a lot of flight options  that’s known for being reliable. But if you’re trying to  vacation on a shoestring budget, you might take a risk with a  budget airline that has fewer rebooking options on the  chance you miss your flight. Robustness measures how much  the connectedness of the network is affected when any given node  gets taken out of the system.

For example, the route from  Indianapolis to Missoula is robust because if a nearby wildfire interrupts  your layover at Salt Lake City, you can still make it to your SciShow shoot by taking a connection through Denver. Hub-and-spoke models tend to do  really well in situations like these because most of the  airports tend to be low-degree, rather than high-degree. If a  completely random airport is hit by something like a storm or a power outage, the odds of it being one of  the hub airports is fairly low.

That means fewer people overall  end up with disrupted flights, because most of the  connectedness is still preserved. A study from 2014 in the  journal Transportation Research modelled a global network of 3,700 airports. They found that even removing 13  percent of airports at random only reduced the connectedness of  the whole network by 22 percent.

There would still be over 3,000  locations you could travel between globally if more than one in ten  airports were randomly shut down. But that same study also discussed a huge problem with hub-and-spoke networks: targeted attacks. Precisely because they make the  network both close and connected, hubs are a huge risk if taken  out intentionally by something like a cyberattack or someone  flying drones near the runway.

If just two and a half percent  of the most frequent layover airports were taken out of  action, then the connectivity of the global air network  system would drop by over half. And if you knock out 13 percent  of airports, like before, but this time targeting the most  connected hubs, then it would pretty much bring the whole  air travel system to a halt. The connectivity of the network would drop from 3,700 airports to just 40.

In other words, the same property  that makes global air travel resilient to random, low-level problems  makes it brittle in extreme events. For instance, when this infamously  difficult to pronounce Icelandic volcano– believe me, I tried– erupted in 2010, the volcanic  ash cloud shut down several major hub airports in Europe,  leaving 8 million passengers stranded and creating nearly  3 billion Euros in losses. But thankfully, network  theory tools can help airlines proactively minimize the  effects of events like these!

A 2021 simulation by a European  Air Safety organization found that since the original eruption, airlines  have improved their coordination. If an event like that icelandic  volcano happened today, its negative impacts would be cut in half  compared to the disruption in 2010. Networks pop up a lot of places  in nature and engineering, from cybersecurity to materials  science to power grid design.

And the math of network theory is  helping scientists plan against disasters or design new materials,  all by “connecting the dots”. [♪OUTRO]