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Go to Brilliant.org/SciShow to learn more. [♪ INTRO]. Thanks to the internet, it's possible to instantly connect with people all over the globe.

And thanks to social media, you can curate a list of friends that can be hundreds, or even thousands, of names long. All these connections make our world seem pretty small. And the math has shown that it kind of is.

You may have heard that any two people on the planet are connected through only a small handful of intermediates, usually six, hence the name for the phenomenon, the Six Degrees of Separation. And it turns out that's probably true. Sociologists and mathematicians have worked together over the past few decades to determine exactly how connected the world is, and they keep arriving at numbers around 6 or less.

But that's not even the coolest part. The mathematical models that explain how people are connected, appropriately called Small World Networks, don't just apply to social interactions. They have uses and implications for everything from the spread of epidemics, to the structure of the internet, and even the neural networks that make up your brain.

This idea of Six Degrees seems to first appear in short story from 1929 where a character is challenged to find anyone on Earth through a chain of at most five people. By the 1960s, scientists had started to get in on the idea. The most famous experiment was conducted by sociologist Stanley Milgram, yes, that Stanley Milgram.

In 1963, the same year he tested people's obedience by seeing whether they'd shock others on command, he sent the first chain letters. He picked people from several US cities and asked them to forward the letters on with the intention of getting them to reach a specific person. The experiment wasn't hugely successful.

Turns out people in the 60s didn't like responding to spam, either, but of the letters that did make it, the average number of jumps needed was about 6. And that ‘six' number has become totemic, mostly thanks to a 1990 play called. Six Degrees of Separation which was loosely inspired by this idea.

Of course, the play says everyone has at most six connections in between them, and that might be a bit of a stretch. For example, in 2016, Facebook did a study of the users on their network and found that the average number of friends separating two active users was 3.5. But even though it's the largest social network,.

Facebook doesn't come close to including all of humanity, and 3.5 is an average. There could very well be people who you'd need a dozen or more jumps to get to. The study didn't try to pinpoint a maximum number of degrees of separation.

Still, even if six isn't the upper limit, the number is probably pretty small. We can also apply the Six Degrees concept to collaboration networks, where the ties are stronger, people who have actually worked with one another. The most famous example of this is the Six Degrees of Kevin Bacon game: for any given actor, you name an actor they've been in a film with, who's been in a film with another actor, who's been in a film with another actor, until you get to Kevin Bacon.

The number of steps needed is the person's Bacon number. For instance, Kevin Bacon and Viola Davis were in the film Beyond All Boundaries, so she has a Bacon Number of 1. But Viola Davis was in Ender's Game with Stevie Ray Dallimore, who was in Paper Towns with John Green, who faced off against Hank in a SciShow Quiz Show that I hosted, giving me a Bacon number of 4!

Bacon once famously claimed to have worked with everyone in Hollywood, or someone who's worked with them, hence his central role in all this. And it turns out he wasn't far off. Research has found he's one of the most connected actors in Hollywood.

The vast majority of actors have a Bacon number of less than 6, with an average of about 3. Scientists have a similar concept called the Erdős number, where you work out how many scientific paper collaborations you are from prolific Hungarian mathematician Paul Erdős. Oh and if you're the kind of overachiever who likes to act and do science, then your Erdős-Bacon number is the sum of the two numbers:.

Natalie Portman is 7, and Carl Sagan's is 6. All of these fun examples illustrate how well-connected our world is, but what they don't tell you is what's going on under the hood, how such inter-connectedness happens. Even in this internet age, you tend to be friends with people near you.

And if that were always the case, it would take thousands of friendship steps to reach around the world. But of course it's not true. People do sometimes meet and form social bonds with people from distant places.

You may not be best friends with that one French exchange student you met in school, but you definitely know them. And that one loose connection ties you to a whole other country of people. In a 1973 paper, a sociologist called special attention to these quote “weak ties”.

He pointed out that the weakest connections can sometimes be the most important. Like, in your friendship network, they might be the ones that open you up to new ideas by giving you the perspective of a different culture, or by introducing you to new people. But it would take another 25 years or so for the real importance of these so-called weak ties to be revealed.

You see, it wasn't until the late 90s that a pair of mathematicians created a model for this kind of network, which they called a small world network. The researchers, Steven Strogatz and his PhD student Duncan Watts, were inspired by the six degrees phenomenon, and wanted to understand where it comes from mathematically. The tools they used are from the field of math called graph theory, which looks at the different ways points can be connected to each other in something called a network.

For instance, let's say you wanted to draw a network representing the structure of a crystal, with points representing the atoms or molecules, and the lines between them representing chemical bonds. The network would have lots of points that are only connected to other nearby points. In other words, it would look very ordered.

And on average, it would take a lot of jumps to get from one side of the network to the other. By contrast, let's say you wanted to draw the network for something like Facebook, with points representing users, and lines representing friendships. It would look more chaotic, random, and messy, with lines going all over the place.

But there would still be some structure to it, for a couple reasons. Firstly, your friends are very likely to know each other, because they have something in common: you. And secondly, people tend to be friends with people who live near them, so you'd see a lot of locational clustering.

So really, it would look a bit like the crystal lattice, but with a few random jumps across the network to faraway points. This small world model was designed to recreate the structure and the messiness of a social network, all before the rise of online social networks. And the technique they used to make the model was simple: they started with an ordered, crystal lattice-like network, and then gradually re-wired it.

Specifically, their recipe was to take one link on the ordered network, and randomly change one end of it to simulate a person having a faraway connection, then repeat this multiple times. They didn't care how strong the connection was between the two points, just that there was one: your best friend and that foreign exchange student count the same. And they found that it only took a small number of re-wirings, just a handful of ‘weak ties', to make the average path length between two points drop drastically.

Specifically, they proved that the average number of jumps scaled logarithmically with the number of points in the network. ‘Logarithmically' is the mathematical opposite of ‘exponentially'. If something grows logarithmically that means it grows really slowly. So even a network of, say, seven billion humans will have a tiny average path length, maybe under 10, thanks to a surprisingly small number of weak ties.

And recent studies have suggested that, with modern technology, these so-called ‘weak' ties aren't even all that weak. Maybe you keep in touch with that exchange student over WhatsApp, so you know them better than you know your neighbor! In addition to giving us a fun Hollywood trivia game, the small world phenomenon has found uses in a number of scientific fields.

For instance, Watts and Strogatz showed how diseases can spread more easily in a small world network because it's easier for the infection to reach faraway places quickly. It's important for scientists to know this sort of thing so they can model diseases more accurately, and that helps them to work out where to deploy resources, and how to stop a pathogen from spreading. And now that scientists are looking for them, it seems like these small world networks are cropping up all over the place.

In fact, Watts and Strogatz found that as long as it's possible to make any long-range connections in a network, the resulting network was almost guaranteed to have small world properties. And that means understanding how small world networks work is essential to understanding the behavior of these networks and how signals travel through them. Computer scientists have found that the hyperlinks between web pages look like this too, for example: almost any web page is only a handful of clicks away from any other one.

So understanding small world networks can help scientists understand how information moves on the internet. And understanding these network properties can help scientists understand biological systems, too. Take brains, for instance.

Some researchers have claimed that the network of neurons in vertebrate brainstems resembles a small world network, with lots of clustering, and the odd link to faraway places. And that affects how signals travel through the brain. One study showed that the small world networks of neurons are better at rapidly synchronising activity across the whole tissue because those ‘weak ties' between distant regions help signals spread far and fast.

And it's also been shown that neural activity lasts longer in these networks, because it's easier for a neuron that's been activated by a signal to be re-activated. In the end, models of the human mind may look a lot like Kevin Bacon's colleague network. As for the Six Degrees idea, well, it may take a few more connections than that to reach some people on the planet, but you're probably about 6 degrees away from most people.

The small world phenomenon reminds us that everyone around us is closer to us than we think. Each person you walk past is a potential new friend, and a potentially meaningful connection, helping to tie over seven billion humans into one messy little network. We might not be able to draw out the whole network for humans on this planet just yet, but computer scientists can do a lot of cool things with neural networks.

And if you want to understand how and why, you might want to try out Brilliant.org. You see, Brilliant offers interactive courses in math, science, engineering, and computer science. So whether you're looking to brush up on subjects you took years ago or learn something new, they've got you covered.

For example, their Artificial Neural Networks course has everything you need to know about how these computer-generated networks work. You can brush up on perceptrons or dig deep into advanced network architectures, even when you're on the go, because Brilliant's courses can now be taken offline on their new iOS app. And to sweeten the deal, the first 200 people to sign up at Brilliant.org/SciShow will get 20% off an annual Premium subscription.

So you can save money, sharpen your math and science skills, and support SciShow all in one fell swoop. [♪ OUTRO].