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  • Writer's pictureAsaf Shapira

Episode 9: Six Degrees of Intuition

Updated: Dec 10, 2021

What's the rumpus 😊

I'm Asaf Shapira and this is NETfrix – The Network Science Podcast.

When we are first presented with the concept that everybody in the world is just 6 steps of acquaintance from everybody else, we usually go through the same 7 stages:

  • Initially you are skeptical. It sounds weird.

  • Then, you try it on yourself, usually calculating in your head how close you are to the president of the United States.

  • Then comes amazement – Wow, I'm just 3 to 5 steps from the president! The human network is amazing! We are all part of the same fabric. Maybe there's some truth in Buddhism that we are all one. Incense sticks anyone?

  • Then comes disillusionment: OK, so I'm 3 to 5 steps away from the president. It doesn’t mean I'm going to pay less tax or be appointed ambassador.

  • Then comes realization – "6 degrees" is a nice concept, but it doesn’t help you in any way..

  • And lastly - the final stage –you run to impress your friends with the idea of the 6 degrees of separation to show them you're not the shallow person that they thought you were.

So, is that it? Is the concept of 6-degrees-of-separation just an elaborate party trick?

We'll engage this question in the episode but that's not what this episode is about. Networks are a simplified model of reality. Because most of us believe we understand reality, so - by definition – we believe that we understand networks as well. But networks are not intuitive, and you can project it on our understanding of reality. And that's why this episode will be dedicated to improving our intuitions about networks. We'll test our ability to mentally visualize networks by using the 6-degrees concept and we'll learn what to look for in a network. On the way, we'll address some of the insights from previous episodes but also add some new ones, including networks' laws we haven’t touched yet.

In episode 2 about the "Small World", we covered the idea of 6-degrees in networks and tried to see how much of it is real and what's a myth and what can we learn from it about life in general and networks in particular. To avoid spoilers, I'll just address it in short by presenting the concept in a popular and somewhat naïve way : The concept of 6-degrees became popular through the famous study of Milgram, the renowned researcher from the field of social psychology. The study showed that, on average, it took about 6 steps for a letter to reach its destination from a random source to a random destination. Milgram conducted his study in the 1960's. Although there was no internet back then, still, it was modern times and the world enjoyed a surge of technological developments. So maybe we should ask if he could have gotten the same result if he would have conducted his research in a different era?

Once upon a time, people lived in small villages, located in valleys, where the land was fertile. Everybody in the village knew everybody, but the outside world was pretty much outer space.

Because of the distance and poor transportation, people usually didn’t travel much, and so spent their lives at the same place, from birth to death.

They knew there was a king with a palace on some remote mountain, because they met his tax collectors, but to reach the palace, not to mention the king, was deemed as possible as a trip to the giant cheese ball in the night's sky.

With today's modern transportation we can go anywhere, hell, with online communication we can reach everyone with a tap on our keyboard, making the world a small village filled with happy go-lightly villagers.

So, what's with the "Once upon a time" intro? That's a homage to fairy tales, as this tale was.

First, even in the distant past, people were more interconnected than we usually give them credit for. How can we know this? Unfortunately, we have the data:

The Bubonic plague that hit Europe in the 14th century and erased a third of its inhabitants, couldn’t' have gone far in such a loosely connected network of ties.

Second, have online communications really revolutionized our network's topology?

It would be a gross understatement to say that the online network has increased the volume of communication. And in the physical sense, we can now send our selfies to the president of the US just as easily as to our closest friends. Easily, but not readily. Why? Haven't we established that the president's just a few steps away from us on the human chain?

When trying to explain this paradox, Barabashi, the famous network scientist, said that even though a network path can be found between two people, the edges or links are too weak to utilize them for meaningful communication. I can relate to this explanation and I used it in a previous episode, but it sounds like a cop-out. What does it mean "weak links"? How can we measure it? And what about the famous paper by Granovetter "The strength of weak ties" which we mentioned in a previous episode? Doesn’t it state that weak ties are actually an important tool to utilize when looking for a new job? So, we have two main issues to resolve here.

The first is to re-address Granovetter's paper but this time, though we'll start the journey on the familiar and popular path, we'll soon discover that the road has some surprising twists and turns.

The second issue we'll for now be our contingency plan in case the theory about weak ties won't suffice us as an explanation to the paradox we are faced with. So, let us first surrender ourselves to the wisdom of Granovetter:

Granovetter's iconic paper "The strength of weak ties", was published in 1973, and and has about 60,000 citations. That's a lot.

The paper is known in popular culture for the research presented there by Granovetter, in which he inquired as to how people find out about vacant jobs. His famous finding was that the weak social ties of a person, like acquaintances from an old job or from school, are the ones that are the most important in the job-searching process.

This finding was groundbreaking because it was counterintuitive. Back then, studies were more focused on the individual's strong ties, like family and close friends. It stands to reason to assume that the people surrounding us influence us more than distant acquaintances. The reason Granovetter gave to this alleged paradox is the circulation of information:

In our tight-knit small network, made up of strong ties, there's a very small chance that we'll get news we don’t already possess. For example, when I hear a strange noise from the kitchen, I already know that's that my child experimenting her invention of carrot soup and a cleanup is due. No news there. But to get new information, we need our weak ties. Weak ties propagate the information between the cliques or communities in the network.

A great example for this is the spread of fake news as seen on Seinfeld: The friends on Seinfeld have kind of the same set of values, ideas and knowledge. It's only through Kramer's friend, Bob Sacamano, that dubious medical information is propagated in their network.

All this is nice and dandy as we say in the fifties, but there are a few problems we face here when discussing Granovetter's research about job finding:

The first is that it's not the subject of the paper.

Say what?! Yaap. It's not. Actually, his job-search research is covered only by about 10% of the paper.

So how come this is the most famous part of the paper? The reason probably lies in another interesting fact: The paper was first handed-in in 1969 - and got rejected. The most cited paper in Social Science got rejected. And by the way, it's not the only one.

The famous "Small World" paper by Strogatz & Watts which earned about 45,000 citations also almost didn’t make it. The absurd reason for it was that the paper was relevant to too many fields. Because each field has its own publishing channel, it was hard to find a relevant academic publisher that saw it as their duty to publish it.

And that's the network science curse that people don’t talk about. Too much relevance. It's like being the prettiest person in the room and so everyone's too shy to approach you. Or the other way around. You decide.

Ok, so back to Granovetter's rejection. I don’t know why it was rejected in 1969 and accepted in 1973, but I believe there's a loose correlation we can use and that's the unemployment rate in the States. In 1969, when the paper was rejected, the unemployment was about 3%. In 1973, when it was finally published, unemployment more than doubled, making a paper about how to get a job more relevant and a more popular piece of information.

Maybe that's the reason that to this day, that's the more discussed part of Granovetter's paper.

But before we cover the other relevant parts of the paper, we need to address the second problem we have with the paper. And the thing is that Granovetter's conclusion is inaccurate. And don’t get me wrong. I love Granovetter. In fact, I owe him this podcast.

When I first planned on publishing Network Science stuff on social media, I decided I'll write a blog and start with Network Science Classics. And what's more classic than Granovetter's paper? A few twists & turns later it occurred to me that this is podcast material and I never looked back. But for those with a lazy ear, I do keep transcripts available, so feel free to use them.

Anyhow, I say that Granovetter was inaccurate because I rely on Granovetter's own words. He unveiled his own mistakes in a paper he published in 1983, 10 years after his original paper, and you got to admire him for it. The paper was titled: "The Strength of Weak Ties: A Network Theory Revisited" This time, he spent a third of the paper to the study of job search and at first, he found confirmation for his early study. For example, He cited a study about a Canadian company, that out of over 2000 employees, 40% of them got the job through relationships, though it was against company's policy.

But when he delved through more recent studies, he found that the idea behind "weak ties" as a steppingstone for a new job was a bit misleading. He discovered that blue-collar employees found most of their jobs via strong ties, meaning family and good friends. In the few cases where weak ties were utilized to the job search, it landed worse jobs meaning a smaller paycheck. The reason he gave for it was that weakened individuals have a hard time to establish weak ties. Most of their time is devoted to their family and friends for plain survival, and they haven’t got old college friends to rely on. The few weak ties they do have are to other weakened communities, which probably favor their own before others for pretty much the same reasons. So it goes that the strong rely on their weak ties and the weak on their strong ties. It's like as every newbie startup knows, the first people you turn to for funding are not the weak ties you made through LinkedIn but rather the 3 strong F's: Friends, Family and Fools.

The part of the paper that is most relevant to our episode is Granovetter's discussion about the circulation of information in the network. Granovetter demonstrates that information about new jobs hardly propagates beyond the first or second circle of ties in an Ego Network. That means that one can find a job through a friend, or a friend of a friend, but most of the information from further parts of the network don't make it to our ego network. Why is that?

For this, we'll turn to Robin Dunbar, who studied primates' in the wild. Dunbar noticed an interesting phenomenon: He found that there is a limit to the number of primates that can act together as a group. After a certain number, the group splits to more tiny groups. When he expanded his research on to humans, he found that the number remained pretty much the same. The number was 150, which is now known as "The Dunbar Number", but most of the times it was even lower.

His contention was that our cognitive ability does not allow for the preservation of a significant amount of connections greater than this threshold. That means that the glass ceiling for a group where everyone knows everyone is about 150. Of course we can know more than 150 people but those that have crossed our Dunbar Number's limit we will know in a very superficial way. For example, we will know them as "the seller in the supermarket" or "the nice neighbor from apartment 12B". Dunbar continued his research well into the age of online social networks, showing that although we can have thousands of online friends, in practice, our strong ties will not cross the Dunbar Number and the online friends above this threshold would be considered friends only in the technical definition of the word but not in the intimate sense. Dunbar suggested that's that why a military company consists of about 150 soldiers. The intimacy provided by the Dunbar Number helps to build strong ties between the company's soldiers. That's also why civilian companies that grow beyond their Dunabr Number need to change their manners, for example, to hire HR personnel and split to divisions, because intimacy is lost and the span of management doesn’t cover all the employees anymore. Dunbar concludes that people will know their first circle of friends very well but will only know about 5% of their friends-of-friends, meaning their 2nd circle and will know no one beyond that.

Though it sounds unintuitive, we can check it ourselves on our LinkedIn profile. LinkedIn, a social network that focuses on networking and jobs, has a feature that lets you explore people from the different circles. A user is supposed to be familiar with almost all of the 1st circle of friends but will not be familiar with about 90% of the 2nd circle and probably zero with the third circle and beyond.

In regard to Granovetter's paper and the circulation of information about jobs, Dunbar suggests an explanation for the difficulty weakened classes face in maintaining weak ties. Large families that live together and the strong ties associated with survival, take up most of the Dunbar Number limit, leaving little room and energy for maintaining or even gaining new ties.

To sum all this up, we'll turn to an unusual use case for the importance of network's intuition regarding the propagation of information in a network. The use case is the NSA's hearing in congress in 2013. The NSA (The National Security Agency in the States) denied conducting surveillance of American citizens, since it focused its efforts on terrorist's networks. But during the hearing they noted that the "agency can perform 'three-hop queries' through Americans' data and records". Meaning, they can track an ego network 3 steps away.


I believe now it's time to turn to our contingency we mentioned early in this episode because I'm still not convinced. Granovetter and Dunbar give us great empirical findings about the network, but I feel something is missing. They do show that traveling through our network is not an easy thing to do, but they don't say why. We know it's something that has to do with energy but what does it mean?

So, we did our warmups regarding our intuition about networks. Now it's time to try and illustrate it using our network's vocabulary and some terrain metaphors. When we initially try to think about how we are tied to the president of the US, we look for people in our immediate circle that know powerful people. We imagine those powerful people must have met someone from the White House and from there on it's easy street to POTUS. When we backtrack our steps, it seems so simple. We can see the target on our network's horizon, but we seem to encounter invisible barriers on the way. Of course we know that POTUS cannot be reached so easily, so why is the network playing tricks on us?

The problem with our intuition about networks is that we visualize networks as we draw them, meaning in 2 dimensions, when they are actually in 3D: The network is made of valleys and mountain tops that we seamlessly flatten when we play the 6-Degrees Game. And the height of the mountains can be measured by their Degree Centrality. Degree Centrality is the basic centrality measure in Network Science and it's simply the count of the edges or links connected to the node. The more edges the node have, the higher it is in our 3D network topology.

2D representation of a node with a Degree of 5. The same network in 3D

A high ranking node can be considered "a mountain" that blocks our view beyond it, meaning the links it possesses are hidden from those who stand at the foot of it.

And if this wasn’t enough, then here is some more bad news: The mountains are Power Law distributed. We met the Power Law distribution, or more accurately, the Long-Tail distribution, on a previous episode. The Power Law states that we'll have a few high degree nodes in the network and many low degree nodes. In a ratio of about 1 to 99. This might seem great at first: Our network is 99% plateau and so should be very visible. But the fact is that the high mountains are really high. Really, really high. And because they own most of the links in the network, we will encounter them soon on our 6-Degree's travel. And here is the invisible barrier explained:

The travel is a non-linear travel. The first few steps are trivial because most of us are part of the plateau or modest hills at best. But when we soon encounter the mountain, it's a nonlinear climb.

For example, though I can easily connect to Kim Kardashian on Twitter by becoming a follower, it doesn’t mean I can see or "access" her links on the network. It's not so hard to get to the Himalayas. Climbing them is another thing altogether.

But why? Can't Kim spare me some slack? I drove all the way over! The reason is "The Dunbar Number" we mentioned earlier. A person can keep up to 150~ connections and a high degree node quickly exhausts all its energy.

So, tapping to such a node isn’t an easy task. But it is possible. How? By investing energy of our own.

For example, if I want to access Kim Kardashian's network, let's say I think her audience is perfect to promote my Network Science podcast, I can pay her a few millions to tap to her audience, say she'll mention the podcast in one of her posts. She would probably do that. But I'll need to waste quite a lot of energy to get those millions. But the good news is that once you are high up at the mountain top you could probably see the rest of the way you need to go and the getting down part is also non-linear, meaning, it's easy to access the nodes at the foot of your mountain when you come down.

I believe Moses at Mount Sinai was the only one that screwed it up. So isn’t there an easier way? Theoretically there is. Mountain tops have shortcuts and they can be found by using Eigenvector Centrality. We mentioned Eigenvector Centrality on episode 4 about Centrality Measures. In short, this centrality measures not the node's own degree but that of its neighbors. The cool idea behind this algorithm is that it states that if I'm connected to central nodes, I'm also central. That's why Eigenvector Centrality is a theoretical shortcut. Going through a node with high Eigenvector Centrality can ease our way to the high Degree nodes.

But why "theoretically"? Because, usually there's a correlation between centralities. In other words, you know what is closest to a mountain's peak in the Himalayas? The peak just besides it.

The harsh truth about 6-Degrees of separation

But this is not a law of nature. There can be some anomalies and that's why anomalies in networks are interesting. The nodes that score high on one Centrality Measure but low on the other have a very interesting story to tell. And what are analysts if not the storytellers of the 21st century?

Did you enjoy, and want to share? Have you suffered and you don't want to suffer alone?

Tell your friends or rate us on apple podcasts, follow us on Twitter or like our Facebook page. If you're from Israel, rate us on Podcasts of Israel. See you in the next episode of NETfrix (:

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