The theory most commonly called "Six Degrees of Separation" or "Six Degrees of Separation" claims that every two people on Earth are connected by a chain. social a connection whose length is typically no more than 6. One would say that the theory is of recent date and was only formulated when the huge social networks provided a large enough sample to practically test this assumption (billions of users).
And yet, it's not like that - social connection and the intertwining of social ties have been thought about for a hundred years. Back in 1929, the Hungarian writer Fridjes Karinty wrote the short story "Chains", in which the main character asks others to find him a person on the planet that he will not be able to reach using a series of up to six acquaintances ("My friend has a friend whose friend knows that person..."). How Karinti arrived at the number six is unknown, but it turns out that his layman's judgment was more than solid.
It would be almost four decades before someone decided to test this hypothesis. In the first issue of the magazine "Psychology Today", Steve Milgram published the paper "The Problem of a Small World", in which he presented some common sense arguments in support of Carinthia's claim.
In 1967, Milgram conducted an interesting social experiment to confirm that the world is much smaller than we think: he selected 300 volunteers in Kansas and Nebraska and tasked them with delivering a package by mail to a stockbroker in Boston. The participants knew the man's name and what he did, but they did not know his address. Considering that direct sending was impossible, the participants of the experiment could only send the package to a relative or friend who, perhaps, had "better connections" and a higher chance of delivering the package to the final destination. The next person in the chain could do the same, until the process reached the individual who was fortunate enough to know the Boston broker personally.
The largest number of packages were never delivered because someone in the transmission chain stopped participating in the experiment, but a dozen packages still reached the right hands. Milgram found that each successfully delivered package typically passed through 5–7 hands, while the average value was about 6. Although all participants were American, Milgram was convinced that an identical experiment would produce similar results anywhere in the world.
A MAN WHO KNOWS MAN
One experiment was not enough to prove or disprove the assumption of six degrees of separation. Is it possible that in a world of almost eight billion people you can reach any person - the Dalai Lama, Shakira, Nikola Jokic, the Mongolian equestrian champion or Miss Cambodia - in just six jumps? And what impact does it have on our lives, in the way (mis)information, epidemics, world crises spread? New research has shown that our world is not only small, but that the other possibility does not actually exist.
Start with a simple thought experiment: suppose you have a hundred friends. Let each of your hundred friends have another hundred friends that you don't know about. That's already 10.000 people you can reach in just two steps. Let each of those 10.000 people know a hundred new ones... Keep multiplying by 100 and in just five steps you will reach every person on the planet. But this way of thinking is deeply flawed - it assumes that your hundred friends are randomly scattered across the planet, so to speak, and that this assumption applies to all other people as well.
If you represent each person with a circle, and the acquaintance between them with a line connecting them, the whole of humanity can be represented by a so-called graph - a set of points connected by lines. If your 100 friends are randomly distributed on the planet, you get a "random graph", a completely chaotic network in which the lines run diagonally, end to end.
Paul Erdos, the famous Hungarian mathematician (more about him later), was the first to study such networks and concluded that, as soon as the average number of connections (acquaintances) per node (person) exceeds unity, the distance between two arbitrary nodes (the number of acquaintances you have to use to reach any person on the planet) becomes drastically smaller - from several hundred that number drops to only five or six. Moreover, that number changes slightly even if the size of the graph (social network) increases many times over: a ten times larger network might raise the average distance from six to seven, but not more than that.
However, this model is not correct, today's social networks do not arise in that way because friendships and acquaintances are not completely random: people are, first of all, grouped geographically. Most of our friends are in our immediate surroundings, with fairly strong barriers at the borders of cities, provinces, countries or continents. In addition, there is a great chance that among our friends there are people who also know each other. But even this picture is too one-dimensional!
Imagine a completely regular world where all your friends are in the circle of people who are geographically closest to you, about fifty people to your left and about fifty people to your right. A graph showing such a population has a much more regular structure. If you wanted to contact a person from the other side of the planet using a line of acquaintances in such a society, a society in which each individual is limited to his immediate environment, you would need about 80 million intermediaries. A drastic difference compared to the mentioned six.
What is it then? catch, not to say "stunt"? It has probably happened to you at least once that you bump into a complete stranger in a restaurant, on the beach, on a train or plane, start a banal conversation and then, five minutes later, discover that this person knows your best school friend... It's a very small world, isn't it? The point is that real social networks are somewhere in the transition between complete order (where everyone knows only family and the closest to them) and complete disorder (where everyone has friends all over the world). What makes it possible to bridge huge distances through friendships and acquaintances in social networks are the so-called "weak ties" - that school friend of yours who connected you with a stranger on the beach whom you didn't even know existed until yesterday. This is your godfather who went across the pond a quarter of a century ago, built a new life there and created a new social circle around him. Thanks to him, you can say that you know a man who knows a man who knows the CEO of Boeing or Google. A weak link is also that acquaintance, actually a friend of your husband, who helped you get a job in a new company. Weak ties are actually much stronger and more important than it seems at first glance.
SCIENTIFIC "EARTHQUAKE"
During the nineties, two American mathematicians, Duncan Watts and Steven Strogac, investigated in detail the phenomenon of the emergence of a "small world". They started from the simulation of extremely organized social networks, in which relations between people are formed only at small geographical distances. And then they began to add coincidences to such networks, the aforementioned "weak ties" that represent unexpected acquaintances you made by exchanging cooking recipes with someone, playing chess, cards or dominoes, or drunkenly fraternizing after a bottle of "metaxa" in a Greek tavern. A key conclusion of Watts and Strogac was that the average number of degrees of separation starts to decrease drastically even when the number of weak ties is relatively small. Moreover, they showed in their computer simulations that for every 10.000 social connections it is enough to introduce three random connections between people so that the number of degrees of separation drops from several million to less than 10, as if it were a completely random graph, one of those studied by Paul Erdos. And that's what makes the world small.
Watts and Strogac were the first to formulate a realistic model of human interactions and published all this in a paper of only three pages, which appeared in the famous magazine "Nature". Over time, it became one of the most cited scientific papers of all time (58.000 citations). Compare that with the work of Watson and Crick on the topic of the structure of DNA (20.000 citations) or the prediction of the existence of the Higgs boson (13.000 citations) and you will understand what a scientific "earthquake" this work caused. It turns out that the model defined by Watts and Strogac (a seemingly regular network with a few "shortcuts" here and there) is ubiquitous: it has been observed in the nervous system of almost all living beings, according to this model the energy networks of entire countries, roads... Social shortcuts allow news to spread, but they also have a negative side: thanks to them, disinformation travels as quickly as epidemics.
However, the Watts and Strogac model is not the only one - Albert-László Barbáši showed as early as 1998 that just by using links on Internet pages it is possible to move from one to any other using a maximum of 19 clicks (at that time there were about 800 million public web pages). It could be said that the phenomenon of "small world" has been valid for decades. However, the structure of Internet presentations is completely different: a huge number of pages have very few links, but that's why there are few pages that contain thousands of them ("Yahoo", "Google" and others). These pages represent "hubs", focal points, places that play a key role in connecting the far corners of the cyber world.
Barbashi, in addition, showed that in all large systems, in all large networks, hubs arise naturally. No large system was created all at once, in the Big Bang. On the contrary, every network, be it small or large, grew node by node. How does that new node connect to the already formed part? Take a new user of "Facebook" or any social network. There is a much greater chance that he will establish contact with a person who already has a large number of relationships than with a person who is completely lonely. The world's airports work on the same principle: those with a large number of existing connections (Frankfurt, London, Chicago) tend to become even larger because they already allow to reach every point on the globe with the fewest transfers. And with each new line, those airports become even more attractive. But each hub also represents the Achilles heel of a large network. We have already seen it several times: when several of the world's largest airports are shut down due to strikes, wars or weather, the consequences are felt in every corner of the world.
The explosion of the Internet and social networks has finally made it possible to accurately test the small-world hypothesis on a population sample whose size is measured in the hundreds of millions. An early study by Microsoft focused on about 30 billion messages exchanged by users of its chat platform, 240 million of them. The calculations showed that the average social distance, i.e. the number of degrees of distance is around 6,6. When social media exploded, that distance began to shrink as well. "Facebook", for example, disclosed that, based on a sample of 720 million users, the social distance decreased to only 4,7, while in the case of "Twitter" this value fell to only 3,5.
"THE SIX DEGREES OF KEVIN BACON"
What happens when you focus on only one group of people, on one profession, from the level of the planet and its eight billion inhabitants? The small world becomes even smaller.
In a 1994 interview, actor Kevin Bacon stated that he had worked with "everyone in Hollywood or someone who worked with them." On the electronic forums of the time, a somewhat mocking discussion was started under the title "Kevin Bacon is the center of the Universe", and a little later three students from Albright College proposed a game under the title "Six Degrees of Kevin Bacon". The object of the game was to match the chosen actor with Kevin Bacon in a maximum of six steps, using pairs of actors who starred in the same film. Let's say, start with Marilyn Monroe. She acted in the movie Some Like It Hot from 1959 with Jack Lemmon. Kevin Bacon and Jack Lemmon played in the company JFK in 1991. Marilyn Monroe's degree of separation from Kevin Bacon, her "Bacon number", is therefore 2.
When the game was presented to the public, Bacon was not enthusiastic, it seemed to him as if he had become a target of ridicule, but over time he came to love the game and gentlemanly accepted the fact that he is the second "strongest" actor in the cosmos (after Chuck Norris). He used the popularity of the game in the right way: he started a very successful humanitarian organization called "Six Degrees" to "help anonymous people who want to become famous" by donating money to noble causes. Additionally, the Bacon number concept has remained popular to this day, to the extent that it has been integrated into Google search and countless sites where you can see your favorite actor's Bacon number.
30.000 FILM TITLES TESTED
By chance, I spent a good part of my free time collecting books and movies, read and watched a lot, and memorized a few. For my film collection, I created a software in which I record, quite detailed and richly, all the people who participated in the creation of a film or series. OK, maybe I didn't record cameramen, make-up artists, stuntmen, chefs or sound engineers, but that's why all directors, actors, producers, screenwriters and composers are in my database. With around 30.000 selected movie titles, I still can't compare to our "Kinetheque" and its 100.000 movie units, but I'm "there somewhere": I still have a representative sample from which I can extract various interesting statistics, perhaps because I know a little bit about programming. There are photos and biographies for about 640.000 film actors in my database, enough for meaningful analysis.
Let's start with the definition first: if two people appear in the same movie (one can be an actor, the other a producer), their degree of separation is 1. In the mentioned graph, those two people are nodes (circles) while the movie that connects them is a dash. Each subsequent dash (film) increases the degree of separation by 1. What can be seen from my large sample? That people from the film world are much more connected than it seems at first glance. Take John Wayne and Greta Garbo: neither of them ever made a movie together, nor did they have similar acting affinities. Even so, their degree of separation is only 2: Greta Garbo starred in a movie with a certain Robert Mackenzie The Painted Veil, while the same Mackenzie collaborated with John Wayne in the film Tall in the Saddle.
PHILIP SCHWARMM AND CHARLES BRONSON
OK, this is not unexpected - big stars are, in graph theory, actually "hubs", nodes that gather a huge number of people around them, which helps to bridge large distances. However, what happens if on one side of the chain you have a person who is not so well-known in the film world, for example the editor-in-chief of "Vremen" Filip Švarm, who occasionally ventures into film and documentary waters? In my collection, Philip is represented by only two titles (while in reality there are more of them), so I reckoned that any attempt to connect him with some "celebrity" would be hard work. Using my programming "work" I tried to find the "secret connection" between Philip and, um... Charles Bronson. Why Charles Bronson? Next question…
It turns out that the degree of separation between Philip Schwarm and Charles Bronson is only 3 - the relationship goes through the movies Ustanicka Street, Eyes wide shut i This Property Is Condemned, through Radet Šerbedžija and Sidni Polak. Moreover, it is even more interesting that the degree of separation between Sidney Pollack and Philip Schwarm is only 2! Even when we replace Charles Bronson with the legendary Chuck Norris, the score remains the same (3), only James Cromwell appears instead of Sidney Pollack.
Let's make things all the way complicated. In the new experiment, I chose Dragan Nikolić as the starting "node", and Chan-wook Park, probably the most famous South Korean director, a man who made many excellent and one brilliant film (Oldboy, 2003). And again, the number of degrees of separation did not exceed 3, as can be seen from the attached proof.
In addition, I spent a good part of the afternoon trying to make a chain of length 4 and in the end I just barely succeeded by putting Frank Lloyd, the now forgotten but once popular American director and producer who made some solid films in the 1930s and 1940s, at one end of the chain. At the other end was, hmmmmm…. an Indian actor named Kota Srinivasa Rao. Never heard of him? Not me, maybe not him either - he ended up in my collection somewhat by accident, thanks to a small cameo role in a Bollywood hit (or "hit"). Sarkar from 2005. That's it - four degrees of separation, five I've never been able to reach. After several dozen experiments in which I dealt mainly with episodics and anonymous people, I would say that the average degree of separation in the film industry is somewhere between 2 and 3.
The world is really much smaller than it seems to us... Of course, there are exceptions, for example the actors of the famous movie Hands, the Hands of Fate. For decades, that 1966 film was considered the worst feature film ever made. For a long time it was at the top (actually the bottom) of IMDb's "Bottom 100" list of tragically bad movies that nevertheless achieved significant ratings. The director of the film, one Harold P. Warren, was a film dilettante, a dealer in artificial fertilizers, who for a bet wanted to show how a good film could be made for less than 10.000 dollars. And he succeeded in that, but the film was so bad in all its segments that none of those who participated in its making ever got a chance to participate in anything like it again. For most viewers Hands is a film in which the devil unleashes sinners for whom a boiling cauldron is not enough punishment, and against the actors from mannose even Steven Seagal acts as a champion of the Royal Shakespeare Theatre. That's why they are actors mannose eventually ending up as an endlessly remote island in an otherwise tight-knit film world.
A FRIEND AND A HUNDRED EUROS
The fact that there is a relatively short chain of friends that will lead us to someone who has the power in their hands to change our lives for the better does not mean much. The world may be small, but that doesn't mean it's easy to understand. In 1990, the American playwright John Gehr wrote a play under the title Six degrees of separation, which was made into a solid movie in 1993 starring Will Smith and Donald Sutherland. In it, one of the actors says the key thing: "I read somewhere that any two people on the planet can be connected by a chain of only six links. The president of America, a gondolier from Venice, put any name: it is a great comfort to me that we are all so close. But in order to really reach someone, closeness is not enough - you need to find the right six people for the connection to really be established."
And that is the essence of the whole story: the fact that the connection itself exists, does not mean that the connection is easy to find. And when you find it, a chain is only as strong as its weakest link. Some chains may exist only on paper. A friend is someone who is ready to lend you 100 euros and does not ask you to return it. Whether a series of six such persons exists on this planet is a question science has yet to answer.
Erdös number
photo: wikimedia.org...
Paul Erdos (pictured) was a brilliant Hungarian mathematician who published an incredible 1500 scientific papers, more than any of his contemporaries or predecessors. None of those papers were trivial and almost all of them were published in prestigious mathematical journals. In terms of the size and variety of his oeuvre, Erdos can be compared only with Leonard Euler, who published 800 works collected in 19 volumes (Euler's works were often, in accordance with the spirit of the time, very long).
While Euler mostly worked alone, Erdos published at least one paper with over 500 people, believing that doing mathematics is first and foremost a collective activity. If he rode the train more, he would certainly have published a work with the conductor. Ever since he, as a Hungarian Jew, left his native Hungary in 1934 under the pressure of growing anti-Semitism, legend has it that he never slept in the same bed for more than a week or two. He traveled the world, a little in Europe, mostly in America, with two suitcases half full of things. In the second suitcase, there was also a radio that Erdos carried with him everywhere so as not to miss the news that Soviet communism had finally collapsed.
Blissfully ignoring almost all the conventions of "normal" behavior, Erdos would simply travel from one mathematical conference to another, and during the breaks between them, he would most often drop by one of his colleagues without any prior notice and solemnly say at the front door: "My mind is open." And that meant he was ready to do math with his host.
No one ever turned him down - moreover, many secretly hoped that this disheveled, sloppily dressed mathematical "priest" would invite them into their homes. The meeting with Erdoş could help them to skip some unsolvable problem that has been a bone in their throat for years or to discover new, unexplored mathematical horizons with an abundance of research material. Even though Erdoš was a "guest from hell", completely unindependent, "rolled up" enough to starve to death next to a full refrigerator, no one was bothered by his unannounced stay. Even if Erdoş would have a brilliant idea in the middle of the night, he would immediately celebrate by walking around the house banging his ladle on the pan.
Erdos was a true "Martian" on planet Earth: he didn't care about money, he had no savings, the little money he earned lecturing at prestigious American universities he would leave to his hosts as symbolic compensation for hospitality. Sometimes he would make $25.000 in two days, usually by solving a problem that the big telecommunications companies couldn't solve for years, but he would only keep a small amount of it to get him through the next few days, while donating the rest to some charity, usually one that cares for vulnerable children. He often didn't have money for a bus ticket to the neighboring town and then he would have to borrow from his host, but he always returned his friends, god knows how, given his very modest income.
Although poor, he made a long list of unsolved problems and offered a monetary reward, sometimes considerable, to anyone who "started" at least one of them. He owed nothing to anyone, and his admirers continue to pay that reward to this day, many years after Erdoş's death in 1996. He never had a place of residence, never had a job, never had an emotional relationship, wife, children... Realistically, he never had anything except mathematics, and that's why he had everything.
He believed that every mathematical proof had a timeless value, and he especially appreciated those that were simple and elegant. For such proofs, Erdoš claimed that they come from "God's book", some cabbage patch in which God jealously guards the most beautiful and elegant solutions to all mathematical problems. Although he called God the "supreme fascist" and never respected him excessively (because God hid his glasses, messed up his socks, tore his pants and made stains on his shirts), the desire to "steal" some leaf from God's book of perfect proofs and share it with humanity never left him until the end of his life. And he was very successful in that.
Before his death, he asked for his epitaph to be the inscription "Finally I stopped being more and more stupid". His wish was not granted because Erdos maintained his mental condition until the last day - he died during a break between two lectures at a mathematical conference in Warsaw. Not long after, his admirers established the so-called "Erdoš's number" as a measure of the "academic distance" of a mathematician from the great Erdoš. Erdos himself had the number zero, and those who published at least one joint paper with him received the number 1. Those who wrote a paper with one of Erdos' direct collaborators but not with Erdos personally received the number 2, and so on... Quite a number of papers were written about the Erdos number, from the graphs that connect people with a similar Erdos number, one can see how certain mathematical ideas spread through the scientific community and how mathematicians grouped themselves wouldn't they solve some challenging problem. The winners of the "Fields Medal", the highest recognition that a mathematician can receive (it is awarded only every four years and to mathematicians who have not reached the age of forty), have an Erdos number no higher than 3. One of them, Terence Tao, probably the greatest mathematical mind of today, has an Erdos number of 2.
The largest Erdos number known today is 15, with most active mathematicians having a number of 7 or less (which fits quite well into the concept of "six degrees of separation"). If only mathematicians with Erdoğan's number are taken into account, the average value is about 4,5, which once again confirms that if you limit the world to one of his professional niches, the number of degrees of separation drops significantly below six. After Erdoş's death, the smallest Erdoş number that a newly minted mathematician can get is 2, and it will be like that as long as at least one of his closest collaborators is alive. Even mathematicians such as the legendary Indian mathematician Srinivasa Ramanujan were given Erdoğan's number, although Ramanujan died when Erdoğan was only seven years old, thanks to the fact that Ramanujan's Cambridge mentor, GH Hardy, had Erdoğan's number 2.
There were also funny situations: some claimed that the legendary baseball player Hank Aaron has Erdoş's number 1 because he signed a baseball with Erdoş before the start of a game. Erdosh's number was also given, quite by chance, to a California horse named Smarty. His owner (aka a mathematician from Auckland with an Erdoş number of 2) wrote a letter of protest to a local magazine in which a short but offensive text appeared about the breed of horse to which his Smarty belonged. As the author of the letter, the professor signed his own horse, which was the "offended party". Given that the professor was a "ghost writer", according to American copyright law, he was considered a co-author of the text, along with the signed author, Smarty, who thus, although a horse by profession and blissfully uninterested in mathematics, received Erdosh's number 3.
