Three and a half degrees of separation?


Last month a team of researchers at Facebook posted an article where they update the “mean degree of separation” of Facebook users.  You have most probably heard of the “Six Degrees of Separation” legend: between you and me, as between anyone in the world, there is a chain of acquaintances that connect us; this chain is at most 6 steps long. In other words, you know somebody, who knows somebody, …, who knows me! Apparently, this idea dates back to Frigyes Karinthy, a Hungarian writer from the first half of the 20th century, but it was then investigated by social scientists and, with the arrival of social networks and Big Data, people have started using online social networks to test it experimentally.
In 2011, researchers at Cornell, the Università Degli Studi di Milano, and Facebook computed the mean degree of separation across the 721 million people using Facebook at the time and found that it was 3.74. Here the separation is defined in terms of intermediate individuals between a given pair, instead of the number of steps. The news is that Facebook users grew to 1.59 billion and the mean degree of separation shrank to 3.57. If you visit the page, a fast algorithm calculates your own mean degree of separation.
However, one may ask how representative is Facebook of real-world social acquaintances.  Maintaining a real-world social relationship is expensive in terms of time and energies while Facebook “friendship” comes almost for free.  Therefore, one can argue that Facebook connectedness overestimates the real connectedness of individuals.
It is reasonable to believe that some of those links are so weak to be non-meaningful. Also, social contacts change with time, while one may expect that most Facebook users wouldn’t regularly prune their inactive links.
Finally, albeit large, the Facebook world is still a sample of the whole humankind, and it is certainly not a random sample.  Just to mention a few reasons, access to the Internet in Africa is still much more difficult than in the other continents (even if things are changing fast), the population on Facebook is less represented for elderly people, etc.
As a result, it is possible that the Facebook sample has a lower mean degree of separation than the world population as a whole.  But it is still a very large sample.
All these remarks are quite intuitive, but the network scientist Duncan Watts has criticized both the work and the approach, putting in evidence the counter-intuitive behaviour of the so-called “small-world networks”.  In a famous paper written with Steven Strogatz in 1998, they proposed a very simple model that showed how adding a few “shortcuts” in a network (links connecting random pairs of individuals) quickly reduces the shortest path length (i.e. the degree of separation) and, quite interestingly, that doesn’t get much shorter if you keep adding random links after this initial drop.  The argument, therefore, is that the world has already become small decades ago, and it’s quite unlikely to shrink much further.
This, in my opinion, opens the question if a mathematical model could quantitatively fit the measured reduction in Facebook’s mean degree of separation from 2011 to 2016. It could help us in understanding better which topological features of the network are relevant in the process.
Anyway, there are many more details in a real-world social network that still need to be understood besides the degrees of separation.  In a recent paper, the sociologist Robin Dunbar has investigated the relationship between the number of “friends” reported in Facebook and the number of the ones personally perceived by individuals.  In some earlier papers, he and his co-workers had identified progressively self-contained layers of closeness in human acquaintances.  They showed that, typically, human layers of social closeness approximately contain 5, 15, 50 and 150 individuals, plus two external layers of 500 and 1500 alters.  In his recent paper and references wherein, Dunbar shows that online social networks can realistically approximate not only the 150-friends layer but also the two most internal layers.  He also claims that the number of online contacts is usually not larger than the one of offline contacts.
Indeed, there is a minority of users who report a larger set of online friends with respect to the offline world, but it is hypothesised that whose extra online connections are weak acquaintances, that in online social networks cannot be typically distinguished from close friends.  This can only be seen by investigating the traffic among individuals by counting the direct posts on Facebook or replies on Twitter.
Besides online social networks, mobile phone networks are a valid alternative to measure social acquaintances.  Phones are still diffused in areas with low Internet connectivity, communication often carries a cost and it is possible to measure the traffic between individuals.
In Idiro we investigate these problems every day and are able to detect layers of social acquaintance to improve digital marketing strategies and provide value for our customers.

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