The Six Degrees of Separation

This blog and this specific article are from the POV of an undergraduate course on Complex Networks and this writer here is now half way through the course, having seen a bit of the maths, a bit of the logic, and also, a bit of the motivation for this course. 

So, if I really have to start with writing a blog, I would prefer for it to be related to my understanding of the course and how it relates to my changing view of this course with the concepts that I've understood.

Complex Networks as a topic always brings to mind a spiderweb of nodes and edges, a very highly detailed and multiplied form of the graph structures we study in maths using the nodes as vertices and edges connecting them. Now, one of the most famous studies conducted by social scientists ever, which has somehow entered our mainstream and pop culture so well that we mostly seem to have forgotten about the basis of it, has its idea based on this topic. This idea, the degrees of separation (immortalised by pop culture to be six) is most famously known for the experiment by Stanley Milgram. 

This idea had spawned because of ideas of proper city planning based on scientific methods just after World War I, which at that time included the idea that how connected are people really in a random scenario in terms of knowing each other on a first name basis so that if cities were planned, the people would always be in an optimal structure for the right expansion of their social circuits. The first such experiment showed that any person could reach any other person by a maximum of two hops or acquaintances who only knew the people who they were pointed to by. 

'In the experiment, Milgram sent several packages to 160 random people living in Omaha, Nebraska, asking them to forward the package to a friend or acquaintance who they thought would bring the package closer to a set final individual, a stockbroker from Boston, Massachusetts. Each "starter" received instructions to mail a folder via the U.S. Post Office to a recipient, but with some rules. Starters could only mail the folder to someone they actually knew personally on a first-name basis. When doing so, each starter instructed their recipient to mail the folder ahead to one of the latter's first-name acquaintances with the same instructions, with the hope that their acquaintance might by some chance know the target recipient.'

The idea behind the original Small World Experiment

Almost 30-40 years later, when Milgram conducted the experiment, he came to the conclusion that any other person could be reached to by any person by a minimum of six people, which came quite as a stroke of luck since his sample size was really small and more than half of his originally sent letters came unanswered and got lost.

Little had Mr. Milgram known how influential his idea about connectedness in a social construct would become in the future. Before his time, the idea of social capital had only come into use in the context of city planning but today this idea has morphed into a general idea for understanding any complex network, starting from a human network to a citation network where it is used to show a 'collaboration distance' between any two researchers from a very important researcher like Erdos or in actor circles, an important actor like Kevin Bacon.

Connectedness as a property is almost as important as the various types of centrality in that, by understanding the concept of centrality, we can identify nodes in a complex network the removal of which could cause the graph to lose some connectivity feature. Similarly, if we see the concept of connectedness, it is basically the idea that let's say, for an example, if there were some food essentials to be distributed and assuming people to be ideal and not steal, we'd be pretty sure to be able to divide all the food by giving the distribution of food to a few people knowing that the degree of separation for this connected network ensures that in a specific number of hops, the entire population could be covered. 

Similarly, this example gains importance in the idea of disease spreading where using centrality, we pinpoint travellers as very high-risk individuals because of the simple reason that they have a much much greater chance of being a central node in a given human interaction based complex network. In a much similar way, we could mark certain sub-graphs or communities of the network as extremely high risk because of their connectedness because that gives us an idea of how many people we interact with on a daily basis.

Now, the fun part about any idea gaining pop culture significance (courtesy John Guare's play, 'The Six Degrees of Separation') is that people find more and more interesting ways to work with the idea and then draw more conclusions from the work done on those ideas. In the IT age of today, people have tried to work out ways to perform the same experiment again in the multitude of networks present in our ever connecting world. 

For examples, researchers from Columbia University tried to modify the original small world experiment and came up with the idea of message chains of emails/ Microsoft Messenger to see if the degree of separation's concept still remains valid. Out of a very high number of participants, only about 0.4% actually reached their destination but those that did and even those that were responded to though they weren't able to reach their final goal landed very curious ideas and conclusions about the interaction patterns for people.

There have been games based on this idea in which you have to reach a certain character in a game world by asking game characters about the person. 

Linkedin works entirely on the concept of connections in which people are listed as being 1st, 2nd or 3rd connections based on the degree of separation among the common followers. This gives ideas as to how a network might be expanded for maximum benefit, most simply by not reaching out to your 2nd connections but to further out people because that brings you in reach of more and more people!

Twitter, being a follows/following based community is very closely connected in terms of this concept and was tested out by researchers to be on average around 4.67.
In another similar study of a random sample of 1500 users, they found that people on twitter were close to each other by a degree of 3.435.

Even Facebook, which works on the idea of having friends and/or followers which make it a bit easier to form connections; has been researched upon on this idea and the result announced by Facebook was that the value of the degree of separation for the largest social network in the world was 3.57.

More importantly, people have connected this idea to the mathematics behind random graphs for eg, in the Erdos-Renyi Model, the average path length between randomly picked nodes is ln N/ ln K where N is total number of nodes and K is the number of neighbors per node, and assuming the human population interactions model a random graph, this calculation modifies to the following.

 If N = 6,000,000,000 (90% of the World population) and K = 30 then Degrees of Separation = 22.5 / 3.4 = 6.6. 

 This surprisingly fits pretty well with the title of this topic even though its origins had been in the lore from a not so properly detailed experiment. At this note, the idea which this writer had set out to explain and illustrate has been discussed so here we end.

Here there be borders. (I know, I know! I'm not really good at puns :3 No other line strikes my mind right now. :P ) 

Edit: A youtube video by popular youtuber Veritasium that I found relating to this topic-