Its a Small World , Is it ?- Part 1

In the this series, i will try to summarize all my learning's of the past month,about the science behind Social Networks. I must warn the readers beforehand that this might end up pretty long , but it will be definitely interesting ride and a time well spent.

So What is it ?

Small World Problem : The world when viewed as a set of acquaintances, is in a certain sense "small"  that is any one person in the world can be reached through a network of friends in only a few steps. This is named from the phrase that we often use when we usually meet stranger  in a party and find out  a mutual acquaintance, we remind each other "what a small world it is". So the Small world problem is more general . "Even When i know someone who knows you , I still know someone, who knows someone, who knows someone who does knows you".
             An experiment was conducted by Milgram in 1967, known as small world method, which is a message passing technique.He gave letters to few hundred people randomly selected from Boston and Omaha,  and the letters were to be sent to a single target person , a stock broker who worked in Boston,But the letters came with a unusual rule, Recipients were to send the letters only to somebody who knew on a first name basis. i.e., If the recipient knows the target person directly he can sent it to him or if he doesn't know, he has to send it to the someone who they did know who they thought as someone closer to the target .
When asked people how many steps would it take to reach the recipient, most of them thought it would be in hundreds, bu the result was a surprising six (yes 6), hence the famous phrase "Six degrees of Separation - Everybody on this planet is separated by only six other people".
           If we try to do an reasoning on this, mathematically   it is like a pure branching network, Let say if I know only 5 people.but within two degrees of separation , I can reach 25, within three degrees 105, and so on.. Scaling this this 100 friends, within 6 steps i can easily connect myself to the entire population of the planet. So maybe its obvious it sis a really small world.

Six Degrees-Branching network

          But there is a fatal bug in the above reasoning. Think about your 10 best friends, and ask yourself who their ten best friends would be , Chances are you would come up with the same people , this feature is called clustering, which is really just to say that most people friends are also some extent friends of each other. This how social networks are in general, little clusters based on location, interests joined to each other by the overlaps created  when individual belonging to one group also belong to another group.

Actual Social network

This characteristic of a networks is particularly relevant to the small world  problem,the more your friends know each other , the less use they are to connect to someone who you really don't know.The paradox of the social networks that Millgrams experiment highlighted is that, on one hand the world is highly clustered,yet on the other hand we can still manage to connect to anyone at all in a very few small steps.
     
  After 30 years of this experiment, the actual nature of the world remained in question,and the paradox at its heart remained just that ,a paradox. however recent works has helped  to resolve the "Small world phenomenon". The idea that broke the stalemate was found by coming at the old problem with a new direction.Rather than going out in to the world and measuring it , we can construct a mathematical model of the social network and solve with the power of "Computers and Mathematics".

  Too much for intro.. next part I will write about the representing the model and understanding the basic properties of such models(Its all about Graphs :-)).