In the recent episode of Crime Patrol, an Indian soap showcasing real life criminal cases in India, the husband confesses murdering the wife as the sleuths closed in with evidences. When asked the motive behind the murder as both were getting along well, he responded – ‘when the test results came, it was obvious that I was the reason for we being childless and the doctor opined that I’m impotent but curable.’ He went on to add further that ‘I was scared that my wife will tell her friends about it, who in turn will tell their friends, they theirs and hence it will spread to the whole complex in a week and to whole town in a month. And I will not be able to take that social stigma of being branded as an impotent.’

The director of the soap could have added few recent social network medium into the dialogue, e.g. ‘my wife would have twitted about my impotence’, ‘posted on the Facebook walls with the medical report uploaded’, ‘sent out bulk sms on my impotence’ etc. Well I left out ‘broadcasted to our Circles in G+.’ The Google+ is a recent phenomenon in Social Networking. Will it be the nemesis to Facebook as Facebook was to MySpace and Orkut?

Going back to the first paragraph, how would have news of husband’s spread across the population. Let’s say she disclosed it to 2 of her friends, in turn the 2 friends told to their 2 friends each after an hour, and hour after that those 4 friends would whisper to their 2 friends each…..and so on. Mathematically, the total numbers of people that are aware after say 10 hours are 511 including his wife and is arrived at by:

1+ 1*2+ 2**2+2**4+……..+2**10

This is called a Geometric Pogression or in short GP. I’m using ** notation to represent exponential, or as we commonly call it ‘to the power’. You don’t have to break your head on understanding GP, just assume it is true and proceed.

But if the wife was to use social network, or broadly let’s call it digital or electronic means, the news could have spread much faster, like a virus. Present day marketers have taken a leaf out of the viral epidemic outbreaks and coined the term ‘viral marketing’ to draw a parallel between the huge number of targets that their marketing campaigns will reach successfully, similar to the spread of the virus to a large number of people in an epidemic.

The spread of the virus in an epidemic and the spread of messages in the digital world have the same dependency on – the infection period (T) and the spread factor (S). Only when one becomes infected, then only one can spread the virus; so is the case in the digital world – only when you get hooked, say start using Facebook, you can infect others by the way of recommendation, referral, invitation or exerting peer pressure. The infection period (T) say for a viral fever is 3 days, i.e. you are down in 3 days after catching the virus and then you become a carrier, i.e. you can infect others around you. The spread factor, S, is the number of people around you whom you can possibly infect. Say in this case of a viral flu, the S is 7, i.e. you can infect 7 people.

There is a difference between how the words spread compared to that of a viral spread or a digital spread. We assumed that in the manual mode, the friends of the wife will spread the word in intervals of one hour, but no such things will happen in case of a viral break out. It will spread continuously as the viruses don’t wear a watch or are mindful of other activities except the sole motive of spreading themselves. Mathematically, while the spread of words that happens in intervals are represented by a geometric progression, the viral spread is denoted using something called as Euler’s Number, which is denoted by ‘e’.

In case of a viral spread the equivalent of the GP formula for spread as depicted above becomes:

C = I * e**[(S-1)*N] I have used the notation * for multiplication and ** for exponential or what you popularly read as ‘to the power’.

C: The number of carriers or infected persons after N infectious periods.

N : [Elapsed Time])/T, T is the infection period

I : The initial number of infected persons S and T are off course the parameters that have already been defined above.

Now let’s turn to the real McCoy.

Facebook took over from MySpace. MS was launched in 2003 and by Aug 2006 it had 100 million active users. In 2007 it was valued at 2 billion US$. The current number of users stands at 53 million and few weeks ago it was sold off for just US$ 35 million. That’s the non conventional economics of digital world or should we say the social networking world – with the decline in active users by half, the net worth declines by 40 times!!

And this decline was brought in by the advent of Facebook, undoubtedly. For MySpace the mathematical equation looks somewhat like this:

53 = 100 * e**[(S-1)*60]

The value of N is 60 because in 5 years the decline happened to its current level, and the infection period is assumed to be 1 months. So N will be equal to 60 months divided by 1 month, which is 60.

Here the moot point is why T, the infection period is assumed to be 1 month? Why not 2 weeks or 2 months? I have no logical explanation for that. I polled 51 people from all continents who are my friends belonging to all possible major races from various countries of the world and conclusion that emerged is to get infected by a social networking site, 1 month is the norm!

The spread factor, S, from the above equation comes to 0.9894 which is less than 1 as expected since there is decline in the number of active users.

For Facebook, which was started in 2004, the current number of active users or should we say carriers are 600 million. If we take the T as 1 month, then the equation for Facebook looks like:

600,000,000 = 1 * e**[(S-1)*84]

N in the above equation is 84, as seven years means 84 months and T is 1 month. So in the formula that we have for N, putting elapsed time as 84 months and T as 1 month, gives us N as 84.

Computing the value of the spread factor S for Facebook we get S = 1.24.

The spread factor as expected is obviously more than 1 as the Facebook virus has spread widely starting from just 1 user, Mark Zuckerberg himself in 2004 to 600 million over a span of 7 years. And can you draw an obvious conclusion? You as a Facebook aficionado infected or brought in at least 1.24 users every month to Facebook that made billions for Zuckerberg in wealth!!

OK, now more real McCoy!

Google+ will inflict similar trend as Facebook inflicted on MySpace. Facebook will inherit the negative spread factor of MySpace, the damage that it inflicted on MySace; while Google+ will inherit the positive spread factor of Facebook.

So when will Google+ overtake Facebook?

We can derive this by using the following equation:

600,000,000*e**[(0.9894-1)*N] = 1 * e**[(1.24-1)*N]

This gives us N as 80.656. We can approximate that to 81 months and conclude that in 81 months Google+ will take over Facebook.

There is more and more real McCoy!

What is the value of the mathematical quantity as below?

600,000,000* e **[(0.9894 – 1)*1]

Here I used N as 1 month, i.e. just one infectious period for any social networking site as I had assumed earlier; have taken S as 0.9894, the same less than 1 spread actor that MySpace faced when Facebook came into being.

The answer is approximately 593.7 million; and the difference between this answer and 600 million is around 6.3 million; Facebook has currently 600 million users.

What’s the big deal? Actually the deal is much bigger than just being big.

As reported in various sources, the number of active users that Facebook lost in the month of May is 7 million. Hope you got the message now?

The decline in Facebook users number is in line with the mathematical explanations. In 81 months Facebook would not be as popular as it is now though it may not be gone from the scene…it will be on chopping blocks for say 20M$ compared to the current market capitalization of 12 billion $.

I have always maintained that mathematics is a conclusive science and it will deliver the knock out punch……

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Chittaranjan Jena currently heads Mahindra Satyam South Africa Divison. He has published two books and is an avid blogger. You can find more information at http://crjena.wordpress.com

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