The mathematics of Web 2.0: Why don’t ALL social networking sites experience phenomenal growth?


When I spoke at Santa Clara/ Stanford last week, I had the pleasure of meeting Dr John Yu – a serial entrepreneur and VC in the valley. The original inspiration for this post came from John.

This is an ambitious post but I also acknowledge that I need some help especially in getting some numbers / stats.

The key observation is : Social networking sites like MySpace and YouTube are growing at rates faster than the growth of the Internet itself – but not ALL social networking sites are showing very high rates of growth.

How can we explain that?

In this post, we consider three things

a) The growth rate of the Internet

b) The growth rate of the top web 2.0 sites

c) The reasons why not all social networking sites show the same effect

Let us first consider the Internet

The growth rate of the Internet is governed by Metcalfe’s law (also called as the network effect ) which states that the utility/value of a network is proportional to the square of the number of users in the network. Mathematically, it is a 2nd order polynomial.

As per Metcalfe’s law, beyond a certain level of members(called critical mass), the network effect kicks in. At critical mass, the value obtained from the good / service is greater than the price paid for the service.

Thus, the userbase determines the value(but not its rate of growth).

A fax machine can be used to illustrate this concept. A single fax machine is not useful but the value of the fax service changes as more people start to use fax machines. The Internet operates according to this principle

Now, let us the new Web 2.0 sites such as MySpace or YouTube

In this case, the growth rate is higher than the network effect. This growth rate is exponential.

It can be compared to the growth of bacteria in a culture(as opposed to the proliferation of fax machines)

For example, suppose we start with a population of cells such that it’s growth rate at any time is proportional to its size. The number of cells after t years will then be at (an exponential function) for some a>0.

So, we are saying that some Web 2.0 sites are showing exponential rates of growth(in this case I am using the term ‘exponential’ mathematically and not in it’s conversational sense. I am also saying that it contrasts to the Network effect – which is itself a high growth phenomenon but at a much smaller rate than exponential growth)

For example, Wikipedia (a Web 2.0 site) believes that it’s growth rate is exponential

As per above link ..

One common model of Wikipedia growth is that:

more content leads to more traffic

which leads to more edits

which generate more content

Thus, the average rate of growth should be proportional to the size of the Wikipedia, that is, the growth should be exponential

Let us consider MySpace

Here is a hypothesis

MySpace is primarily driven by music/bands. Thus, the effect driving MySpace is that of bands inviting their respective fan bases. Also, it percolates in the site itself when there are many groups around a single theme(283821 ‘music’ groups for example)

In contrast, the network effect is of the order of two i.e. one on one (pairwise) interaction (i.e. square of the number of users) in comparison to ‘set wise’ interaction we see in Web 2.0 sites(for instance between music groups)

Let us consider some numbers. As at March 2006, MySpace had 67 million members since its launch in 2004. It was then growing by an average of 250,000 new members daily

Just about a year ago, as at July 2005, MySpace had 22 million members and a growth rate of 2 million members a month.

This means at July 2005, MySpace was growing at 66,666 members per day (at a membership of 22 million) BUT in March 2006, it was growing at 250,000 members a day (at a membership of 67 million members)

Note that: 66,666 = 2million members per month divided by 30 days per month

So, in 2005 it was growing daily at 0.30% each day and in March 2006, it was growing at 0.37% of its membership

These figures indicate a growth rate which increases as membership increases.

Having said that, I don’t know if qualifies as exponential. Which is where I need help with better (maybe more granular figures) and a cross check on my calculations/thinking.

Similarly, the ‘unit’ of growth of facebook is a ‘college’ and potentially this may also lead to interaction by such related groups

To really prove this theory we need a interim numbers(which I don’t have and I seek any help from anyone who does – anyone from MySpace/Facebook reading this? :)

Truly exponential growth would appear as per the graph as above(source: wikipedia ) and those being derieved as per the principles of exponential growth

Some more numbers to point to a phenomenal growth (year on year – albeit UK specific)

As at July 2006, Wikipedia in the UK had 6.5 million visitors (up 253 percent versus year ago), (up 467 percent to 5.2 million visitors), (up 393 percent to 4 million visitors), (3.9 million visitors), and (up 328 percent to 3.9 million visitors).

In a nutshell, participation leads to exponential growth. More the users, the more pictures to share, videos to upload and comments to add. Thus, the growth rate of a Web 2.0 site is proportional to the number of members in the site at a point in time(the classic definition of exponential growth – la bacteria in a culture)

This leads us to the final part of the question ..

Why does not the same exponential rate of growth occur in sites such as Ryze, ecademy or Linkedin ?

After all, they have been there for longer .. so logically they should show more members.

Is it because they are not ‘free’, they are ‘business’ , they are not ‘fun’( i.e. less social – more business)?

It may be all of the above .. but I think it is more due to a severely limited architecture of participation.

Typically, such sites have a membership – often a tiered membership and / or a paid membership.

There is nothing wrong in that except that it cripples the architecture of participation (the very thing driving the exponential growth of Web 2.0 sites). The effect of tiered membership/ restricted membership is : the whole is split up into components thus reducing the number of potential interactions between members i.e. How can users participate? with whom?

In any case, interaction is never potentially with ALL members’ (which it is in case of Web 2.0 sites) leading to the split

Again, we can explain this using some mathematics(any comments welcome on this section)

According to the principles of permutations and combinations : if you have ‘n’ different objects and ‘r’ members amongst these are to be arranged – then the number of permutations is given as


where n is the number of different objects and r of them are to be arranged.

If n = 4 then n! = 4 * 3 * 2 * 1

Suppose we have (hypothetically) 10 members then we have

10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3628800 combinations (i.e. with no combinations)

Now suppose we start ‘combining’ these members into groups of three (r = 3)

(By ‘combining’ I mean subgroups of members who cant interact easily beyond their own subgroup – membership level – ‘links’ etc)

We get

= 10! / (10 – 3)! = 720

This is a dramatic drop in comparison to 3628800 (in fact the interaction is only 0.01%)

Thus, I would then argue that they don’t even see the network effect(the rate of which is lower than the exponential growth of Web 2.0 sites).

This may explain the high valuations of Web 2.0 sites and also the concept that not all communities can be deemed to be demonstrating Web 2.0 (MySpace like) growth and by extension may never command similar market valuations.

That’s not to say that they are not useful or valuable but are severely limited in growth potential and by extension market valuation

Seek thoughts?

see my book Mobile Web 2.0