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


  1. Alchemist says:

    I address some of the same points over at my blog (link below) although more from the user participation point of view.

  2. Ajit Jaokar says:

    thanks Alchemist. I like your views. lets keep in touch. rgds Ajit

  3. Paddy Byers says:

    There are some pretty extensive mathematical models that relate to the proliferation of an infectious disease in a population. At the early stages, where the level of infection is low, most of those in contact with infected people are not themselves infected, but become infected (with a certain probability); thus at the early stages the rate of growth is proportional to the infected population and growth is exponential. Later, when a significant proportion become infected, this model no longer holds and the rate of growth slows.
    The most effective social networking sites grow by a similar (ie viral) effect. Members invite those they are in contact with, and if the proposition is attractive, there is a reliable conversion rate and exponential growth.
    Exponential growth will not happen in those situations where the model no longer applies, ie:
    1 – the “infection” rate is not a simple constant proportion of those contacted, eg where there is a more complex function, perhaps inhibited by paid membership requirements;
    2 – when the size of the community grows to become a significant proportion of the overall population.
    Perhaps you could think of a competing site (eg facebook vs myspace) in the same way as an antidote – ie if you are a member of one already, you’re much less likely to become a member of the other community. There are models for this situation too.

  4. PaulSweeney says:

    Hi there. I think I remember an explanation of this over at I am sure if you drop bruce an email or comment he could point you in the right direction.

  5. Ajit Jaokar says:

    thanks Paul. shall do. rgds Ajit

  6. Chris Worth says:

    My feelings here relate to differing definitions of ‘communities’ and ‘audiences’.
    Many so-called communities – e.g. “the Myspace community’ – are actually more like audiences than communities. Are the people attending a band’s concert a ‘community’? No, they’re an audience.
    The advantage of audiences is they can scale to pretty much any size. It’s only dependent on the popularity of the performer and the size of the hall.
    Explosive growth, like Myspace, seems to happen when NOT being in that group causes more pain than being on it (e.g. it’s really difficult these days not to have a phone.) When it became essential for every new band to have a myspace page, the audience started exploding.
    A community, however, follows some more basic laws. Human societies are wired for a few basic units of society: the family (2-5 related people), the extended family (6-20 people) the village (150 people) and the town (2000 or so I think.) Town level is the point where not everyone knows everyone – i.e. the urbanised society.
    Amusingly, you’ll see these numbers everywhere. Even among urbanites, with most people belonging to a group of 150 or fewer people they know. Villages in Africa average 60-200 people. Business in Britain tend to hit the stops at 150 employees. A London survey found most under-40s have 12 close friends and a further 120 associates (equivalent to family and village).
    Ecademy is a community; there’s no star performing for us at the front. Content is created by a percentage of active members and disseminated by a further group who pay to be here.
    I bet the active members number very close to that 150 figure, and the extended group very close to 2000. So Ecademy’s been at its natural limit for some time.
    Community doesn’t scale – that’s why it’s called community. The bigger it gets, the less there is to make it special.

  7. Hi,
    Great article. Very mathematical approach. You touched on exclusivity and the ability to share and interact within the community which determine its growth rate.
    I believe the growth rate also relates to the following factors:
    1.relevance/affinity of the subject(common element that binds the group) for the members(audience)
    2. Addictiveness( mostly driven by communication possibilities and fresh objects(content, members))
    I believe there is an internal growth rate and an external growth rate. Internal is determined by the architecture. External more with the influence of the network outside of the network.. My example: If social network is a big part of your life you are more likely to talk about it(mouth to mouth, IM to IM, Email to email) to others. This will result in fresh members. Innovative technology or new features could also set-off new growth..
    These are my first thoughts. If I have more I will bother you with more comments.
    Keep the sharing going!!

  8. Maggie Baldry says:

    A slight slant on the mathematical theory. Perhaps the use of Eigenvectors
    (within the context of matrix theory) may help with the mathematical
    There’s a long page on Wikipedia about this and the links below should help
    with navigation to the pertinent points.
    The final link on infinite dimensional spaces mentions “exponential growth
    or decay provides an example of a continuous spectrum”.
    Consider if a social network exists within an infinite dimensional space
    (the Internet) but for we humans who exist as part of a network there is
    only a finite amount of time for one human life. Therefore the most
    ‘attractive’ social networks will be the ones that will allow us to make
    best use of our time and resources – which will in turn be based on why
    decide to join or leave a network.
    Consider the network itself is the string and it’s the oscillation of the
    members of the network that make it move and resonate. The members of the
    network will need to be moving with the same rhythm as other members of the
    network to keep the network (string) oscillating at the same resonating
    rhythm and keep attracting more members (and retain existing members).
    Whether or not the oscillation of the string could be equated to market
    value; or whether Web 2.0 companies make the string oscillate more; are 2
    more questions that could be considered.

  9. Graham Jones says:

    Ajit, fascinating stuff.
    In my role as an Internet Psychologist I’m also interested in why some sites expand more rapidly than others. If MySpace expands at an exponential rate, why hasn’t Ecademy.
    From a psychological perspective I think it’s quite simple. Ecademy doesn’t replicate what we do “normally”, whereas MySpace does. By that I mean Ecademy has formal structures, has rules provided by an “authority” and is focused (mostly) on business. LinkedIn doesn’t have any real social interaction, so it is unlikely to grow as rapidly as MySace either.
    In contrast MySpace has few authority figure rules and is comparatively open. It is also unfocused and is more akin to the haphazard nature of “normal” life. That makes it instinctively more appealing to all people.
    Interestingly, MySpace’s own figures show that the bulk of their audience is over 35 years old – not the young audience the media would have us believe. So it may not be popular music that is driving the growth. It could be the basic human need for socialising. That’s much easier with MySpace than Ecademy. And that may be part of the reason behind the differences in growth levels.
    In other words, I think it has less to do with free vs paid, or design, than it has to do with how the sites replicate (or otherwise) normal social activity.

  10. Ajit Jaokar says:

    thanks Raimo, Maggie, Chris, Graham.
    Must do a followup to this one
    kind rgds Ajit

  11. This is an interesting approach although I sense from my own research it’s more complex than npr. I’ve always taken an overly simplistic position arguing that search is the key – that has been historically very poor at Ecademy, Ryze and OPenBc. This is supported by quite a lot of emerging research by Wolfram and more recently Matteo et al. The most relevant research is from Sarkar, Butler and Seinfield on Intermediaries and Cybermediaries that is based on Transaction Cost Theory. If you read this through, you’ll get an understanding of why other networks have not seen exponential growth. Professor Michael Mainelli and I attempt to explain why/how networks are monetised in this Ecademy blog, a while back.
    One of these days, I’ll get round to writing a paper called “we don’t want your money, we want your data” to describe the future economic models of social networks.

  12. OIM says:

    Mobile social networking could easily outperform these numbers. In the end, a mobile phone is a communication device and one very personal as well. Two key aspects that can be a catalyst for growth.

  13. Good points, Ajit -
    - but my impression is that applying Metcalfe or any mathematics (‘exponents’ are good for PR headlines, n-th order polynomes may describe some very generic networking effects) to phenomena like MySpace or YouTube hides a potential risk of oversimplification – and therefore misinterpretation.
    In what has (for lack of resistance by all of us) become widely called ‘social networks’ there are a lot more factors at play than the number of nodes or the nature of joints between them.
    One thing to keep in mind when analysing growth is that these online communities grow through, and because of, interactions outside the community, not inside it. It is the social (or other kinds of) networking that happens outside MySpace that brings new members. Interactions that happen off-line, as well as on other online platforms (that we know so-o-o little about) that generates the flow of new subscribers to the observed platform/network/community.
    One may argue that rich interactions within the community make it valuable to members and creates willing advocates that spread the WoM. This may have some contribution, but it’s negligible – the evidence is in your example with Ecademy. There is abundant, meaningful and highly rewarding interaction within that network, yet it doesn’t experience ‘exponential’ growth. Not more than LinkedIn, where there is practically no such interaction, the model being a pure ‘dating engine’ fascilitating contacts and discovery of people, but not desiged for high interactivity.
    So, the external interactions have (much) bigger contribution – but are so difficult to analyse that putting the MySpace growth in any simple mathematical function/equation is premature and would yeald primitive results.
    Ironically, many MySpace members have arrived to the portal from other communities (including, but not limited to online/virtual ones) that have not experienced ‘exponential’ growth! Or haven’t enjoyed the same level of hype, at least :)
    One shouldn’t ignore the ‘Web 1.0′ or even ‘Old Media’ factor – awareness and interest in MySpace and YouTube generated not by networking, but by plain old ‘shouting’ – mass marketing and PR. Those one-to-many, one-way communications, while sneered at by modern marketeers, still have a formidable power to influence, which is not to be underestimated. If it didn’t cause the exponential effect, it certainly raised the order of the polynome. Media hype has the tendency to become self-fulfilling porphecy. When there is so much noise about YouTube it is hard to resist the curiosity and visit to see what it’s all about.
    It is debatable whether MySpace were lucky to become the ealry nucleus of a media hype snowball, or everything was the result of very cunning marketing plans. Perhaps a little of both. But I have no doubt that, had Ecademy had similarly cunning marketing (or luck) it would have seen growth way further from the linear one it enjoys (which is not bad, either, other portals have flattened or are shrinking).
    If we are to applly mathematics to everything, it would be interesting to model the inevitable implosion of MySpace. Wherever hype has played a role, there follows a correction – bubbles burst etc. There are already signs that the theory of diminishing returns has kicked in – as the critical mass of members has ballooned, those already inside are starting to get less and less value: quality members become rare, as is quality content, there is too much ‘rubbish’ as everyone and his hamster try to outshout the noise and publish the works of their narcissistic ‘genius’. The growth of spam, spoof and outright fraud is ‘more exponential’ than the community itself. The dripping leaks of churn are starting to undermine further growth. At best it is soon likely to slow down. At worst – see DotCom 1.0 :)
    No model is ever perfect – but social networks (and that’s more than MySpace) are somewhat more complex than a simple mathematical notation, albeit an exponent.

  14. Dr Vladimir Detrovoski says:

    I love reading web waffle, and this is one of the best waffles on the web.
    Great post – This post gets 10 out of ten for beating around the bush for no reason

  15. I agree that tiered memberships reduce the chances for interaction between members and the possible network effect. On the other had social networks aimed at young people looking to have fun or date will naturally have more activity than general business sites. (An exception might be sites like intrade where there is a strong and immediate profit incentive involved. In the future, business focused networks will have to have contents other than profiles, e.g. jobs, business news, investment advice etc, if they are to continue to grow in relative importance for their members. I think that a Japanese business sns Bonnect may be on the right track in that regard.

  16. Brian says:

    I think it depends on a lot of variables. Many of the social networking sites with the most unique visitors cross brand, meaning there is more exposure to an audience. This allows them to be more leveraged to promote the sites. I think these will become a fad though in the long run.
    Sites which focus on a simple targeted audience I think will be longer lasting; i.e., LinkedIn for the business community, Erotas Online for the adult community, for linking with old friends.