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Thursday, September 13, 2012

Online Population Projection

For some reason, this came to my attention. Because, math was wrong? What?

Ben Foster anticipated Facebook's billionth user would login last month, in August 2012. He wasn't alone. Here was NBC News' technology blog, reporting on a prediction by iCrossing:

And Bloomberg Businessweek would only call it for "later this year:"

Which all seem to presume linear growth from here on out. Indefinitely? I don't know. Which just makes me wonder, to what extent can we apply usual population growth-type logic to online populations? If Facebook were growing within an environment with biological limiting factors, we would have expected what we've already seen, for example, exponential growth at first. For quite a while, Facebook was growing at a lovely exponential clip of approximately 10% a month. This shows their growth for December 2004 through December 2009, with an exponential regression to fit:

However, maintaining that growth rate was clearly unsustainable. If it were, Facebook's population would have reached seven billion In June of 2012. Obviously, that didn't happen.

In biological populations with finite space and resources, we expect growth that looks exponential at first, but due to limiting factors, levels off eventually. And, indeed, Facebook's growth did not continue exponentially.

And I suppose it has looked rather linear for a while, but I'm not sure that's the best model. The rate of increase has slightly decreased the past year or so (shout-out to the second derivative!) 

If we apply a logistic model to the data so far, we get:

Which has Facebook reaching a billion users in April of 2013, and predicts its eventual population will top out at less than 1.1 billion.

But this all raises more questions than it resolves. Facebook may be approaching its maximum realistic number of users within the United States. However, as far as I understand, it has lots of room to grow in other huge markets. So this logistic growth model is flawed as well. I'll cop to not understanding how graphing calculators come up with logistic regression equations, like, at all. At least not with nearly the depth that I understand how they calculate linear regression equations. I simply know how to apply it as a blunt instrument to a table of values. On the other hand, linear growth, as the news organizations have used, has not panned out - as we're past August 2012, and have not reached a billion users yet. Have worldwide, internet ecosystem limiting factors unavoidably kicked in already? Should we expect another period of exponential growth in the future, if it catches on in India? Are there reasons to think this linear-looking growth will continue for a long time? And for how long? I don't know if any of these are answerable! But I do love the questions.

Here is a Geogebra file, if you want to play.