Alex's Notes

There’s no doubt about it: Canada loves Facebook. Toronto was the first city to break the one million user mark, and in some cities non-Facebook users are in the minority. Members have made their influence felt on both provincial and national level politics, prompting government to treat Facebook as a serious political tool. This article examines evidence that Facebook is reaching saturation levels in Canada.

Canadian Facebook growth – finished already?

Various mathematical models exist for explaining population growth. The logistic function is a natural model to apply here. It describes a system where the population rate of growth is proportional to:

1. the current population (facebook members), and
2. the remaining resources (non-members who will eventually join)

Differential form of the logistics equation:

N(t) = number of Facebook users at time t
r = rate of growth
K = saturation level

Initial logistic growth is nearly exponential, which applies if site growth is driven mostly by referrals (one person tells two friends, who each tell two other friends, etc.). Followed by a nearly linear period of rapid growth, growth slows to reach a saturation value. At this point everyone who is willing and able to join has already done so.

The graph below shows the basic logistic function fitted to actual Facebook member data. The best fit results in a saturation value of 11,069,190 members, or 33.0 percent of the Canadian population. It clearly suggests Facebook membership — currently at 32.9 percent — has little remaining growth potential.

Facebook Market Estimation

The 33 percent saturation value represents everyone who is both:

1. technically capable of joining, and
2. sufficiently influenced to create a membership.

The first requirement can be considered a technical coefficient. 78 percent of Canadians are “current internet users” (CIP study), accessing the internet at least once in the past three months. 72 percent of Canadians fall between 13 and 64 years of age, where 13 is the lower cut-off for registering on facebook. The below graph of age distributions show that Facebook has low popularity with the 60 to 64 age group, so 64 will be considered the maximum age of potential  members. The current technical coefficient becomes 78% * 72% = 56.2%.

The second component is a social coefficient. This represents the fraction of the entire population who would like to register, either because they feel it would benefit them (internal motivation), or because of recommendations by friends, family, and media effects (external influence). The social coefficient can be found assuming a national saturation value of 33 percent:

Market Potential = (Technical Coefficient) * (Social Coefficient) * (Population)
(Market Potential) / (Population) = 33% = (56.2%) * (Social Coefficient)
Social Coefficient = 58.7%

A Facebook saturation levels of 33 percent implies a social coefficient of almost 59 percent. This value will rise if Facebook develops a higher perceived-value among non-members (which will happen due to network utility effects), or if external influences increase.

What if everyone wants to join Facebook? A social coefficient of 100 percent means the technical coefficient is the only limitation, and Facebook saturation might occur at 56.3 percent of the Canadian population.

Canadian City Projections

Edmonton, Alberta – a nice city

We can’t be certain that members of a Facebook city network actually live in the stated city, making a meaningful comparison of users to city population difficult. This is especially true with large metropolitan areas and where city boundaries meet. Edmonton makes a good sample city, being a large but isolated city with a greater metropolitan population of 1,081,300.

Growth in Edmonton differs fundamentally from the national growth data. The initial growth rate is very high, and no exponential growth is seen in the data. Exponential growth may have happened prior to the first point (April 24, 2007), however the inflection point — the point where accelerating growth becomes decelerating growth — also happened before the first data point. Consequently the growth is not S-shaped, and a logistics function cannot model it.

Instead we’ll use a model where growth rate is proportional to:

1. a constant value, and
2. the remaining resources (non-members who will eventually join)

Differential form of simplified NUI model:

N(t) = number of Facebook users at time t
a = rate of growth
K = saturation level

This is actually a special case of the Bass model for diffusion of innovations, and it fits the available data very well. The significance is that the member-proportional growth term r which causes initial exponential growth represents word-of-mouth effects, something seen when a product or service is spread predominantly by personal referrals and recommendations through existing social channels, as we might expect in the case of Facebook. Replacing this effect with a fixed-rate growth term (a) means that external influences dominate the growth. External influences typically models advertising; the more advertising, the higher the growth rate a. But it can also reflect “buzz” in a population, where everyone “knows” about something because of multiple rapid and pervasive communication channels. This model suggests that Facebooks’s rapid growth in Edmonton may have been due more to buzz (since Facebook hasn’t engaged in traditional advertising) than to interpersonal social interaction.

The Bass diffusion model also helps us predict Facebook saturation. The best-fit curve has a steady-state value equal to 55 percent of Edmonton’s population.

Halifax, Nova Scotia – most penetration, rapid growth

Of the 23 Canadian cities examined, Halifax leads the pack in Facebook penetration. 71 percent of the population appears to belong to the Halifax network, and its average population-adjusted growth rate was second-highest (slightly behind the smaller city of Kelowna, B.C.).

71 percent penetration seems impossibly high. Halifax’s age distribution leans slightly younger than the national average, with 74 percent between 13 and 64. Therefore the opt-in rate (“social coefficient”) is 96 percent (compared to the national 58.7 percent) assuming every resident has internet access.

The Bass model predicts that 85 of the population will eventually become members. This would require every resident age 10 to 75 to join.

Toronto, Ontario – biggest network

Toronto is currently the largest Canadian city network. The Greater Toronto Area encompasses several city networks, so the difficulties with matching members with their actual cities of residence are particularly bad here.

Because the earliest data point was already at half a million members, it’s possible a strong exponential growth occurred prior to that time. A different function that can model exponential growth was applied to test that theory. The resulting, albeit brief, “exponential” growth is seen in the graph at the lowest membership levels.

The modified logistics function (with exponent q) is commonly called the Non-Uniform Influence model (NUI). For 0 < q < 1 it represents a variable word-of-mouth effect; one which is important at first, but diminishes as member-base grows. Here, q = 1/5

N(t) = number of Facebook users at time t
r = rate of growth
q = growth exponent
K = saturation level

Although Model 1 (Bass) fits the available data better, both models predict the same saturation level of 30 percent.

Montreal, Quebec – untapped potential

For its size, Montreal had far slower initial growth than any other examined city. Early growth exhibits clear acceleration, suggesting that word-of-mouth referrals played a bigger role than in other cities. Buzz may have been less, possibly because the French media gave less Facebook coverage than English media.

Montreal is currently near its peak Facebook growth rate. Despite Toronto having 50 percent greater population, the model predicts Montreal will become Canada’s largest city network by July 12, 2009.

Provincial Penetration

Could some provinces be far ahead of others in Facebook adoption? Since provincial data should be more reliable than city data (less ambiguity regarding networks borders), these trailblazing provinces could be a strong indicator of where the country is heading.

Yukon is clearly in the lead, with 67.5 percent of its 31,530 population having facebook profiles. While examining the possible reasons for this are beyond the scope of this casual analysis, we can rule out Yukon’s age distribution as a factor. While 76.5 percent fall between 13 and 64 (higher than the national average of 72), that distribution is heavily skewed to the right.

On the other end is Quebec with only 21.5 percent penetration. If Quebec follows Montreal, this French province will be a significant growth market within Canada. One barrier may be the 15 percent gap in internet access between English-speaking and French-speaking Canadians. Furthermore, a 2007 study concludes that social networking sites in general have a greater appeal for English-speaking Canadians (43%) than for French-speaking Canadians (24%).

Conclusions

Using a logistics model applied to limited data, the Canadian Facebook saturation level was found to be 33 percent. However this data is a superposition of all city data; when cities were examined individually a saturation level above 50 percent was common. Three provinces were found to have current penetrations greater than 40 percent, with Yukon at 67.5 percent. This raises the possibility that national saturation could eventually reach such levels.

Quebec was found to have a very low penetration, but strong growth potential. Despite a smaller population, Montreal should surpass Toronto as the largest network by the summer of 2009.

About the Data

Facebook doesn’t publish membership numbers on the city or provincial level, so the data in this article was culled from various other sources. Because of the uncertainty of the data (low resolution, network counts versus facebook.com/advertising numbers) this casual analysis focuses more on trends than absolute numbers.

Historic provincial data: http://www.canadianmarketingblog.com/archives/2008/02/facebook_stats_primer.html
Historic city data: http://www.thoughtballoons.net/index.php/2008/04/28/one-year-look-facebook-growth-canada/
Historic Canadian data: http://blog.facebook.com/blog.php?post=2398302130
http://themeaningofweb.com/facebook-user-profile-canada-2008/
All current data: http://www.facebook.com/advertising
Population data: Wikipedia

Upcoming related articles:

• The social network adoption curve
• What’s powering your network: network utility functions