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A scientific approach to viral marketing

If you Google "viral marketing", you'll find a ton of results. The majority of the websites, though, deal with the qualitative aspects. Strangely enough, even Wikipedia's page, while it is very insightful and interesting, does not report the quantitative definition of viral (as of November 2011).

What is viral?

A viral message is supposed to spread from person to person like a disease. Suppose that a random person, called John, is the first who gets a new type of flu this winter. Betwen the time when he gets it and the time when he's healed, he's likely to pass it on to other people, who will then pass it on to others, etc...

Numbers now: if 10 people have the flu and, on average each of them passes it on to 0.8 people, there will be only 8 more people with it. Then, from these 8, there will be on average 6.4 people. Less and less. The flu will slowly die out. If the average is 1.2 rather than 0.8, from the initial 10 there will be 12, then 14.4, etc... More and more people. No matter how small the "seed" was initially, the flu will spread.

We are now ready for the mathematical definition of strictly viral marketing: a product or message can be defined viral if, on average, everyone who gets it passes it around to at least one more person. I wrote "strictly" because the term viral is often used for slightly different situations as well (see the last sections of this article)

The mathematics of viral

Let's define R for a given message this way: a recipient of that message passes it on to R people on average. Then, the definition of viral is:

R ≥ 1

In epidemiology, R is called "basic reproduction number" or "basic reproductive rate" or "basic reproductive ratio".

Let's now decompose R into factors. Let's suppose that a person, which we'll call Jack, sees a post on Google Plus about your website:

Let O (for Opening) be the probability that Jack clicks on it and visits your website.
On the landing page, Jack might like it and stay, or bounce away. Let's call the probability that he likes it L (for Like).
Jack can decide to post about your website on Google Plus. Let's call the probability P (for Posting).
Then a number, which we call N, of his friends will see the message about your website.

The loop is now closed: we went from Jack seeing the post to the average number of new people seeing the post. We can rewrite the definition of viral:

O x L x P x N ≥ 1

Optimizing the "virality" of your message: where Plus One To Download helps

Let's look at each factor in the expression above:

O: guess what? The title, thumbnail and description in Google Plus posts need to look nice! Obvious. More interestingly, you can control the look of the posts about your website by optimizing the <head> section of your homepage. In terms of value, this number is usually few percent points (or even less than 1%)
L: make sure your landing page is clear and easy to navigate. The bounce rate for a website usually ranges from 20% to 60%
P: Plus One To Download dramatically helps you here, because it motivates your visitors to post on Google Plus about your website. If your "reward content" is good, you can get close to 100%
N: this number is close to the number of friends the average person has on Google Plus, unless a user decides to only share with a set of people in their circles. Plus One To Download helps you here too, because it asks users to share with "Public", that is with everyone. The number of friends someone has on Google Plus can be anything from a few to hundreds or thousands

What happens if I don't reach the viral threshold?

As explained on Harvard Business Review, achieving an R ≥ 1 is extremely challenging. However, even if your R is below 1 your viral efforts help you increase the visibility of your website. The average user will still forward your message to others. It can be mathematically proven that your message will reach S / ( 1 - R ) people, where S is the number of people who got it in the first place (seed). In other words, while with a strictly viral message (R ≥ 1) even a very small seed will bring your message to the World, with a "close to viral" message, you still have a multiplication effect. Suppose that you're getting 100,000 hits per day from organic search and you set up a Plus One To Download page with some content your visitors want. Suppose that your R is 0.8 (lower than 1, than not really viral). As per the formula above, you'll reach 100,000 / ( 1 - 0.8 ), that is 500,000 people instead of the initial 100,000, which is a great result.

Approximations used in this description

Everytime math is applied to a real life problem, the discussion is not over without a word about the approximations and simplifications that have been made. In this case, the most relevant one is that we have neglected non-linear phenomena, such as:

Once a person has already received the message, she/he is out of the game. Just like someone who got immunized to the flu and will not contribute anymore to spreading it around.
When you hear a message about a website from a friend, you can open it or not. If you don't open it, but you hear about it from another friend too, the probability that you open it is a lot higher the second time

However, the big shots in the universities don't consider these elements either, then I guess we're safe