The viral coefficient is a measure of how many new users are brought in by each existing user. It’s a quick and easy way to measure growth: if the coefficient is 1.0, the site grows linearly, and if it’s less than that, it will slow down. And if the coefficent is higher than 1.0, you have superlinear growth of a runaway hit.

In an invite-only situation (e.g. gmail closed beta), it’s easiest to calculate this directly, based on how many people are being invited by each new user, and how many of the invitees create new accounts themselves. The viral coefficient is simply:

v = new user invites accepted / new users
   = accepts_t / δp_t-1

where δp_t denote the number of new users who join in time slice t (ie, the increase in population between p_t-1 and p_t). 

But most sites have an open account creation policy. For those, we’ll have to estimate population acceleration from raw population deltas. Let’s assume that each accepted invite is quivalent to creating a new account at the next time slice. Then we can estimate virability as:

v ≈ δp_t / δp_t-1
   = (p_t – p_t-1) / (p_t-1 – p_t-2)

which is an acceleration metric, easy to compute from historical data. 
 

Population Forecasting 

Viral estimate calculated as momentary acceleration will fluctuate over time. But we can use it for some short term forecasting.

To calculate expected future population p_t some t steps in the future, given the viral coefficient and present population p₀, we first invert the above:

δp_t = v δp_t-1 = … = v^t δp₀

and plug this right back in:

p_t = δp_t + p_t-1
    = Σ_k≤t  δp_k + p₀
    = Σ_k≤t  v^k δp₀ + p₀

This describes a geometric series. When v ≠ 1, p_t  = δp₀ (1 – v^t+1) / (1 – v) + p₀