robert zubek / blog

(Edit: Welcome reddit readers. The following is a howto on setting up an embedded Jetty server in Clojure, and writing a minimal servlet that serves up dynamic content.)

Last week I discovered a very nice language named Clojure. It’s based on Lisp, but hosted on the Java platform, and running inside the JVM. It has some lovely features, like native support for Java collections, and a clean API that ditches a lot of the old ANSI CL library cruft.

But the best part for me was language interop: Clojure can effortlessly load and use any external Java library. And there is a ton of good ones out there, including a rich set that comes with the Java platform. (This may seem like a minor point, but most free Lisps lack good libraries; only the non-free Allegro CL has really extensive, cross-platform ones.)

So I decided to take the language for a spin, especially exercising the language interop bit. For example, by building a little web server to serve up some dynamic content. You know, web development 101 kind of stuff. This turns out to be really easy to do.

Since Clojure is backed by Java, we can use an existing server library to do the heavy lifting for us. Several high-quality ones are available - I went with Jetty, a servlet-based engine that’s really easy to embed in custom applications. Using a servlet container will take care of all the mundane bookkeeping for us.

In the following posting, I’m reproducing the steps required to get a Clojure-based web server up and running, in much less than 100 lines of code. Hope you find it interesting!

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Viral Coefficient and Growth

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