There seems to be a little feeding frenzy going on regarding how much cash Twitter burns. After reading Paul Boulin’s original post, I was going to write up some thoughts. I see, now, that the conversation has progressed with Paul posting some details of the feedback he’s been getting. However, there is still something bugging me about the write up. It’s this bit:

Finally, the biggest problem of all is the network effect of Twitter’s popularity. As the number of people using a network goes up, the number of connections between them rises much, much faster. Most Twitter users now have several times more followers than they did a year ago. As a result, every update sent to all of a user's followers costs the company more to send to everyone who’s subscribed by phone.

It’s an exponential function: The more people use Twitter, the faster the number of messages grows.

Let’s imagine you have 4 people in a network. How many connections are there between these people? Each person can have at most 3 connections, so we can approximate the max size of the connections as 4^{2}. In general, the max number of links in a network of size N is approximated by N^{2}.

The word ‘exponential’, like the word ‘literally’ seems to be losing its meaning. Backing its invocation with a link to Wikipedia doesn’t really help.

[If you are interested, the term ‘literally’ is used to remove the metaphorical or idiomatic meaning from a phrase. Perhaps the best comment of this is a search on Google, which appears to turn up more articles about the word’s misuse than anything else!]

True. Links to wikipedia don't help a lot here :)

But still, exponential often means, by extension, "growing faster than n"...

Which is a true problem for twitter.

If you consider that the number of messages you send every day is also growing with n (because the more contacts you have the more you are interested in sending them updates), it comes to n^3 right?

:)

Posted by: Nico | January 28, 2009 at 10:31 AM

The growth of the Twitter is very likely to be exponential, but the explanation is wrong. Moreover, in a realistic world, users are poorly interconnected. Usually, there is a bunch of leaders whose auditorium grows constantly and the rest of users, who have a couple of friends forever. Therefore, the traffic will grow almost linearly with the number of users. On the flip side is the total number of users AND the number of posts that grow exponentially.

Posted by: Itman | January 29, 2009 at 02:42 PM

Indeed, this isn't the first place I've seen this error.

I'm afraid that in the popular usage exponential is going to be used to mean anything that's substantially more than O(n) (probably doesn't include O(n log n) but does include O(n^2)).

Even before this, most "exponential" growth was probably just the exponential part of a logistic growth curve anyway.

Maybe it's time to create a new word that means "literally" exponential?

Posted by: Brad | February 03, 2009 at 05:23 PM

Weighing in my (mis) understanding of exponential.

In my naivity, I had thought that linear growth was O(n), that geometric was

O(n^^a) and that exponential growth was O(a^^n).

However, I don't think that is correct any more :-(. So what is growth of the form O(a^^n) called?

(I am taking ^^ to mean "raised to the power of" in the above)

Posted by: Chris Bird | February 15, 2009 at 10:40 AM