It seems that the primary objective of the Lucretius, ver. 21c blog is to compile statistics on the twitter accounts of the top physicists, philosophers, and astronomers.

Occasionally, there are posts that show the growth of different accounts. In particular, I like this short analysis.

When I look at this, I have two thoughts (yes, just two). First, growth rate over this time scale is surprisingly linear over the given time frame. Second, it seems like Twitter users with more followers have a greater follower increase rate. Oh, also notice that followers mostly just increase and not decrease. Yes, that could be considered my third thought.

How about I make a plot of the follower rate increase vs. number of followers to see if there is some type of relationship? I will use the data from the graphs on the Lucretius, ver. 21c blog instead of gathering my own data (since it's already there).

First, there are two users without graphs. @ProfBrianCox and @michiokaku both have over 100k followers but they are the only ones. They have listed growth rates of 920 and 260 followers per day. Brian Cox currently has 1.34 million followers and Michio Kaku has 235K followers (as of March 7 2014). If I assume a constant follower rate, then I can estimate the number of followers they had at the date of the analysis (December 20).

Now for an analysis. I first recorded the number of followers at both the beginning of the time period and the end. From this, I can get the average follower rate as (I will call follower rate r):

There were a few users that didn't have follower data until after the start of the time period. I have removed these from my analysis except for @veritasium (because he has a large number of followers). But what I want is a plot of rate (r) vs. followers (f). However, if the follower number is changing, which number do I use? How about the follower number in the middle of the time interval given on Lucretius21, ver. 21c? This seems to be 42 weeks of data, so the midpoint will be at 21 weeks.

Let me call the beginning date t 1 , the midpoint t 2 and the last date t 3 . For the users that I have starting and ending followers, the followers at the midpoint would be:

For the two super users, I will just use the rate and the final follower number to estimate the follower number at the midpoint in time.

Now for the data. Since I don't know that this will be a linear relationship, I will plot the natural log of follower rate vs. the natural log of followers.

Here I fit a linear function to the log of these two variables. This gives a slope of 1.104 and an intercept of -7.810. Yes, I left off the units. Why? Because I am cheating. You can't actually take the natural log of some value with units so I divided each by some constant to get rid of the units. Just calm down. Everything will work ok in the end.

So now I can write the following relationship between f and r with m as the slope and b as the intercept.

There you have it. There is my relationship between the follower rate and the number of followers. Maybe I should write this with my values of m and b.

Let's just check this function a little bit. What if I have zero followers? In that case, f = 0 so r would also be zero. That sort of makes sense, doesn't it? If no one is following me how would other people find out about me in order to follow?

What is my follower rate? Well, right now I have 4,317 followers (but you can increase this by following me @rjallain). Using the model above, my follower rate increase should be 4.18 followers per day. That seems a bit higher than what I have noticed, but not crazy.

Notice something else - everyone would have a positive increase in followers. This of course isn't always true. Users sometimes lose followers. However, overall I think it is fine. Usually users grow in users.

When will I have one million followers? —————————————

Why make a model if you can't use it for something? I want to calculate how many days it will take for me to one MILLION followers. Why? Well, I have heard that if you have that many followers you will have great powers. I want those powers. Don't worry, I will only use my powers for good and not evil (but I might make a new definition for "good").

If I stayed at my current follower rate of 4 followers a day, it would take about 250,000 days to get to one million followers. That's a pretty long time. However, it shouldn't be quite that long. As I get more followers, my follower rate should also increase.

But how do you calculate this? It's not such an easy problem since the follower rate depends on the followers and the followers over time depends on the follower rate. Sure you could set up a differential equation - it might even be solvable. For this case, I will just use a numerical method.

Here is my code. It's fairly simple.

That's all it takes. Maybe you want a visual representation of the output. Here is a plot of the projected followers as a function of day.

That's just 4295 days from now (under 12 years). I guess I can wait. Of course, I suspect over the next 12 years, twitter will change. Maybe it will be no longer in use. Maybe more people will start to use it such that my parameters for the follower rate would change.

Oh, if I started with one million followers I would also increase in followers over this same 12 year period. How many would I have at the end? 9.5 x 1011 followers. That's more than the population of the whole world. This can only mean one thing. Within 12 years we will discover life beyond Earth. This extraterrestrial life will use twitter.

I guess I should end with some practical advice. Everyone wants more twitter followers, right? Well, here is the tip:

The best way to increase the rate that you gain more followers is to have more followers.

Simple, right?