NS Census: population growth

[Ballotonia]

Using 22 nation XML feeds (covering all nations in NS), Ballotonian number analysts decided to have a closer look at the population growth of nations. Ancient myth is that this after starting at 5 million, it grows 5 days at 1 million, 5 days at 2 million, 5 days at 3 million, and a random 5-8 million after that. Turns out the myth is wrong!

In this analysis we're focusing on the low end of the population growth. Using the data mentioned above, we have roughly 20000 data points to calculate with, tracking the growth of new nations. Yes, that does indeed mean there are lots of new nations each day. Between 597 and 1584 for the data samples used.

Here are the results:

Day 1: start at population: 5 million
Days 2-6: 5 days growth of 1 million: 6, 7, 8, 9, 10,
Days 7-11: 5 days growth of 2 million: 12, 14, 16, 18, 20,
Days 12-15: 4 days growth of 3 million: 23, 26, 29, 32
Days 16-17: 2 days growth at 4 million: 36, 40
Days 18-19: 2 days growth at 5 million: 45, 50
Days 20-21: 2 days growth at 6 million: 56, 62
Days 22-23: random growth of 5, 6, or 7 million per day, with even chance.

Distribution population on day 22:
67 million: 7227/21105 = 34.2%
68 million: 6817/21105 = 32.3%
69 million: 7061/21105 = 33.5%
This is consistent with each three possibilities for population growth having an even chance (33.3%).

Distribution population on day 23:
72 million: 2356/20947 = 11.2%
73 million: 4576/20947 = 21.8%
74 million: 7039/20947 = 33.6%
75 million: 4660/20947 = 22.2%
76 million: 2316/20947 = 11.1%
This is consistent with each three possibilities having once again an even chance (33.3%). Due to the spread of the prior day being already present, we now have 5 possibilities.

Beyond that it gets difficult to discern easily, since the population spreads of each day start overlapping. Figuring out whether or not this growth rate sustains itself beyond this point requires a more thorough crunching of the numbers, where many individual nations must be tracked. I hope to get back to this question soon.

Ballotonia

[Lordieth]

Interesting stuff.

I have a question though; when we get to Day 22-23 and beyond, then we move to random number distribution territory, and I imagine the Perl code uses pseudo random generation correct?

So in theory, shouldn't future growth become more predictable, assuming uniform distribution?

Edit; I meant uniform, not linear.

[Ballotonia]

Interesting stuff.

I have a question though; when we get to Day 22-23 and beyond, then we move to random number distribution territory, and I imagine the Perl code uses pseudo random generation correct?

So in theory, shouldn't future growth become more predictable, assuming linear distribution?

Most likely each next step of growth is random, independent of the next day. If the random selection of 5-7 million growth continues forever onward (and let's be honest: I've seen nothing whatsoever to contradict this, I just haven't verified it yet), that would mean that the growth of an 18 billion population nation is exactly as uncertain as that of a nation on day 22.

Perhaps you're thinking of a percentage difference? The percentage different between the actual population and the expected value based on the mean growth rate will decrease (due to the law of large numbers) as a nation grows older, while the absolute spread of the population values will become larger. The day 23 curve is a pyramid, and as time goes by more independent random values are added and the spread becomes more closer to a bell-curve. The absolute bell-curve becomes wider and flatter as the nation grows older. Only when described as a percentage of mean it becomes smaller. See the graph on that Wiki page: the average of all die rolls goes towards 3.5 as more dice are rolled. But the possible range of the SUM of all die roles will only increase as more dice are rolled.

Ballotonia

[Oliver the Mediocre]

Whatever invective we might hurl at each other over gameplay issues, I do enjoy watching you geek-out statistically. Keep it up, this is awesome. ^_^

[Luna Amore]

I've recorded the total growth of my 15 largest nations over the last 86 days (so far), and the game actually seems to average higher than what I expected. I'd figure it would average out to 6.5 million per day. For that 86 day period, 13/15 were above 6.5, with the highest being 6.74, and the two that were below were at 6.43. It could just be that my sample of nations is too small,86 days is too short, or I could just be misunderstanding something(:p), but, by those numbers, the NationStates average daily population growth rate is 6.585.

Sorry to go on a tangent, saw Lordieth's post, so I thought I'd post that. Nice work with the numbers, Ballotonia.

*edit- Please, ignore the man with the faulty numbers*

[Lordieth]

Interesting Luna! I wonder what the average would be over a far longer period? My earlier point was in regards to random number generators not being 'truly random' as computers are procedural in nature and can't produce truly random numbers without an external source to provide them. What I am curious to know is; given enough time would the average of daily growth eventually come exactly between the lowest and highest growth rate, or does the code bias a higher or lower growth rate than an average produced truly randomly?

[Ballotonia]

I've recorded the total growth of my 15 largest nations over the last 86 days (so far), and the game actually seems to average higher than what I expected. I'd figure it would average out to 6.5 million per day. For that 86 day period, 13/15 were above 6.5, with the highest being 6.8, and the two that were below were at 6.43. It could just be that my sample of nations is too small,86 days is too short, or I could just be misunderstanding something(:p), but, by those numbers, the NationStates average daily population growth rate is over 6.6 (6.615).

Sorry to go on a tangent, saw Lordieth's post, so I thought I'd post that. Nice work with the numbers, Ballotonia.

6.8 ?!? That's an absurdly high growth rate. So far I'm looking at the overall statistics of all nations over a small time period, but your comment tells me I should be looking at something else too: the cumulative growth rate of nations over a long period as well.

BTW: I was planning to write an update today, but ran into some odd numbers for nations with > 16 billion in population (statistically significant lower growth rate than what would be expected). So, I'm gonna gather a bit more data and will have to get back to this in a more thorough manner later.

Ballotonia

[Lordieth]

I will await your results with interest Ballotonia, for the reasons I mentioned in my previous post.

[Luna Amore]

Your shock made me double check my numbers, B. Sure enough, the high average wasn't 6.8 but rather 6.74. I must have accidently merged the population gained (580 million) with average (6.74) while I was typing. Here's my numbers for the 15 nations for 86 days growth;

total population gained (millions)/average per day (rounded):

553/6.43
553/6.43
560/6.51
561/6.52
562/6.53
564/6.56
565/6.57
568/6.60
568/6.60
571/6.64
572/6.65
572/6.65
575/6.69
575/6.69
580/6.74

*Edit- Nothing to see here folks. Move along*

[Lordieth]

That's not what I expected at all Luna. I thought that as a population grew it would average out closer to 6.5 over time.

[Maltropia]

Quite a few members of my region joined the same day (December 19th, 2009), yet the populations vary by quite a bit: I have 3.16 billion, one has 3.169 billion, and another has 3.191 billion.
Just thought I'd throw those statistics out there.

[Bears Armed]

That's interesting.
Back when I tracked the growth for a whole bunch of my own nations that I'd created during a particular period, a few years ago, the increase seemed to go directly from the (5 days @ [+ 3 million per day]) stage to the randomised stage, without the stages that you've listed as

Days 16-17: 2 days growth at 4 million: 36, 40
Days 18-19: 2 days growth at 5 million: 45, 50
Days 20-21: 2 days growth at 6 million: 56, 62

So did I get it wrong, or have the Admins actually changed the process at some point since then?

[Ballotonia]

So did I get it wrong, or have the Admins actually changed the process at some point since then?

If you've kept your source data from back then, it could be checked. Or an Admin could weigh in, but sadly I do not have the data to be able to verify this one way or the other.

@Luna Amore: Could you please double check your '86 days' math? Cause looking at your numbers and mine I'd expect that to be closer to 94...

Ballotonia

[Luna Amore]

@Luna Amore: Could you please double check your '86 days' math? Cause looking at your numbers and mine I'd expect that to be closer to 94...

Ballotonia

I fouled it up. Should've double checked the day count. Instead of 86, it's 95, which I'd imagine produces averages closer to what you're seeing. Somewhere closer to 5.96.

Sorry for the bad data. Not a good day for me I suppose.

[Riemstagrad]

i was looking for that stuff earlier this year. (it's on the beneforum, ballo )

i found the following growth:

5,6,7,8,9,10
12,14,16,18,20
23,26,29,32
36,40
45,50
56,62

and from then 5-7 at random

wich is the same as ballotonia found.

i found this by comparing some data of puppet nations wich i rather randomly collected since august 2003.
in all these data, i NEVER saw nations leaving the 21 day fixed growth path neither saw i nations older then 22 days growing slower then 5 or faster then 7 per day.

EDIT: i have a large gap in the data from mid 2004 to end 2008. but from mid 2003 to mid 2004 and from end 2008 to present, things are the same.

i then calculated the average growth from day 22 to present for some (very old) nations who remembered their founding date and knew they never had their growth disturbed (population bug, update stealing/missing, cted) and i found averages from 5.90 to 5.98, but more in the 5.96-5.98 range... it's not significant, because i only did this for 4 nations.
then i searched for those nations on the wayback machine, and did the calculations again. when they were in their first year, they showed average growths of 5.56 to 5.80 (only for 2 nations, so that's far from significant)

[Bears Armed]

So did I get it wrong, or have the Admins actually changed the process at some point since then?

If you've kept your source data from back then, it could be checked. Or an Admin could weigh in, but sadly I do not have the data to be able to verify this one way or the other.

Alas, I copied it from the original discs onto a memory stick a couple of years ago but that stick has since failed and (although I should still have them somewhere] amongst my stuff) I can't now find the floppies again...

[Ballotonia]

Having promised to get back to this, here is my follow-up: I've decided to keep track of nation growth over the past week and a half (10 growth steps), to determine two things:
1. does the 5-7 range continue for all population ranges forward?
2. is each day's worth of growth independent of the other days?
The answer to both is yes.

The data samples covers all 67579 nations which all went through the major update 11 days in a row (June 8 through 18) and hence grew in population. Note that currently there are 88,909 nations. Many nations apparently miss an update due to (temporarily) dying off, while continuously being replaced with many new nations (see also the OP). All growth steps are in the 5-7 million range. Each day the distribution is even.

When looking over the entire period to observe the interdependency of growth steps, the following reasoning is used: If growth steps for each day are independent the expectation is that the cumulative growth over time for all nations will approach a bell-curve. If there is a dependency (like some nations always have a lower growth, others always higher) the bell-curve would be disturbed. It wouldn't be smooth, may even have multiple peaks, etc...

The observed 10 days growth results in this bell-curve graph:

X: growth in millions of populations, Y: number of occurrences in the data sample.

Ballotonia

[Romili]

i have gone up by 4 million after day 22 before, i was waiting to hit the 3 billion mark so I was looking at population closely at the time. Also I have heard that some have gone up by 3 million before.

[United German Regions]

I've recorded the total growth of my 15 largest nations over the last 86 days (so far), and the game actually seems to average higher than what I expected. I'd figure it would average out to 6.5 million per day. For that 86 day period, 13/15 were above 6.5, with the highest being 6.74, and the two that were below were at 6.43. It could just be that my sample of nations is too small,86 days is too short, or I could just be misunderstanding something(:p), but, by those numbers, the NationStates average daily population growth rate is 6.585.

Sorry to go on a tangent, saw Lordieth's post, so I thought I'd post that. Nice work with the numbers, Ballotonia.

*edit- Please, ignore the man with the faulty numbers*

[Frisbeeteria]

:lol2:

Not a horrible thread to bump, but please refrain from bumping 2+ year old threads with no content other than a smiley in future.

[United German Regions]

:lol2:

Not a horrible thread to bump, but please refrain from bumping 2+ year old threads with no content other than a smiley in future.

Ok

[Rifty]

So is discussion starting up for this...or do we just want it in the front page? xD

[Consular]

Not a horrible thread to bump, but please refrain from bumping 2+ year old threads with no content other than a smiley in future.

Ok

You effectively just bumped it again. D:

[Zenya]

Ok

You effectively just bumped it again. D:

Yeah! srsly stawp bumping D:

[Rifty]

You effectively just bumped it again. D:

Yeah! srsly stawp bumping D:

And now you do it - Gah~

Oh your sig is oversized; 10 lines for me when it can only be 8. Tisk tisk tisk. Might wanna get to that before moderation does