Wikipedia:Reference desk/Archives/Mathematics/2008 May 23

= May 23 =

wikipedia and power laws
I know that if you consider a network model of Wikipedia (in which nodes are pages, edges are links), then that network is (or is close to being) scale-free. That is to say, the network degree follows a power law. But I can't find any reference that says what the slope of the power law relationship is (the variable k in the power law article). Anyone have any thoughts? AndrewGNF (talk) 01:49, 23 May 2008 (UTC)
 * Well, let's gather some data. I clicked "random article" 20 times, and got articles with the following number of blue links (not counting boiler-plate text):
 * 2, 3, 5, 5, 6, 6, 7, 7, 10, 11, 14, 17, 20, 27, 28, 30, 33, 41, 44, 55
 * These are outgoing links, not incoming links (since Wikipedia is a directed graph). Among articles with at least one link, the probability that an article has exactly one link is $$\frac 1{\zeta(\gamma)}$$ (where $$\gamma$$ is the parameter in the scale-free network article). At $$\gamma \geq 2$$, we have $$\frac 1{\zeta(\gamma)} \geq \frac 6{\pi^2} = 0.607927 $$, so at least a 61% chance of an article with exactly one link -- and since even in 20 random articles, there are no articles with exactly one link, we can be fairly sure that if Wikipedia is scale-free, then $$\gamma$$ is less than 2 -- probably very close to 1.  Someone with more statistics knowledge (or more data) could probably conjure up a reasonable estimate for $$\gamma$$.  Eric.  217.42.199.10 (talk)  —Preceding comment was added at 10:50, 23 May 2008 (UTC)

Variation in statistics
I have a few questions regarding statistics and my textbook (Stirzaker) isn't helping.

Say you have a set of data that shows daily price movement in some commodity over ten years, with 3652 or so prices (one for each day).

Given that it is normally distributed, I understand that once you calculate the daily variance over N days (σ^2_N, that is sigma squared subscript N), you can find its 95% confidence interval: σ^2 ∈ $$\left [ \frac{\sigma^2_N}{1+1.96\sqrt{\frac{2}{N-1}}},\frac{\sigma^2_N}{1-1.96\sqrt{\frac{2}{N-1}}}\right ]$$

where N is the number of days (which in this data set would be 3652), and σ^2 is the daily variance.

(apologies for not using the html code but it 'fails to parse' the σ symbol)

I understand that you could find the yearly variance by multiplying the daily variance by 365 (and the annual standard deviation or 'volatility' by multiplying the daily standard deviation by sqrt(365).

But if you wanted to find the 95% confidence interval for the annual variance, how would you change the formula? Could you just replace σ^2 with 365*σ^2, and N=3652 with N=(3652/365)? In which case the N would be the number of years (10)? Because when I do that I end up with an awfully wide interval. Or would you let N=365, the number of days in one year?

I'm interested in seeing whether or not the annual volatility changes over time. I was thinking that once you had found the confidence interval of the annual variance, one could split the data set into smaller intervals, calculate the annual variance of each interval (by multiplying the variance of each interval by 365), and see if 95% of the resulting set of annual variances falls within the confidence interval. Would this be a valid method of tracking changes in annual volatility? —Preceding unsigned comment added by Damian Eldridge (talk • contribs) 11:30, 23 May 2008 (UTC)
 * I TeXed your formula for readability. Algebraist 11:40, 23 May 2008 (UTC)

Counting problem
In how many ways can 15 students be seated in a row such that the 2 most talkative children never sit together? I've been given the answer 14! * 13, but I don't compute this. Seems like it should be 13! ways for every two positions that are not adjacent. 70.250.149.151 (talk) 19:27, 23 May 2008 (UTC)john


 * Well, if we considered it as a circle, then we place the most talkative student first (15 options). That leaves 12 places for the 2nd most talkative (can't use the seat taken, or the 2 on either side). After that you have 13 seats available which can be used in any order, so here the answer is 15.12.13! = 1.12E12.
 * In a row, things are more complicated - there are two cases to consider. First suppose that the most talkative student doesn't sit at the end (13 options). Then there are again 12 possible places for #2 to sit, and then we arrange the other 13 students as we like, so 13.12.13!. If we sit the most talkative student at the end (2 options), then there are 13 places for #2, and then 13! ways to arrange the rest, so a tital of 2.13.13! Putting this together, we have 13!(13.12 + 2.13) = 14.13.13! = 13.14! as required. Nice problem. -mattbuck (Talk) 19:47, 23 May 2008 (UTC)


 * Another way to see it: there's a one-to-one correspondence between pairs of nonadjacent seats chosen from a row of N and arbitrary pairs of seats chosen from a row of N−1 (just add/remove a seat between the chosen seats). So there are 14·13 choices for the two most talkative students and then 13! for the rest. -- BenRG (talk) 21:06, 23 May 2008 (UTC)


 * Instead you might want to consider the way to seat the two most talkative together. So making them a block of 2 which can be arrange 2 (or 2! ways) you have 14 items, 13 children and the two most talkitive together. So these are arranged 14!*2 ways. Then you can just minus this from the total 15! ways or arranging. I think this is a more simple way of getting your 13*14! answer. Hope this helps Rambo's Revenge (talk)  07:41, 24 May 2008 (UTC)


 * This is the lovely thing about combinatorics - no matter how clever you think you're being, there's always someone else who can find an easier way. That and it deals with proper numbers. But then, I like complex analysis so who am I to talk? -mattbuck (Talk) 11:03, 24 May 2008 (UTC)

What is the formula for exponential growth/decay?
What is the forumla for exponential growth/decay?  Marlith  (Talk)   23:56, 23 May 2008 (UTC)
 * See exponential growth. Paragon12321 (talk) 01:32, 24 May 2008 (UTC)