Wikipedia:Reference desk/Archives/Mathematics/2011 May 21

= May 21 =

Dead people
Time for a bit of fun....

Over on the Humanities desk User:Dismas has just asked "What percentage of the world's population dies every day? I'm just wondering if the 3% figure that the 2011 end times prediction article states is close to the daily average."

Go for it. HiLo48 (talk) 01:44, 21 May 2011 (UTC)


 * My own contribution, which I acknowledge comprised "... incredibility rough calculations involving several outrageously simplistic assumptions...", was...


 * Assume 50 years average life span. So, 100% of those people die in 50 years. That means 2% die in one YEAR. So in one day the percentage is 2% divided by 365, approximately 0.005%. HiLo48 (talk) 01:51, 21 May 2011 (UTC)
 * According to mortality rate, "as of July 2009 the crude death rate for the whole world is about 8.37 per 1000 per year according to the current CIA World Factbook." So more like 0.002%. Algebraist 09:54, 21 May 2011 (UTC)

Inequality proof
How do you show that $$4xy-6(x^2+y^2)-4(x^3+y^3)-x^4-y^4 \le 0$$ for all real x and y? Widener (talk) 05:02, 21 May 2011 (UTC)
 * Rewrite as $$-(x^2+2x)^2-(y^2+2y)^2-2(x-y)^2 \le 0$$--RDBury (talk) 06:38, 21 May 2011 (UTC)
 * Thanks. I can not see how to solve these types of questions without already knowing the solution. Could you provide some tips on how to solve these problems generally? Widener (talk) 07:37, 21 May 2011 (UTC)
 * I more or less just used trial and error. You might want to look at Hilbert's seventeenth problem and related articles though.--RDBury (talk) 11:13, 21 May 2011 (UTC)

statistics/probability of getting to an article via patterns?
I recently saw something posted on a forum that stated you could click on Special:Random and follow a certain pattern to always arrive at a certain article. The instructions were: from a random page, click the first link that is not in parentheses, if you continue to do so, you will arrive at the Philosophy article. Being naturally curious, I decided to try it. Surprisingly, my first 3 tries actually worked (assuming I didn't mis-click something, link trails below.)

Dilson Torres > Major League Baseball > Baseball > Bat-and-ball games > Playing field > Grass > Monocotyledon > Flowering plant > Embryophyte > Plant > Life > Physical body > Physics > Natural science > Knowledge > Philosophy

KLWO > Radio broadcasting > Radio waves > Electromagnetic radiation > Energy > Physics > Natural science > Knowledge > Philosophy

Socialist state > Socialism > Economic system > Outline of industrial organization > Microeconomics > Economics > Social sciences > Scholarly method > World > Human > Science > Knowledge > Philosophy

While obviously this can't hold true for every page, is there any way to reasonably calculate the statistics or probability of this happening? Avic ennasis  @ 09:28, 17 Iyar 5771 / 21 May 2011 (UTC)
 * To clarify the instructions (going by your examples), you follow the first link in the main text (excluding the interface, but also DAB links). This is an interesting phenomenon! Special:Random took me to Leslie Goodman, and from there it was a roller-coaster ride (I was at Meaning (philosophy of language), but was taken out as far again as world before finally settling in). But thinking about it as a rewrite system, there only are three possibilities. Either you end at an article without any outgoing links, you go in circles, or you end at philosophy. Since Wikipedia is finite (if growing), the only way of getting an infinite derivation is a loop. Since definitions tend to define things in more general terms, loops will be rare. And articles without outgoing links are also rare, and tend to be for low-interest, specialized topics and hence unlikely to be linked to in a definition. But I have no idea how to quantify this. Note also that philosophy is not unique with this property - it certainly also holds for Reason and Rationality (which do form a loop with philosophy). We can probably deduce that philosophy, reason, and rationality are the foundation of our worldview... --Stephan Schulz (talk) 09:48, 21 May 2011 (UTC)
 * The first link in an article is likely to be to a more general subject. So the chain will usually lead to more and more general subjects. The most general subjects are things like science and knowledge which link to Philosophy. Using a different scheme which avoids increasing generality will most likely bounce around more or less at random, though since the number of articles is finite it will eventually terminate (no links) or start to cycle.--RDBury (talk) 10:56, 21 May 2011 (UTC)


 * Thanks for the additional instructions - I forgot that part. :) It's very interesting, I think. I've also seen other variants in my time - another one I remember is the six degrees of Hitler, where from a random page, you can usually get to the article on Adolf Hitler within 6 clicks. I did find a few pages that did this, but also some that didn't. The pattern above though seems a lot more amazing to me - that a chain 16 links long from a random article could be predicted to reach a specific article. I'm not handy with math by any means, which perhaps adds to my wonder. Avic ennasis  @ 11:02, 17 Iyar 5771 / 21 May 2011 (UTC)
 * As for your wonder: This is a pretty amazing thing, and some of the math behind it is still fairly new. The general mathematical structure behind this phenomenon is the topology / network topology of the wikipedia link network. Our article on small-world networks claims WP has the small-world property. The claim is totally plausible, though un-cited, and if true it explains these short paths to philosophy as an expected consequent of the gross network properties. This question is also closely related to kainaw's question below, because a few key articles are 'nearly' roots of the network. SemanticMantis (talk) 19:21, 21 May 2011 (UTC)
 * The topology will be a Directed graph with the additional property that each node has exactly one node leading from it. The graph cannot be a directed acyclic graph. Interesting question how many links does it take to get from Philosophy back to Philosophy? So there is a least one cycle, are there any others? I can imagine there may be case there two articles both link to each other.--Salix (talk): 23:55, 21 May 2011 (UTC)
 * Quite short, actually: Philosophy > Reason > Rationality > Philosophy. Heh. Very interesting. "All roads lead to Rome Philosophy." :) Avic ennasis  @ 01:11, 18 Iyar 5771 / 22 May 2011 (UTC)

I've done some experiments with random graphs. The graph consists of a number of nodes and each node has one link to another node (not including itself). Its relatively easy to calculate how many Connected component the graph has. On a run with 100 graphs of a million nodes each I got the following frequencies - (number of components: frequency) 1: 1, 2: 1, 3: 5, 4: 16, 5: 11, 6: 17, 7: 16, 8: 7, 9: 13, 10: 8, 11: 2, 12: 1, 13: 1, 14: 1. So it looks like the mean number of components is about 7. I guess wikipedia is far from a random graph as there are some articles with very large numbers of incoming links mathematics names of countries etc.--Salix (talk): 06:44, 22 May 2011 (UTC)
 * The graph is actually a "maximal directed pseudoforest" seems like some work has been done studying these.--Salix (talk): 17:37, 22 May 2011 (UTC)


 * To analyse this kind of stuff you need to read about Markov chains and Markov matrices. These are the kind of techniques that Google use, which is in a very loose sense like a web-wide version of this game. In the language of Markov chains, being on an article page would constitute being in a state. Then you click on the first link, meaning that the probability of moving from that state (article) to the other state (article) is 1; while it is zero for all other states (articles). Given that we have a finite number of articles at any one time, it would be possible to compute the probabilities (although we might have to do it for a fixed starting article and not a random one). In fact, because we would stop the game once we found the philosophy article, we have a terminating Markov chain, and we could calculate an average, expected number of clicks between starting on a given article, and getting to Philosophy. One spanner in the works would be articles that had no links at all. — Fly by Night  ( talk )  23:21, 22 May 2011 (UTC)

Thanks to all for your responses. :-) On a minor note, I found that the most recent XKCD comic has the alt-text: "Wikipedia trivia: if you take any article, click on the first link in the article text not in parentheses or italics, and then repeat, you will eventually end up at Philosophy." Avic ennasis   @ 08:30, 21 Iyar 5771 / 25 May 2011 (UTC)
 * At least one other loop Skeletal striated muscle -> Striated muscle tissue. --Salix (talk): 17:00, 25 May 2011 (UTC)

Data set/graph terminology
(Background) I have many data sets that I am trying to describe. Each set contains multiple "transactions". A transaction contains three elements: user, resource, and time. A user selects or uses a resource at a specific time. I decided to plot these sets by treating the resources as nodes and placing an edge between nodes to indicate that a user selected one resource and then selected another one - so the edges indicate the order in which the resources are selected by the users. Now, the terminology issue:

(Question) Given a graph of nodes and edges, some of my graphs show that from just about any node you pick, there will be a short path to a very popular node. It appears that all traffic converges on a single (or very few) node. I call this "convergent" - but I'd like to know the proper term. Another thing I've found is that some graphs have multiple sequential paths that run parallel with one another. The paths pretty much go from node to node to node. The paths have very few edges between them. I call this "sequential" - but I'd like to know the proper term. Also, I would like to know what the proper term is for absence of what I've called "convergent" and "sequential". I'm calling that "random" right now. -- k a i n a w &trade; 17:20, 21 May 2011 (UTC)

To help explain this better, I added a picture of the three types of graphs to http://kainaw.com/tmp/graphs.png -- k a i n a w &trade; 01:53, 22 May 2011 (UTC)


 * This is difficult because it seems as though you want some 'fuzziness' in the definitions (i.e. converging to one OR 'very few' nodes.) Thus, I am not confident that there are any widely used, well-defined terms for what you describe. In any case, the best I can think of is that each of your 'convergent' nodes can probably be defined in terms of being root nodes of some particular spanning tree of the total network. HTH, SemanticMantis (talk) 19:28, 21 May 2011 (UTC)


 * For the convergent case, you could call that a "hub and spoke" system. StuRat (talk) 05:26, 22 May 2011 (UTC)


 * Right, or a star network. Note that the top image at hub and spoke is inconsistent with the description. All the verbiage considers only a single hub (e.g. n-1 edges to connect n nodes). So, we have a few good words for the single-hub case, but I don't think either of these terms is properly applied to a graph with, say 100 nodes and 3 hubs, which (as I read it) kainaw does want to include as a 'convergent' graph. SemanticMantis (talk) 16:25, 22 May 2011 (UTC)


 * He said "all traffic converges on a single (or very few) node". I take that to mean that multiple hubs are allowed. StuRat (talk) 17:50, 22 May 2011 (UTC)


 * Correct. I'm not creating the graphs to meet the description. I'm trying to describe graphs of actual user data. I want to use proper terminology when saying "For data set A, the traffic converges on two resources, but in data set B the traffic does not converge to any notable degree." -- k a i n a w &trade; 18:38, 22 May 2011 (UTC)