Wikipedia:Reference desk/Mathematics and Talk:Scale-free network: Difference between pages

Add links
From Wikipedia, the free encyclopedia
< Wikipedia:Reference desk
(Difference between pages)
Content deleted Content added
VisualWikitext
 
SineBot (talk | contribs)
Bots
2,546,196 edits
m Dating comment by Kmote - "→‎The name "scale free": "
 
Line 1: Line 1:
I opened this discussion page because I have seen many active flip-flop of versions in a fairly short period recently. It seems that some like the Lun Li, et al. paper and definition, but some don't. I like the frankness of the paper although it may appear as "opening fire" to many recent papers contributed in the last six years. I think '''critique''' is good to healthy science and helps us in knowing the truth. The current version [http://en.wikipedia.org/w/index.php?title=Scale-free_network&oldid=21315826] seems to put the Lun Li, et al. paper in a weaker position in the exchange of ideas about the science of scale-free network, which I oppose.
[[Category:Non-talk pages that are automatically signed]]{{Wikipedia:Reference desk/header|WP:RD/MA}}


{{Wikipedia:Reference_desk/Archives/Mathematics/2008 October 7}}


:I agree, I'm putting it back in especially considering the edit was made by an unregistered user. Also to answer the question below, the formal definition captures the notion of self-similarity, while Barabasi's definition really doesn't (From my understanding he basically just picked the name because it sounded good and would catch people's attention. Presumably he had some ideas about self-similarity but I haven't seen him talk about it) [[User:Meekohi|Meekohi]] 13:56, 12 December 2005 (UTC)
{{Wikipedia:Reference_desk/Archives/Mathematics/2008 October 8}}


:I consider that paper too scandalous and inaccurate. E.g. HOT model obviously has more dimensions than SF model. It is just a more specific case. You can't say that "horses are better than mammals" or "horses are faster than mammals". It is... well... a bit incorrect. On the formal definition. Just give me formal definition of a cat first. OK, the topic of degree correlations is actively studied, so it will settle to something in some years... let's consider that paper a "critique". [[User:Gritzko|Victor Grishchenko]]
{{Wikipedia:Reference_desk/Archives/Mathematics/2008 October 9}}


== The name "scale free" ==
= October 10 =


This is my first time using the talk page, so please forgive any missed wikiquette.
== The real deal with fitting lines to observed data points ==


The article was most helpful to me, however I am curious about one thing. Why are these networks called "scale free"? I assume that this is a reference to self-similarity, but what exactly is self-similar in a scale-free network? The sample diagrams of networks I have seen are not obviously fractal.
I have a set of (x, y) data points with corresponding experimental uncertanties (dx, dy) which are different for the different points. I want to estimate the slope of the line that best approximates them. I want an estimation method with the following properties:


David M.
# It should take into account the errors in both x and y (unlike linear regression).
# It should be symmetric under the exchange of x and y (unlike linear regression).
# The method should be scale-invariant; that is, even if x and y have different units, the best-fit line should not change if they are measured in different scales (unlike [[total least squares]]).
# More accurate points (those with smaller uncertainties) should contribute more to the fit.


:scale-free is not self-similar. Scale-free (as I understand it) is that there is no meaningful "average" for the behavior of the system. [[User:ChaTo|ChaTo]] 17:19, 12 December 2005 (UTC)
This doesn't seem like much to ask, but I can't find any method that actually satisfies these.


::For the old definition of Scale-free this was true, but the Lun Li definition actually includes the concept of self-similarity. [[User:Meekohi|Meekohi]] 15:34, 13 December 2005 (UTC)
Something called "diagonal regression", "geometric mean functional relationship", "least areas line", "least products regression", "line of organic correlation", "neutral data fitting", or "reduced major axis" (do we have an article on it?) seems close, but I need a weighted version of it to satisfy the last criterion. —[[User:Keenan Pepper|Keenan Pepper]] 06:54, 10 October 2008 (UTC)
:I numbered your points. Point 3 is satified even by linear regression. To allow for point 2, try minimize Σz<sub>i</sub><sup>2</sup> where z<sub>i</sub>=f(x<sub>i</sub>,y<sub>i</sub>)=ax<sub>i</sub>+by<sub>i</sub>+c and a<sup>2</sup>+b<sup>2</sup>=1. This is symmetrical in x and y. It provides the same result as linear regression, however, unless the line is vertical. Take account of point 1 = point 4 by minimizing a weighted sum Σa<sub>i</sub>z<sub>i</sub><sup>2</sup> where a<sub>i</sub> is the weigth of the (x<sub>i</sub>,y<sub>i</sub>). [[User:Bo Jacoby|Bo Jacoby]] ([[User talk:Bo Jacoby|talk]]) 08:02, 10 October 2008 (UTC).


:As I understand it, the term "scale-free" is closely related to the term "[[scale-invariance|scale-invariant]]", which is perhaps a more intuitive phrase. It means (roughly) that regardless of how much you "zoom in or out", the structure is going to look essentially the same. [[User:Kmote|Kmote]] ([[User talk:Kmote|talk]]) 21:05, 8 October 2008 (UTC)
No no no. This clearly does not satisfy 3, because if x has units of length and y has units of time, then a must have units of 1/length and b must have units of 1/time to make the first equation dimensionally consistent. In the second equation, however, a<sup>2</sup> and b<sup>2</sup> are added together, so they should have the same units, which is a contradiction. Therefore your method is not dimensionally consistent, and if I change the scale of x or y, the best-fit line will change.
Also, there must be some mistake when you say it's symmetrical, yet provides the same result as linear regression. The result given by linear regression is NOT symmetrical (i.e. the slope of y as a function of x and the slope of x as a function of y are NOT inverses), except in the special case when all the points lie exactly on a line already. —[[User:Keenan Pepper|Keenan Pepper]] 15:11, 10 October 2008 (UTC)


::I was only partially correct. Here is the real answer, as described by Barabasi himself in <ref>Linked|Linked: The New Science of Networks, Albert-László Barabási,Basic Books, 2003
::: Have you looked at [[Errors-in-variables model]]? [[Special:Contributions/155.91.45.231|155.91.45.231]] ([[User talk:155.91.45.231|talk]]) 17:03, 10 October 2008 (UTC)
ISBN 0738206679, p.70</ref> "The power law distribution thus forces us to abandon the idea of a scale, or a characteristic node. In a continuous hierarchy there is no single node which we could pick out and claim to be characteristic of all the nodes. There is no intrinsic scale in these networks. This is the reason my research group started to describe networks with power-law degree distribution as <i>scale-free</i>." [[User:Kmote|Kmote]] ([[User talk:Kmote|talk]]) <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|undated]] comment was added at 15:54, 13 October 2008 (UTC).</span><!--Template:Undated--> <!--Autosigned by SineBot-->
:[[User:Keenan Pepper|Keenan Pepper]] may be right but I am not quite convinced. Do you have a simple example where 'the slope of y as a function of x and the slope of x as a function of y are NOT inverses' ? [[User:Bo Jacoby|Bo Jacoby]] ([[User talk:Bo Jacoby|talk]]) 23:20, 10 October 2008 (UTC).


== Original research, against Wikipedia guidelines? ==
::{(0,0), (1,1), (1,0)}. If you apply linear regression, i.e. minimize:
:::<math>\chi^2 = \sum (y_i-a*x_i-b)^2</math> or its inverse <math>\chi^2 = \sum (x_i-a*y_i-b)^2</math>
::Then the normal solution is y = 0.5*x, and the inverse solution is x = 0.5*y + 0.5. This is because this way of formulating the problem intrinsically treats uncertainty in x and y asymmetrically. [[User:Dragons flight|Dragons flight]] ([[User talk:Dragons flight|talk]]) 00:00, 11 October 2008 (UTC)


Please forgive me if I am mistaken, but on reading, 'the authors...', it seems that an original viewpoint is being expounded, I thought that this was against Wikipedian principles of no new or original research being included?
::You can get a correct solution by minimzing the function:


[[User:Jim lewis1|Jim lewis1]] ([[User talk:Jim lewis1|talk]]) 22:28, 3 April 2008 (UTC)
:::<math>\chi^2 = \sum_i {(y_i-a x_i-b)^2 \over (\sigma_{y_i}^2 + a^2 \sigma_{x_i}^2)}</math>


:I believe "the authors..." refers to Pennock et al., rather than to the authors of the wikipedia article, I have changed "the authors..." to "they" in order to make this clearer [[User:Echinoidea|Echinoidea]] ([[User talk:Echinoidea|talk]]) 21:09, 2 September 2008 (UTC)
::You'll note that the goodness-of-fit function is dimensionless and changing the dimensions of x or y can always be exactly compensated by the corresponding dimensional changes in a and b. You can also show it works under x, y interchange by writing the inverse form and showing that identifying a' = 1/a and b' = -b/a restores the original fit function, exactly as using y = a x + b => x = a' y + b'. This is the form most often seen in research applications, and satisfies your conditions. You can also minimize the derivatives of <math>\chi^2</math> to get algebraic solutions for a and b. [[User:Dragons flight|Dragons flight]] ([[User talk:Dragons flight|talk]]) 00:40, 11 October 2008 (UTC)


== Link directly to graph theory ==
:::This is exactly what I'm looking for. Thanks, Dragons flight. I think this method is called [[element-wise weighted total least squares]] or [[EW-TLS]]. Does that sound right? —[[User:Keenan Pepper|Keenan Pepper]] 01:19, 12 October 2008 (UTC)
You should link directly to [[graph theory]].


I realize that the article links to [[complex network]] which again links to [[network]] which again links to [[graph theory]]. But the first two pages do not give a definition of the term. They cannot be read and understood by readers that do not know graph theory beforehand.
:Thank you. I stand corrected. [[User:Bo Jacoby|Bo Jacoby]] ([[User talk:Bo Jacoby|talk]]) 09:15, 11 October 2008 (UTC).


== Confusing figure (fixed) ==
== Use of atan2 in a maths article ==
The figures shows a [[directed graph]], but the text does not mention directed graphs. the reader will ask herself. What is the meaning of the arrows? Must a hub necesarily be a source? etc..


:Agreed. I'll contact the original artist and ask if he wouldn't mind making the modification, I think having a figure is very useful for people seeing this for the first time. [[User:Meekohi|Meekohi]] 14:05, 12 December 2005 (UTC)
Bit of a dispute I'd like some advice on
I'd like to use atan2 as used in [[complex number#Conversion from the Cartesian form to the polar form]] in the form there in another article


:changed to undirected graph, thank you for contacting me Meekohi. The source of this image is my Ph.D. thesis on [[Web crawler|web crawling]]: (Carlos Castillo: "Efficient Web Crawling". PhD Thesis. Universidad de Chile, 2004.) [[User:ChaTo|ChaTo]] 17:15, 12 December 2005 (UTC)
:<math>\varphi = \arg(z) = \operatorname{atan2}(y,x)</math>


==Ah, Lun Li, Lun Li...==
In fact I wanted to have the multivaluedness explicit so I could refer to k by saying
There are no such thing as "SF vs HOT story". HOT model just has <i>more dimensions</i>. SF reflects connectivity only, while HOT also involves bandwidth. I may introduce, say, GEO model (and I do) which also involves RTT times. There will be a picture quite different both from SF and HOT but there will never be any "GEO vs HOT" or "GEO vs SF story".
Further, regarding the core hubs debate, here is an AS graph from CAIDA: http://www.caida.org/analysis/topology/as_core_network/pics/ascoreApr2005.png The core looks like a core formed by hubs, not something like "low-degree core routers" like in Li et al HOTnet model. Authors don't address this issue and limit scope of their critics to router-level topology, which is much more subjected to geographical and technical limitations.
So, Li et al are too selective in their critics. Also, they sometimes oppose their own claims (e.g. "...a vulnerability that has presumably been overlooked by networking engineers", p.11) etc etc
Till this moment, nobody did blow up top 5% internet routers to test whether the Internet will survive that. So, Li et al argument on "legendary... high resilience" (p. 13) is also questionable.
Given that level of inaccuracy, that theory better be read with caution (Victor Grishchenko 11 Mar 2006)
:I don't think this article intends to say anything about how that paper relates to router topology. The HOT model's <b>Graph</b> just hapens to be a graph with very low s(g). My main point is just that regardless of whether you or Li are correct about the real nature of the Internet's topology, graphs with low-degee cores always have low s(g), and graphs with high degree cores have high s(g), because it's built into the definition. [[User:Meekohi|Meekohi]] 00:46, 13 March 2006 (UTC)
::1) let G be a router graph with a high-degree core (S(G) close to 1.0) 2) let insert intermediary router ("relay") between every two adjacent hubs 3) Result: packet flows remained nearly the same, but S(G) is very low. Am I correct?
:::Yeah I see your point. Since s(g) is just the sum of all neighboring degrees, the way to maximize it is to have the highest degree nodes connected to each other, and the way to minimize it is to have high degree nodes connected only to low degree nodes. I think one big problem is that defining the "core" of the network is hard to do, and depending on how you visualize the graph it can be misleading. The way I am thinking of the graph you've descibed, I would say the intermediary routers are part of the core... so I wouldn't describe it as having a high-degree hub core either, but by making sure the high-degree hubs don't connect directly, you effectively lower the s(g) value. I might rewrite the article some this afternoon if it seems too misleading. [[User:Meekohi|Meekohi]] 14:13, 13 March 2006 (UTC)
::::Just put more emphasis on degree distribution. At least, it is something most authors are agree on. S(g) is just one of the theories. Me personally is a supporter of the skeleton approach by Kahng et al http://phya.snu.ac.kr/~kahng/paper1.html (I may add a respective paragraph to the article). Also, there are some other theories. (Victor Grishchenko 13 Mar 2006)
:::::Well, I'm still in support of using S(g) as the formal definition. It's a lot better than "any graph with a power-law degree distribution". S(g) at least says something about the network structure, even if it doesn't apply well directly to router throughput. Nobody else has really put forth a meaningful formal definition as far as I know. [[User:Meekohi|Meekohi]] 04:26, 14 March 2006 (UTC)


==History==
:<math>\varphi = \arg(z) = \operatorname{atan2}(y,x) + 2 \pi k</math>
Probably silly question, but if "This model was originally discovered by Derek de Solla Price in 1965 under the term cumulative advantage", how was it "further developed by Herbet Simon in the 1950s"? [[User:Mmt|Mmt]] 03:12, 5 April 2006 (UTC)
:Sounds like a pretty '''good''' question to me actually. Does anyone have a source for this Herbert Simon? I've never heard of him before. [[User:Meekohi|Meekohi]] 03:26, 5 April 2006 (UTC)
:Part of the problem was a typo. Should have been [[Herbert Simon]], but the dates still make no sense so I took that part out of the article for now. [[User:Meekohi|Meekohi]] 03:30, 5 April 2006 (UTC)


== Identical degree distribution ==
And another editor undoes the edit saying atan2 is not mathematical saying
<blockquote>
Now define


<math> S(g) = \frac{s(g)}{s_{max}} </math>
:I understand quite well why the arctangent is not the correct function to use. However "atan2" is a not a commonly used mathematical function. I also know there are several computer programming languages that have this function in their library. Using atan2 in a math article is fully unnecessary and makes a strange impression. The intention of the angle is much clearer explained by defining it as one that together with the radius yields back the coordinates by the cosine and sine functions
and
:I am not against mentioning the atan2 function as a way to calculate the angle, but it should not appear in the definition.
My main point is
:The article is not solely for pure maths people, it is for a general audience including people who want to actually use the information. Defining a value by saying the conditions it must satisfy rather than giving a well defined function that just produces it is not very useful in applied mathematics.. And arg is not as well defined as atan2 as it could be multivalued or map to [0, 2π) even if the principle value is conventionally (-π,π].


where ''s<sub>max</sub>'' is the maximum value of ''s(h)'' for ''h'' in the set of all graphs with an identical degree distribution to ''g''. </blockquote>
What do you feel about using atan2 the way it is in the complex number article? Could you frame it differently to make it more mathematical and less programmerish? [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 22:44, 10 October 2008 (UTC)
I was wondering. What is an ''identical'' degree distribution? Is the amount of vertices of a certain degree exactly the same across these identical distributions? Or is the power-law exponent exactly the same? Or are they all power-law distributions or Poisson distributions? [[User:Saviourmachine|Andy]] 10:53, 6 October 2006 (UTC)
:What the hell is atan2? [[User talk:Algebraist|Algebraist]] 22:47, 10 October 2008 (UTC)
::[[atan2]]. My question is, why is this being asked here rather than at [[Talk:Complex number]]? [[User:Silly rabbit|<font color="#c00000">siℓℓy rabbit</font>]] ([[User talk:Silly rabbit|<span style="color:#FF823D;font-family:Monotype Corsiva;cursor:help"><font color="#c00000">talk</font></span>]]) 22:50, 10 October 2008 (UTC)


== Derek de Solla Price did not "Discover" Cumulative Advantage ==
:::It has been used in the [[complex number]] article for a while, I was going to use it in another article [[logarithm]]. It would be rather prejudicing the issue to raise it in a place where people seemed quite happy with it and anyway it seems a general issue - how do you represent how to get the principal value of the angle when converting to polar form? And should a computerish function be used if there isn't something mathematial which expresses it easily - tan<sup>-1</sup>(''y''/''x'') goes wrong for negative ''x'' if converting to polar form [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 23:00, 10 October 2008 (UTC)
:::Another problem with the tan<sup>-1</sup>(''y''/''x'') is that it blows up if ''x'' is zero. Arg is formally undefined for z=0 but the conventional value is 0. atan2 just gives all the conventional values without the messing around. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 23:06, 10 October 2008 (UTC)


In the entry for [[Derek J. de Solla Price]], it says that his contribution was "his interpretation of Herbert Simon's theory on cumulative advantage processes, or Robert K. Merton's Matthew effect, respectively (Price 1976)." So shouldn't Simon or Merton be credited with the discovery? --[[User:Nickg|Nick]] 17:05, 1 December 2006 (UTC)
:This should really be on [[Wikipedia:Wikiproject Mathematics]], but oh well... I would be against using atan2 in an article, since only programmers are going to have the faintest idea what you're talking about. Don't be afraid to use words when describing mathematics - just explain how to choose the appropriate quadrant in English. --[[User:Tango|Tango]] ([[User talk:Tango|talk]]) 23:08, 10 October 2008 (UTC)


:Sounds reasonable to me, but someone should track down a source for this before making a final statement about it probably. [[User:Meekohi|Meekohi]] 04:49, 3 December 2006 (UTC)
::Sorry, will do that if I ever come across a problem like this again. I've been trying to think of a good way to frame it and I also worry about giving a rigmarole to people when most good calculators have the function on them. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 23:25, 10 October 2008 (UTC)


==Population==
:I just had a look via google and atan2 had more hits than atan, 12 mllion versus 7 million. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 23:36, 10 October 2008 (UTC)
::So? Neither atan nor atan2 are common terms in mathematics. The common term is arctan, which has 25 million ghits. [[User talk:Algebraist|Algebraist]] 23:38, 10 October 2008 (UTC)


::Re Dmcq: I'm not surprised about that; there's a reason atan2 is included in so many programming languages. But mathematicians by and large will never have heard of it. They will be familiar with the ordinary arctangent, which is a function of a single variable, not two.


A scale-free network should has a large population.
::The main concern for me isn't mathematicians, but high school students. Most of them will also not know about atan2, even if they have learned about arctangent in trigonometry class. But I had left the atan2 function in the complex number article when I edited it; I think I may be biased by already knowing what atan2 is. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 23:51, 10 October 2008 (UTC)


A real-world network always obtain a large number of individuals.
:::My concern is people using tan<sup>-1</sup>(''y''/''x''). See for example this one I found on the first page with a google of 'polar form' at [http://www.diracdelta.co.uk/science/source/c/o/complex%20numbers%20polar%20form/source.html] where a science and engineering encyclopaedia encourages people to get it all wrong. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 00:07, 11 October 2008 (UTC)


== Web is not random ==
::::Yes, that page is all bad. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 00:16, 11 October 2008 (UTC)
::::Well, that is the correct formula, you just need to include details of how to choose between the multiple values it returns (and what to do if x=0). --[[User:Tango|Tango]] ([[User talk:Tango|talk]]) 00:25, 11 October 2008 (UTC)
:::::I think of arctan as a normal, single-valued, function. There are other ways of defining the argument of a complex number than using arctan, and I expect that's how a lot of math texts do it. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 01:30, 11 October 2008 (UTC)


''To their surprise, the Web did not have an even distribution of connectivity.'' Why is this surprising? [[User:Kope|Kope]] ([[User talk:Kope|talk]]) 19:42, 18 June 2008 (UTC)
: The reasons programmers love atan2 is that it's so well behaved. According to the C manual page:
:: "''The atan2() function calculates the arc tangent of the two variables x and y. It is similar to calculating the arc tangent of y / x, except that the signs of both arguments are used to determine the quadrant of the result.''"
: Using a regular atan function produces all sorts of annoying practical nightmares. I can understand that mathematicians don't use it - but it's really their loss. It's a perfectly reasonable function - it's just a matter of history that it's not a part of the standard canon. [[User:SteveBaker|SteveBaker]] ([[User talk:SteveBaker|talk]]) 02:23, 11 October 2008 (UTC)
::As a mathematician, I'd say we ''do'' use it, we just call it 'the principal value of the argument' and have it act on complex numbers rather than pairs of reals. [[User talk:Algebraist|Algebraist]] 07:32, 11 October 2008 (UTC)
:::True, but I'd been looking at [[arg (mathematics)]] too and really I think that article at least should refer more to the multivalued form and use Arg for the principal value. Also it refers to the complex numbers, if doing that we might as well remove the square root equivalent of the modulus at the same time, it seems silly to have one and not the other. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 08:01, 11 October 2008 (UTC)
::::Where I come from, arg and log are single-valued while Arg and Log are multi-valued. Are other practices in use? [[User talk:Algebraist|Algebraist]] 08:28, 11 October 2008 (UTC)
:::::The [[Principal value]] article uses the convention that Arg gives the principal value. MathWorld and any books I've seen do the same. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 13:23, 11 October 2008 (UTC)
:::::My university uses Arg and Log for the principal values, arg and log for multi-valued. Eric. [[Special:Contributions/131.215.159.187|131.215.159.187]] ([[User talk:131.215.159.187|talk]]) 03:48, 12 October 2008 (UTC)


= October 11 =

== 4th grade physics question (fluids, force, [[pressure]]) ==

A partially evacuated airtight container has a tight fitting lid (of negligible mass) over a airtight container.

*Surface area of the lid = 77 meters-squared
*Pressure above the lid = 100,000 Pa
*Pressure below the lid = unknown
*Force required to move the lid up = 480 N

My book gives the wrong answer (38,000 Pa), I just need a verification that its [[99,993 Pa]], or am I just using a wrong measurement such as meters instead of millimeters. Thanks, i'm embarassed to even ask this question, its easy algebra. The only reason I ask, is that I missed the first 4 questions in a row before this question (all for careless mistakes) but this one is playing tricks on my mind. 38kPa is the correct answer if the conversion factor between a netwon per square meter and a Pascal was some other amount. This was from yesterday's 11th hour of a study session and it caused me a great deal of headache, and I look forward to starting today's marathon session with closure. [[User:Sentriclecub|Sentriclecub]] ([[User talk:Sentriclecub|talk]]) 10:25, 11 October 2008 (UTC) <!-- If its helpful, I even dreamed about pressure and force last night, no kidding -->
:Are you sure you have the question right? The given answer is right if the surface area is 77 square centimetres (which is also a much more reasonable size for an airtight container!). [[User talk:Algebraist|Algebraist]] 10:57, 11 October 2008 (UTC)
::Thanks. I'm looking at the book right now ''Fundamentals of Physics'' 8E Jearl Walker and I bought the detailed solutions manual. Now I can have my sanity back. [[User:Sentriclecub|Sentriclecub]] ([[User talk:Sentriclecub|talk]]) 11:09, 11 October 2008 (UTC)


= October 12 =

== Multiplying matrices of functions/arrays (in MATLAB) ==

I am looking for a good way to multiply matrices of functions, or matrices of arrays in Matlab. The description of my problem is as follows:

(In the following, uppercase letters denote matrices and lowercase letters denote scalars.)

I want to find Z = A(x)*B(x) where,

A(x) = [a11(x), a12(x), ..... , a1n(x);
a21(x), a22(x), ....., a2n(x);
..............................;
an1(x), an2(x), ....., ann(x)];

B(x) = [b11(x), b12(x), ..... , b1n(x);
b21(x), b22(x), ....., b2n(x);
..............................;
bn1(x), bn2(x), ....., bnn(x)];

Here, (.)(x) means (.) is a function of x.

For example, the first element of Z would be

z11(x) = a11(x)*b11(x) + a12(x)*b21(x) + ... + a1n(x)*bn1(x)

Furthermore, I need to integrate Z(x) from x = a to b to get ZI, which would be a simple nxn matrix.

The functions aij(x) and bij(x) are known numerically, so each one of them is a 1D array (say,
of size m).

As an example, consider 2x2 matrices

A = [a11(x), a12(x);
a21(x), a22(x);]

B = [b11(x), b12(x);
b21(x), b22(x);]

Where,

x = [1,2,3,4,5];

a11(x) = x + 1 = [2 3 4 5 6]

a12(x) = x.^2 = [ 1 4 9 16 25]

a21(x) = x - 1 = [ 0 1 2 3 4]

a22(x) = 1.5*x = [ 1.5000 3.0000 4.5000 6.0000 7.5000]

b11(x) = sin(x) = [ 0.8415 0.9093 0.1411 -0.7568 -0.9589]

b12(x) = cos(x) = [ 0.5403 -0.4161 -0.9900 -0.6536 0.2837]

b21(x) = exp(-x) = [0.3679 0.1353 0.0498 0.0183 0.0067]

b22(x) = [1 1 1 1 1]

Then

Z = A*B = [a11(x)*b11(x) + a12(x)*b21(x), a11(x)*b12(x) + a12(x)*b22(x);
a21(x)*b11(x) + a22(x)*b21(x), a21(x)*b12(x) + a22(x)*b22(x)]


That is, the terms of Z are the functions of x, such that for x = [1,2,3,4,5]

z11 = a11.*b11 + a12.*b21 = [2.0508 3.2692 1.0126 -3.4910 -5.5851]

z12 = a11.*b12 + a12.*b22 = [ 2.0806 2.7516 5.0400 12.7318 26.7020]

z21 = a21.*b11 + a22.*b21 = [ 0.5518 1.3153 0.5063 -2.1605 -3.7852]

z22 = a21.*b12 + a22.*b22 = [1.5000 2.5839 2.5200 4.0391 8.6346]

and integral of Z over x = 1 to 5 will be approximately (using trapezoidal rule)

ZI = [-0.9763, 34.9147;
-1.9556, 14.2103]

I am looking for a simple way to do this in the general case. It must be possible to do it somehow using a matrix with three indices, but so far I have not been able to come up with a way.

Any help will be *greatly* appreciated. [[User:deeptrivia|deeptrivia]] ([[User talk:deeptrivia|talk]]) 02:53, 12 October 2008 (UTC)

:Whether this is simple or not will depend on how your functions are defined, but it certainly possible to store values in three index arrays. The drawback is that they don't have natural arithmetic the way that two index matrices do.

:For example on could implment it as:

A(1,1,:) = a11;
B(1,1,:) = b11;
A(2,1,:) = a21;
B(2,1,:) = b21;
etc...

for k = 1:n
Z(:,:,k) = A(:,:,k)*B(:,:,k);
end
Z = sum(Z*dx,3);

:You'd still have to manually define the lists a_kj and b_kj, but you are going to have to do that anyway in telling Matlab what the functions are for each index. The "3" in the final sum tells Matlab to sum over the third index after multiplying by the increment dx and results in a n x n matrix of the integral you are looking for. You can of course do more complicated approximations to the integral, but generally that will require defining the integration process inside a for loop. [[User:Dragons flight|Dragons flight]] ([[User talk:Dragons flight|talk]]) 04:04, 12 October 2008 (UTC)

:: Thanks Dragon's flight. I have to define complicated expressions, and I would like to use operators like "*" to do multiplication. Is there any operator overloading in Matlab? I just figured out a cumbersome way to do transposes for these kind of "3d" matrices, but I'd like to overload that method onto the " ' " operator. And, I want to overload the method you suggested to the "*" operator. Is that possible? Thanks a ton! [[User:deeptrivia|deeptrivia]] ([[User talk:deeptrivia|talk]]) 05:11, 12 October 2008 (UTC)

:::Recent versions of Matlab do allow you define data classes with overloaded operators, though this is not an area I have much experience with. Each operation in Matlab has a builtin function name associated with it. For example "A + B" is translated by the interpreter to "plus(A,B)", and within a class you can define a new "plus" function that will then become the "+" operation for that class. [http://www.mathworks.com/access/helpdesk/help/techdoc/index.html?/access/helpdesk/help/techdoc/matlab_oop/ug_intropage.html&http://www.google.com/search?sourceid=navclient&ie=UTF-8&rlz=1T4DMUS_enUS209US209&q=class+programming+matlab This] provides an introduction to class programming on Matlab, though it is somewhat cumbersome. Unless you will be working with these things alot I'd think it would be simpler to write functions to deal with your specific cases rather than writing a data class and overloaded operators to deal with general cases. [[User:Dragons flight|Dragons flight]] ([[User talk:Dragons flight|talk]]) 05:53, 12 October 2008 (UTC)

:::: Thanks, Dragons flight. What if I just modified the default mtimes.m to check if the operands are 3D arrays. If so, use my routine, and if not, do the normal thing. mtimes.m is the following

function [varargout] = mtimes(varargin)
if nargout == 0
builtin('mtimes', varargin{:});
else
[varargout{1:nargout}] = builtin('mtimes', varargin{:});
end

How can I suitably modify this? Thanks, [[User:deeptrivia|deeptrivia]] ([[User talk:deeptrivia|talk]]) 16:49, 12 October 2008 (UTC)

== Proving inequality for derivates (defined in Royden's Real Analysis) ==

Hello, I am working on a homework problem and I am not necessarily even looking for a hint but it seems to be wrong and I just want to see what others think. It's number 3 from Chapter 5 of Royden's book, 3rd edition. It says
If <math>f\,</math> is continuous on [a, b] and assumes a local maximum at <math>c \in (a, b)</math>, then
<math>D_- f(c) \leq D^- f(c) \leq 0 \leq D_+ f(c) \leq D^+ f(c)</math>.

Any way, the outer two inequalities are immediate, so the inner two are the problem. In case you do not know what a "derivate" is, because it does not seem to be a very common term, the definitions for the two needed ones are

<math>D_+ f(x) = \liminf_{h \rightarrow 0+} \frac{f(x + h) - f(x)}{h}</math>

and

<math>D^- f(x) = \limsup_{h \rightarrow 0+} \frac{f(x) - f(x - h)}{h}</math>

So, my problem is this. If there is a local maximum at <math>c\,</math>, then <math>f(x) \leq f(c)</math> for all <math>x \in [a, b] \cap N_\delta(c)</math>, for some <math>\delta > 0\,</math>. Then, <math>f(c + h) - f(c)\,</math> should be negative for <math>h\,</math> close enough to <math>c\,</math> so that <math>D_+ f(c)\,</math> should be nonpositive, whereas we are asked to prove it is nonnegative. Similarly, <math>f(c) - f(c - h)\,</math> should be positive so that <math>D^- f(c)\,</math> should be nonnegative, whereas we are asked to prove it is nonpositive.

Am I completely missing something here? I'm not looking for a solution. I just want to understand this part. Thanks [[User:StatisticsMan|StatisticsMan]] ([[User talk:StatisticsMan|talk]]) 03:04, 12 October 2008 (UTC)

:You look right; it seems that either the definition provided or the homework problem have a sign flipped somewhere. Eric. [[Special:Contributions/131.215.159.187|131.215.159.187]] ([[User talk:131.215.159.187|talk]]) 03:43, 12 October 2008 (UTC)

== Differentiation problem ==

Find the coordinates of the stationary points on the graph <math>y = (x-2)^3 - 12(x-2)</math>.

So I differentiated to get <math>\frac{dy}{dx} = 3(x-2)^2 - 12</math>
And then found the x coordinates to be 0 and 4 and therefore the coordinates are (4, -16) and (0, 16)

However then I have to give the points of intersection with the axes which I dont know how to do. I know the intersection with the y axis from above is (0,16) but how do i get the x intersections.

This is what i tried:
<math>0 = (x-2)^3 - 12(x-2)</math>

Multiplied out the using the binomial theorem to get:
<math>0 = x^3 - 6x^2 + 16</math>

Is this correct? If so what now? --[[User:RMFan1|RMFan1]] ([[User talk:RMFan1|talk]]) 12:55, 12 October 2008 (UTC)

:Where a graph intersects the x-axis is where the function equals zero, so your method is right. It's equivalent to saying "find the roots of the polynomial". --[[User:Tango|Tango]] ([[User talk:Tango|talk]]) 13:26, 12 October 2008 (UTC)

Yes I know how you find intersections i just dont know how i can simplify the above. I could also have <math>0 = (x-2)^3 - 12(x-2)</math> so that <math>0 = (x-2)^2 - 12</math> but when I try to solve with the quadratic formula, the discriminant is negative so I cant simplify it. --[[User:RMFan1|RMFan1]] ([[User talk:RMFan1|talk]]) 13:34, 12 October 2008 (UTC)
:If the discriminant is negative then there are no real solutions, so it doesn't intersect the x-axis. That means x=2 is your only intersection point. --[[User:Tango|Tango]] ([[User talk:Tango|talk]]) 14:07, 12 October 2008 (UTC)

:First think visually. Sketch the graph. You already know two points on it, (0,16) and (4,-16). When ''x'' is large and negative then ''y'' is also negative, so the graph comes up from the bottom left. It must cross the ''x'' axis at least once in the range ''x''<0 to get up to (0,16). Then it must cross the ''x'' axis at least once more in the range 0<''x''<4 to get down to (4,-16). And when ''x'' is large and positive then ''y'' is also positive, so there is at least one more intersection with the ''x'' axis in the range ''x''>4. But a cubic equation can have at most three real roots - so we have one root less than 0, a second root between 0 and 4, and a third root greater than 4.
:Now suppose we recast the equation in terms of ''t''=''x''&minus;2. As a function of ''t'', we have
::<math>y=t^3-12t</math>
:The (''t'',''y'') graph looks just like the (''x'',''y'') graph except it is shifted to the left by 2 units - so in the (''t'',''y'') graph the maximum and minimum are at (-2,16) and (2,16). Finding the three real roots of
::<math>y=t^3-12t</math>
:is quite easy - one root is obviously ''t''=0. To find the correspoding values of ''x'', just use ''x''=''t''+2. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 14:12, 12 October 2008 (UTC)

== Linear algebra puzzler ==

Let ''A'' be a 3&times;2 matrix and ''B'' a 2&times;3 matrix. Suppose that
:<math>AB = \begin{bmatrix}5&-1&2\\-1&5&2\\2&2&2\end{bmatrix}.</math>
Find ''BA''. Your solution should also show that there is a unique right answer. [[User:Silly rabbit|<font color="#c00000">siℓℓy rabbit</font>]] ([[User talk:Silly rabbit|<span style="color:#FF823D;font-family:Monotype Corsiva;cursor:help"><font color="#c00000">talk</font></span>]]) 15:15, 12 October 2008 (UTC)

:My first thought would be to consider transposes. --[[User:Tango|Tango]] ([[User talk:Tango|talk]]) 15:24, 12 October 2008 (UTC)
::Not that that seems to work... I can't get anywhere with it (but I may be missing something, the question still screams "take transposes" to me). You could always try it the hard way - write A=(a<sub>ij</sub>), B=(b<sub>ij</sub>), multiply out both ways round and try and solve the horrible mess of simultaneous quadratic equations - I wouldn't recommend it, though! --[[User:Tango|Tango]] ([[User talk:Tango|talk]]) 15:38, 12 October 2008 (UTC)
:I'm pretty sure the uniqueness of <math>BA=6I</math> can be proved by considering all possible [[singular value decomposition]]s of <math>AB</math>. <math>A</math> and <math>B</math>. [[Special:Contributions/84.239.160.166|84.239.160.166]] ([[User talk:84.239.160.166|talk]]) 16:29, 12 October 2008 (UTC)

:On a quick look I'd say it was probably 6 times the 2x2 identity matrix since multiplying the matrix by itself gives itself multiplied by 6. Just did that because I got BA in the middle of AB.AB. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 16:52, 12 October 2008 (UTC)

I suppose I should give the answer, although I was expecting this to generate somewhat more interest.

''Method 1.'' Note that ''AB''<sup>2</sup>=6''AB''. That is ''ABAB''=6''AB''. Since ''AB'' is of rank 2, both ''A'' and ''B'' are also rank 2, and so ''A'':'''R'''<sup>2</sup>&rarr;'''R'''<sup>3</sup> is injective and ''B'':'''R'''<sup>3</sup>&rarr;'''R'''<sup>2</sup> is surjective. Thus ''A'' is cancellable on the left and ''B'' is cancellable on the right of ''ABAB''=6''AB'', and so ''BA''=6''I''.

''Method 2.'' From ''ABAB''=6''AB'', we have
:<math>A^TABABB^T = 6 A^TABB^T.</math>
Since rank(''A''<sup>T</sup>''A'') = rank(''A'') = 2 and rank(''BB''<sup>T</sup>) = rank(''B'') = 2, we have
:<math>BA = 6 (A^TA)^{-1}A^TABB^T(BB^T)^{-1} = 6I.</math>

''Remark.'' I seem to recall that there was a way to solve this problem using orthogonality somehow (e.g., the SVD or QR decomposition). I was unable to recall the details for the purposes of this solution, but feel free to add a "Method 3" if you can get this to work. Cheers, [[User:Silly rabbit|<font color="#c00000">siℓℓy rabbit</font>]] ([[User talk:Silly rabbit|<span style="color:#FF823D;font-family:Monotype Corsiva;cursor:help"><font color="#c00000">talk</font></span>]]) 14:36, 13 October 2008 (UTC)

== cost of transporting goods ==

Con you tell be the cost to transport 100 lbs of material in a car that gets 25 mpg / gas @ $3.50 gal. My husband feels that the 'junk' I was to take is worth less than the gas. We will travel 1000 miles. Many Thanks[[User:Twinkle2toes|Twinkle2toes]] ([[User talk:Twinkle2toes|talk]]) 21:48, 12 October 2008 (UTC)

:Can't be done with mathematics only. If I were to assume that fuel consumtion is proportional to the loaded mass of the vehicle, and the vehicle weighs around 2,000 lbs, then you'd end up with around $7 additional cost. But I'm pretty sure it's ''not'' proportional, and I wouldn't be surprised if the physics behind it is complicated. [[Fuel economy in automobiles]] only says that engine efficiency varies with the mass of the automobile and its load, but not how. We need input from someone who knows something about cars... -- [[User:Jao|Jao]] ([[User talk:Jao|talk]]) 22:51, 12 October 2008 (UTC)
::Not just cars, but that specific car. It's going to vary widely from car to car, for example a lightweight car is going to affected by an extra 100lbs more than a very heavy car (I would imagine, anyway). --[[User:Tango|Tango]] ([[User talk:Tango|talk]]) 23:03, 12 October 2008 (UTC)

:::The cost to transport "''100 pounds of material''" is like talking about the cost to bring along a small teenager or a box or two of treasured family photos and such . The answer is more subjective than practical remembering that "one person's junk is another's treasure". Neither the actual dollar cost of transporting nor the emotional cost (even though you didn't ask) of leaving the "material" behind can't be calculated with the information given. -[[User:Hydnjo|hydnjo]] [[User talk:Hydnjo|talk]] 23:51, 12 October 2008 (UTC)

:::: Fuel efficiency would be more or less proportional to the mass assuming things are ideal (like the engine is a [[carnot engine]]). This makes some sense because both kinetic and potential energy are proportional to mass, and so is any frictional loss during braking, etc. So, [[User:Jao|Jao]]'s estimate is the best we can do unless we decide to go really fancy. [[Special:Contributions/128.118.150.91|128.118.150.91]] ([[User talk:128.118.150.91|talk]]) 00:20, 13 October 2008 (UTC)

:::::Yeah, $5 to $10 seems about right. -[[User:Hydnjo|hydnjo]] [[User talk:Hydnjo|talk]] 00:31, 13 October 2008 (UTC)

:The fuel economy depends on whether the extra 100 hundred pounds is being driven in town or on the highway. Once you're at a steady highway speed (the 1000-mile trip), air drag dominates and an 100 extra pounds isn't going to matter much. Take the stuff out for around-town driving, though. [[User:Saintrain|Saintrain]] ([[User talk:Saintrain|talk]]) 12:55, 13 October 2008 (UTC)

::$5 sounds about right. Let's say you get a 2% drop in gas mileage, down from 25 mpg to 24.5 and that means instead of using 40 gallons of fuel, you will use instead 41 gallons of fuel. [[User:Sentriclecub|Sentriclecub]] ([[User talk:Sentriclecub|talk]]) 15:20, 13 October 2008 (UTC)

= October 13 =

== Volume ratio ==

The ratio of the volume of a cube to that of a sphere which will exactly fit inside the cube is? <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:NAKABBOSS|NAKABBOSS]] ([[User talk:NAKABBOSS|talk]] • [[Special:Contributions/NAKABBOSS|contribs]]) 00:20, 13 October 2008 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->
:Well, the formulae for the volume of a [[cube]] and a [[sphere]] are fairly simple, you just need to know what you use for the radius and the side length. -''[[User:Mattbuck|mattbuck]]'' <small>([[User talk:Mattbuck|Talk]])</small> 00:34, 13 October 2008 (UTC)

::The ratio is: [[Homework|click here]]. -[[User:Hydnjo|hydnjo]] [[User talk:Hydnjo|talk]] 00:39, 13 October 2008 (UTC)

==1/x integral==

Is the integral of 1/x from 0 to 1 infinity or undefined? <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/165.124.138.143|165.124.138.143]] ([[User talk:165.124.138.143|talk]]) 02:06, 13 October 2008 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

Infinity (you might find proving this formally using the epsilon delta definition interesting).

[[User:Topology Expert|Topology Expert]] ([[User talk:Topology Expert|talk]]) 02:20, 13 October 2008 (UTC)

:If you want to get pedantic, then the [[Riemann integral]] does not exist, but the [[improper integral]] (and the [[Lebesgue integral]], IIRC) evaluates to +infinity. [[User:ConMan|Confusing Manifestation]]<small>([[User talk:ConMan|Say hi!]])</small> 04:44, 13 October 2008 (UTC)

== help in editing gelfand pairs ==

I hope it's the rite plase to ask this requst.
I'm curently editing the artical [[gelfand pair]] and I wont to put ther a section abut its aplications. so I rote it and put it in the [[Talk:Gelfand_pair|discation page]]. However my english and riting stayl are to poor to put it in the artical, so I'll b grateful if somebody will fix this section and put it in the artical.

Thenks alot [[User:Aizenr|Aizenr]] ([[User talk:Aizenr|talk]]) 10:47, 13 October 2008 (UTC)

== Calculating a growth (or interest) rate ==

If I have a starting population of X, a number of periods P, and a final population of Y, how do I calculate the population growth per period assuming it's a smooth growth rate? — [[User:PhilHibbs|PhilHibbs]] | [[User talk:PhilHibbs|talk]] 15:41, 13 October 2008 (UTC)
:Assuming you are talking true exponential growth, then its easiest to manipulate your compounding period to satisfy f(1)=e. Setting up the problem with no aim, is difficult and unintuitive. (will add more later-I'm seeking my notes)[[User:Sentriclecub|Sentriclecub]] ([[User talk:Sentriclecub|talk]]) 15:46, 13 October 2008 (UTC)
::Yes, I meant a smooth exponential, e.g. starting with 2000, over 1000 generations, a final population of 42,000,000 requires a growth rate of 1%. I calculated this example the easy way around! 1% growth actually gives 41,918,311. Oh, and this isn't homework, I left school 22 years ago. I could probably have done this in my head back then. — [[User:PhilHibbs|PhilHibbs]] | [[User talk:PhilHibbs|talk]] 15:51, 13 October 2008 (UTC)

Revision as of 15:55, 13 October 2008

I opened this discussion page because I have seen many active flip-flop of versions in a fairly short period recently. It seems that some like the Lun Li, et al. paper and definition, but some don't. I like the frankness of the paper although it may appear as "opening fire" to many recent papers contributed in the last six years. I think critique is good to healthy science and helps us in knowing the truth. The current version [1] seems to put the Lun Li, et al. paper in a weaker position in the exchange of ideas about the science of scale-free network, which I oppose.


I agree, I'm putting it back in especially considering the edit was made by an unregistered user. Also to answer the question below, the formal definition captures the notion of self-similarity, while Barabasi's definition really doesn't (From my understanding he basically just picked the name because it sounded good and would catch people's attention. Presumably he had some ideas about self-similarity but I haven't seen him talk about it) Meekohi 13:56, 12 December 2005 (UTC)[reply]
I consider that paper too scandalous and inaccurate. E.g. HOT model obviously has more dimensions than SF model. It is just a more specific case. You can't say that "horses are better than mammals" or "horses are faster than mammals". It is... well... a bit incorrect. On the formal definition. Just give me formal definition of a cat first. OK, the topic of degree correlations is actively studied, so it will settle to something in some years... let's consider that paper a "critique". Victor Grishchenko

The name "scale free"

This is my first time using the talk page, so please forgive any missed wikiquette.

The article was most helpful to me, however I am curious about one thing. Why are these networks called "scale free"? I assume that this is a reference to self-similarity, but what exactly is self-similar in a scale-free network? The sample diagrams of networks I have seen are not obviously fractal.

David M.

scale-free is not self-similar. Scale-free (as I understand it) is that there is no meaningful "average" for the behavior of the system. ChaTo 17:19, 12 December 2005 (UTC)[reply]
For the old definition of Scale-free this was true, but the Lun Li definition actually includes the concept of self-similarity. Meekohi 15:34, 13 December 2005 (UTC)[reply]
As I understand it, the term "scale-free" is closely related to the term "scale-invariant", which is perhaps a more intuitive phrase. It means (roughly) that regardless of how much you "zoom in or out", the structure is going to look essentially the same. Kmote (talk) 21:05, 8 October 2008 (UTC)[reply]
I was only partially correct. Here is the real answer, as described by Barabasi himself in [1] "The power law distribution thus forces us to abandon the idea of a scale, or a characteristic node. In a continuous hierarchy there is no single node which we could pick out and claim to be characteristic of all the nodes. There is no intrinsic scale in these networks. This is the reason my research group started to describe networks with power-law degree distribution as scale-free." Kmote (talk) —Preceding undated comment was added at 15:54, 13 October 2008 (UTC).[reply]

Original research, against Wikipedia guidelines?

Please forgive me if I am mistaken, but on reading, 'the authors...', it seems that an original viewpoint is being expounded, I thought that this was against Wikipedian principles of no new or original research being included?

Jim lewis1 (talk) 22:28, 3 April 2008 (UTC)[reply]

I believe "the authors..." refers to Pennock et al., rather than to the authors of the wikipedia article, I have changed "the authors..." to "they" in order to make this clearer Echinoidea (talk) 21:09, 2 September 2008 (UTC)[reply]

Link directly to graph theory

You should link directly to graph theory.

I realize that the article links to complex network which again links to network which again links to graph theory. But the first two pages do not give a definition of the term. They cannot be read and understood by readers that do not know graph theory beforehand.

Confusing figure (fixed)

The figures shows a directed graph, but the text does not mention directed graphs. the reader will ask herself. What is the meaning of the arrows? Must a hub necesarily be a source? etc..

Agreed. I'll contact the original artist and ask if he wouldn't mind making the modification, I think having a figure is very useful for people seeing this for the first time. Meekohi 14:05, 12 December 2005 (UTC)[reply]
changed to undirected graph, thank you for contacting me Meekohi. The source of this image is my Ph.D. thesis on web crawling: (Carlos Castillo: "Efficient Web Crawling". PhD Thesis. Universidad de Chile, 2004.) ChaTo 17:15, 12 December 2005 (UTC)[reply]

Ah, Lun Li, Lun Li...

There are no such thing as "SF vs HOT story". HOT model just has more dimensions. SF reflects connectivity only, while HOT also involves bandwidth. I may introduce, say, GEO model (and I do) which also involves RTT times. There will be a picture quite different both from SF and HOT but there will never be any "GEO vs HOT" or "GEO vs SF story". Further, regarding the core hubs debate, here is an AS graph from CAIDA: http://www.caida.org/analysis/topology/as_core_network/pics/ascoreApr2005.png The core looks like a core formed by hubs, not something like "low-degree core routers" like in Li et al HOTnet model. Authors don't address this issue and limit scope of their critics to router-level topology, which is much more subjected to geographical and technical limitations. So, Li et al are too selective in their critics. Also, they sometimes oppose their own claims (e.g. "...a vulnerability that has presumably been overlooked by networking engineers", p.11) etc etc Till this moment, nobody did blow up top 5% internet routers to test whether the Internet will survive that. So, Li et al argument on "legendary... high resilience" (p. 13) is also questionable. Given that level of inaccuracy, that theory better be read with caution (Victor Grishchenko 11 Mar 2006)

I don't think this article intends to say anything about how that paper relates to router topology. The HOT model's Graph just hapens to be a graph with very low s(g). My main point is just that regardless of whether you or Li are correct about the real nature of the Internet's topology, graphs with low-degee cores always have low s(g), and graphs with high degree cores have high s(g), because it's built into the definition. Meekohi 00:46, 13 March 2006 (UTC)[reply]
1) let G be a router graph with a high-degree core (S(G) close to 1.0) 2) let insert intermediary router ("relay") between every two adjacent hubs 3) Result: packet flows remained nearly the same, but S(G) is very low. Am I correct?
Yeah I see your point. Since s(g) is just the sum of all neighboring degrees, the way to maximize it is to have the highest degree nodes connected to each other, and the way to minimize it is to have high degree nodes connected only to low degree nodes. I think one big problem is that defining the "core" of the network is hard to do, and depending on how you visualize the graph it can be misleading. The way I am thinking of the graph you've descibed, I would say the intermediary routers are part of the core... so I wouldn't describe it as having a high-degree hub core either, but by making sure the high-degree hubs don't connect directly, you effectively lower the s(g) value. I might rewrite the article some this afternoon if it seems too misleading. Meekohi 14:13, 13 March 2006 (UTC)[reply]
Just put more emphasis on degree distribution. At least, it is something most authors are agree on. S(g) is just one of the theories. Me personally is a supporter of the skeleton approach by Kahng et al http://phya.snu.ac.kr/~kahng/paper1.html (I may add a respective paragraph to the article). Also, there are some other theories. (Victor Grishchenko 13 Mar 2006)
Well, I'm still in support of using S(g) as the formal definition. It's a lot better than "any graph with a power-law degree distribution". S(g) at least says something about the network structure, even if it doesn't apply well directly to router throughput. Nobody else has really put forth a meaningful formal definition as far as I know. Meekohi 04:26, 14 March 2006 (UTC)[reply]

History

Probably silly question, but if "This model was originally discovered by Derek de Solla Price in 1965 under the term cumulative advantage", how was it "further developed by Herbet Simon in the 1950s"? Mmt 03:12, 5 April 2006 (UTC)[reply]

Sounds like a pretty good question to me actually. Does anyone have a source for this Herbert Simon? I've never heard of him before. Meekohi 03:26, 5 April 2006 (UTC)[reply]
Part of the problem was a typo. Should have been Herbert Simon, but the dates still make no sense so I took that part out of the article for now. Meekohi 03:30, 5 April 2006 (UTC)[reply]

Identical degree distribution

Now define

where smax is the maximum value of s(h) for h in the set of all graphs with an identical degree distribution to g.

I was wondering. What is an identical degree distribution? Is the amount of vertices of a certain degree exactly the same across these identical distributions? Or is the power-law exponent exactly the same? Or are they all power-law distributions or Poisson distributions? Andy 10:53, 6 October 2006 (UTC)[reply]

Derek de Solla Price did not "Discover" Cumulative Advantage

In the entry for Derek J. de Solla Price, it says that his contribution was "his interpretation of Herbert Simon's theory on cumulative advantage processes, or Robert K. Merton's Matthew effect, respectively (Price 1976)." So shouldn't Simon or Merton be credited with the discovery? --Nick 17:05, 1 December 2006 (UTC)[reply]

Sounds reasonable to me, but someone should track down a source for this before making a final statement about it probably. Meekohi 04:49, 3 December 2006 (UTC)[reply]

Population

A scale-free network should has a large population.

A real-world network always obtain a large number of individuals.

Web is not random

To their surprise, the Web did not have an even distribution of connectivity. Why is this surprising? Kope (talk) 19:42, 18 June 2008 (UTC)[reply]

  1. ^ Linked|Linked: The New Science of Networks, Albert-László Barabási,Basic Books, 2003 ISBN 0738206679, p.70
Retrieved from "https://en.wikipedia.org/w/index.php?title=Wikipedia:Reference_desk/Mathematics&oldid=245012918"