Talk:Power law

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Contents 1 Merging with other articles 2 References to terrorism 3 Pareto Distribution: reference to Capitalism 4 (Comments 2003-2005) 5 external links 6 Merging Zipf, Power Law, and Pareto? 7 Financial market power laws 8 A How To Guide? 9 Weibull not power-law? 10 with k > 1 11 "Practical use" 12 Category: exponentials 13 Major revision of article 13.1 Exponent/universality issues 14 Accessability

[edit] Merging with other articles

I propose that our revised power law article become the redirect point for Fat tail and Heavy-tailed distribution. I've placed such a suggestion on the talk pages of each of these articles (which substantially overlap with the revised power law article, and with each other).

Paresnah 20:14, 13 March 2007 (UTC)

Also, the Extreme value theory article, which mentions 'tail-fitting' but points to a non-existent article on the topic, should be connected to the Power laws subsection on estimating the tail exponent of power-law distributions. Paresnah 20:21, 13 March 2007 (UTC)

This latter point (linking with extreme value theory) is now done. Paresnah 01:28, 17 March 2007 (UTC)

[edit] References to terrorism

I concurr with the removal of the two references on terrorism; if the article isn't going to list all the citations that show power laws in weird systems, then it shouldn't priviledge a few while ignoring others. So long as we keep the links to major review articles like Newman's up-to-date, we can let the academics keep track of which systems show power laws -- Paresnah 23:38, 28 August 2006 (UTC)

[edit] Pareto Distribution: reference to Capitalism

The Pareto distribution, for example, the distribution of wealth in capitalist economies

There's no mention of capitalism in the article on the Pareto distribution - the word capitalist seems redundant here since in feudal societies there appears to have been a similar distribution, only more with a higher exponent, and even within Romania, USSR, and most other non-capitalist societies similar distribution of wealth appears to have occured. Any thoughts? (especially from someone who knows more about Pareto distributions!) --Dilaudid 08:10, 12 December 2006 (UTC)

I believe you're right, the Pareto distribution of wealth is pretty universal although the exponent may change. Incidentally, although Pareto proposed a pure power law, nowadays people recognise that the income distribution is more like a Lévy distribution.

Incidentally I think that the Pareto distribution article is pretty parochial, in the sense that they consider "Pareto distribution" to refer to power law probability distributions in general. That's very much an economics point of view. It would be better to merge the technical power-law material away from that article into the present one, and rewrite the Pareto distribution article to talk just about income, making reference to Mandelbrot's article on Pareto-Levy law, etc. —WebDrake 23:26, 12 December 2006 (UTC)

[edit] (Comments 2003-2005)

Should it be y=ax^k, where a is a constant? -- Heron

Yes, I believe so, since Zipf's law is a power law and has a probability distribution of the form of y = ax^-b. -- Lexor 22:37, 18 Aug 2003 (UTC):

A more general relationship would be (i.e. y "goes like" x^k) rather than assuming a literal equation like the above. e.g. consider the equation y = (x − a)^k where a is constant. WebDrake 22:31, 7 October 2005 (UTC)

Minor edit: I changed the internal link on self-organized criticality to point to that page, instead of just self-organization; and I updated the xxx.lanl.gov external link so that it points to the abstract of the paper rather than just the PDF (this allows a user to select download file type). WebDrake 22:31, 7 October 2005 (UTC)

"One rule of thumb, however, is if the distribution is straight on a log-log graph over 3 or more orders of magnitude." => This sentence has an "if" part, but it is missing a "then" part. Is it supposed to mean:

"One rule of thumb, however, is if the distribution is straight on a log-log graph over 3 or more orders of magnitude, then it is a power law distribution."

[edit] external links

I have deleted some external links that were unrelated to this subject and could not be considered external sources of information on power laws. There are lots of published and unpublished scientific works that can be related in some way to the application of power-law techniques, but do not apport any knowledge to the generic properties of power laws or generic applications and methods. 20:31, 21 February 2006 (UTC)

Sorry but I don't agree with 161.116.80.71 about this. Why does the Wikipedia article have to discuss only generic properties? Referring readers of the article to a variety of sources which illustrate the diversity of cases in which power laws occur seems helpful and constructive to me. -- JimR 03:06, 26 February 2006 (UTC)

Since there's been no further discussion on this, I've now restored the previously removed links. -- JimR 09:19, 9 March 2006 (UTC)

[edit] Merging Zipf, Power Law, and Pareto?

This article, the Zipf's Law article, and the Pareto distribution article have significant overlap, for obvious reasons. Is it worth trying to create a central article for this set of material? --Experiment123 02:38, 8 March 2006 (UTC)

I don't think so. Zipf's Law is a particular case from linguistics, and Power laws are not exclusively statistical, but also found in natural sciences. By the way, there is also overlap or a common theme with e.g. 80-20 rule, Lorenz curve and The long tail.--Niels Ø 18:57, 8 March 2006 (UTC)

I re-read the above, and yes, I do actually think a central article clarifying the relationship between these various topics would be good, but no, I don't think the articles should be merged. In fact, a while ago I created an article listing these articles, and linked each article to the list instead of to all the other articles. I never found a good name for my list, though, and it was afd'ed.--Niels Ø 09:49, 9 March 2006 (UTC)

Yes, that's the way I was thinking, too: relatively brief articles for Zipf's Law, Power Law, Pareto, etc., which all contain links to a meatier piece that can discussed shared concepts and talk about the relationships between the subjects. I see what you mean about the name, though. "Power Law Distributions" maybe? Do you have your old article around still somewhere? --Experiment123 12:34, 9 March 2006 (UTC)

It never was a proper article; it was just a list of links to the articles in question. Unfortunately not competent to write such an article, but I'd love to read it! - By the way, add 10/90 gap to the list.--Niels Ø 10:18, 12 March 2006 (UTC)

[edit] Financial market power laws

The article refers to Mandelbrot and Taleb as "recently" popularising the presence of power laws in financial markets. Actually Mandelbrot's statistical analysis of price fluctuations, showing power law tails ("fat tails"), dates back to---and was well-known in---the 1960s. There's also the Pareto law for the distribution of income, which is another fat tail distribution.

More generally the article needs to be more clear about distributions/scalings which are power law for only part of the range.

I don't have time to make the change now but will try to add some material in the near future. —WebDrake 09:57, 29 April 2006 (UTC)

[edit] A How To Guide?

Thank you all - this is the best clearing houses of information on Power Laws I've found. I don't have the most technical of backgrounds, so forgive me if these questions would better be asked elsewhere...

At what point have I confirmed the presence of a power law?

1. I've got my nice straight log-log graph done up in Excel. The y-axis of log of probability runs to -3.5, but my x-axis only carries out to 1.5 (my data set n ~ 1,500).

2. This leads me to my next question, do I have to have a slope of ~-1 for a power law to be shown?

3. My R^2 is 0.9, are there any special considerations for evaluating correlation coefficients for power laws?

How would I compare two power law distributions?

1. My current results give me the equation of f(y) = -0.06x^(-2.4) - What would it show if over time my a and k values were to change?

If I have shown a power law in the data, this means...

1. The presence of a power law shows that there is no mean, and that standard Gaussian or Normal statistics should not be applied to this data set

2. Power laws are evidence of self-reinforcing systems

3. Power laws are evidence of small-world or scale-free networks

4. Power laws imply the presence of random behavior

5. Power laws are simply Cauchy distributions with an asymptote at the mean

6. In a rank order setting the farthest right nodes could be considered to be either / both the oldest and most crucial to a network

Thanks again -- Flybrand 16:43, 30 April 2006 (UTC)

A quite good how-to relating to power laws is found in M. E. J. Newman, "Power laws, Pareto distributions and Zipf's law", Contemporary Physics 46, 323–351. Very brief reply to your questions:

Presence of power laws. Accurately checking for power laws basically comes down to one thing: having enough data. Too little may cause you to not see the law, or get the wrong exponent. When you're binning the data for plotting, you will need to put it in logarithmic bins, otherwise the tail of the distribution will get very messy.

I don't understand what you mean about "a slope of ~-1". Power laws come with many different possible exponents.

Most importantly, standard regression tests are not good for estimating exponents. There's a method described in the Newman paper which is far preferable. Another method of testing involves scaling for systems of different sizes [see e.g. the two papers by Lise and Paczuski (2001)].

Comparing two power-law distributions. I would forget about the a value in y = ax^k. Concentrate on the exponent k. If you're running simulations, you may see that the exponent is lower, or the distribution not really power-law like, if you do not let the system go through a transient period before taking data. Ditto if you have too few data points. If the negative exponent is higher (e.g. if you have an exponent of -2 instead of -1) you will need more data points, as the larger events are more unlikely. This is particularly important with e.g. Lévy distributions where the power-law is only the tail and most points are bunched in the initial peak.

If you have shown a power law in the data, then ... what you can infer tends to be context-dependent. But to go through your cases: (1) whether or not there is a finite mean depends on the exponent. If the exponent is less than -2 a mean exists. If it's greater (e.g. -1.5) the mean is infinite. You can tell this just by basic probability theory. I don't know what you mean by "applying standard Gaussian or Normal statistics". (2) I don't know what you mean by self-reinforcing. (3) Depends on whether you're looking at a network or something else. (4) No, not at all, you can generate power laws by purely deterministic means. (5) I don't know what you mean, but I suspect no. (6) Depends on what system you're looking at. In e.g. the Barabasi-Albert model of scale-free networks, yes, to the "oldest". In other systems, not at all. "Most crucial" is something it's not possible to talk about in general.

These answers were written in a hurry so hope they make sense. I suspect you need to find a complex-systems physicist to work with who can sit down and supervise your work and lead you through things gently. :-)

—WebDrake 14:37, 18 July 2006 (UTC)

[edit] Weibull not power-law?

I'm pretty sure the Weibull distribution is not an example of a power-law distribution, as the current article claims. When Weibull parameter k=1, it's the same as the exponential distribution, which is definitely not power-law. I'm fairly certain that Weibull isn't power-law regardless of the value of k, but in case my understanding is muddled, I wanted to give others a chance to comment. —The preceding unsigned comment was added by Agthorr (talk • contribs) 14 June 2006.

Agreed: it is not a power law (although the formula for the failure or hazard rate h(x;k,λ) given in the Weibull distribution article is a power law). I've removed the Weibull distribtuion from the examples, and put it in See also instead. -- JimR 05:31, 18 June 2006 (UTC)

[edit] with k > 1

I have added "with k > 1", as, e.g., with k = 1 dX/dt = aX^k describes just an exponential, with k = 0 we have a lineal function and so on. —The preceding unsigned comment was added by Athkalani (talk • contribs) 2006-07-13.

Sorry but I don't think this is correct. There are significant cases where k < 0, for example, inverse-square laws and critical exponents. The cases k = 1 (not exponential but simple proportionality) and k = 0 (a constant function) are not very interesting, but they are still degenerate cases of power laws. I've reverted this change. -- JimR 04:24, 15 July 2006 (UTC)

Athkalani appears to have confused the equation y = ax^k with the equation dx / dt = ax^k. It's not an issue. —WebDrake 21:59, 15 July 2006 (UTC)

[edit] "Practical use"

I've removed the comment that "It remains to be seen whether knowing that something follows a power law has a practical use". This seems to me just the sort of very silly POV thing said by people who are cynical about the whole complex-systems-science field. Knowing that something is power-law distributed is as practically useful as any other accurate description of nature. ---WebDrake 10:54, 13 July 2006 (UTC)

The very first sentence mentions that power laws are observed in many fields. No doubt, but some of the listing feel ike they need a reference to me. War, Terrorism? No doubt, there is something in catagories of war and terrorism that does, indeed, follow a power law distribution, but to juxtiposition these next to a list of sciences just felt POV to me. --195.176.176.226 11:40, 27 July 2006 (UTC)

It does seem surprising. But there were already two external links in the article justifying the mention of war and terrorism. I've turned these into explicit reference notes to make the connection clear. -- JimR 05:07, 30 July 2006 (UTC)

I think this is a style issue. The article jumps too quickly into "where power laws are observed" rather than telling you what they are first. As for the juxtaposition of "war and terrorism" versus sciences, power laws really are very ubiquitous — they are observed in hard sciences, biology, social sciences, the works. It would be better to make a short comment ("found in many situations in nature") initially and elaborate later: the trouble with making a list is that everyone and their mother wants to add their personal pet example (it's happening on the self-organized criticality page too). Will try to rewrite soon. —WebDrake 19:37, 3 August 2006 (UTC)

[edit] Category: exponentials

What's the justification for power laws being in the "exponentials" category? —WebDrake 09:30, 19 July 2006 (UTC)

[edit] Major revision of article

I've begun a major revision of the Power law article, which I've placed for the moment at Talk:Power law/Revised article. This is to enable two things:

• A gradual rewrite — it will take time to create a complete article of high quality.
• Collaboration — I do not wish to make such a large-scale change without scrutiny and others' input.

The aim of the rewrite is mainly to bring better structure to the article, and give a better overview of what power laws are. For example the present article's introduction starts by telling you where power laws are found, and only later gets round to describing power laws themselves.

I also propose to clear up a bit the cited references and external links, since it seems that some of the choices are a bit arbitrary.

I'm looking forward to people's thoughts and ideas about this. —WebDrake 18:28, 15 August 2006 (UTC)

I tweaked a couple of parts of your draft, and added the maximum likelihood estimator for discrete power laws (from the Goldstein article). Generally, I like what you've done with the article. I'm a bit concerned that the "power laws in nature" section could get really lengthy, since power laws basically show up everywhere. -- Paresnah 23:42, 28 August 2006 (UTC)

I'm delighted by your contributions, they are very helpful, and you really should feel free to add your own material as well as revise what I've written. Regarding power laws in nature, you're right, it could get huge, and there is a tendency for everyone to add their pet topic to a never-ending list (this is happening on the self-organized criticality page as well, which I will probably have to do another rewrite of at some stage). What I plan for that section is to give a general overview, some cautionary words about spurious "not-really-power"-laws, and some prominent examples as per the reviews cited. We might try to craft a policy regarding what qualifies for being added here, and create a "list of power laws observed in nature" page, or something like it, if too many people want to crowd in, so as to reserve the main page for a concise writeup. —WebDrake 09:54, 29 August 2006 (UTC)

I've done a major revision of the proposed revised article. There are still some things to tidy up, particularly the sections on power-law functions, but I think this new version should be quite close to something we can use to replace the currently public version of the article (which, with every passing day, becomes more and more shameful). Paresnah 11:28, 11 March 2007 (UTC)

Made another pass through the article to flesh out the sections that needed attention, and to more fully integrate its links with other related wikipedia articles. I think this revised version is now very close to being ready to post. Paresnah 03:31, 14 March 2007 (UTC)

I replaced the old article with the new one (and made a couple of small adjustments). Paresnah 01:29, 17 March 2007 (UTC)

[edit] Exponent/universality issues

Following a couple of brief exchanges between Paresnah and I on our talk pages, it's perhaps best to continue the debate here. The discussion is on how to phrase comments relating to universality.

Anyway, here's what Paresnah wrote:

I think that the section on universality needs to make a couple of things explicit. In some systems in physics, e.g., spin glasses, the universality of power-law behavior rests on firm empirical and theoretical grounds. For other systems, such as complex networks or various sociophysics applications, claims of universality seem to be mere speculation. It seems to me that a lot of statistical physicists who work in these areas have imported the familiar (to them) idea of universality, but are applying it in a way that may not be congruent with its original meaning. For instance, in statistical physics, critical exponents describe the functional behavior of a variable, while in these "complex systems", the exponent of a power law describes the probabilistic behavior of the variable. Even if we accept that the idea of universality is applicable to probability distributions, significant problems remain.

For instance, universality requires the coincidence of the scaling exponents. But, in most of these "complex systems", the estimates of the scaling exponent's value are extremely coarse (this is at least partially due to the MLEs not being widely used). People often claim universality after seeing a power law with an exponent in the range 2 < α < 3, rather than by precisely matching exponents as was done for the more traditonal physics systems. Also, 2 < α < 3 is a huge range over which to claim a precise matching, and it's expecting a lot to assume that, as the fitting methods improve, the values will continue to match in a precise way among these disparate systems. And, there are a large number of stochastic mechanisms that all produce power law distributions with exponents in exactly the observed range, so there doesn't seem to be any reason that we should a priori expect any universality class, in the traditional sense, to apply to these probabilistic systems (more on this in a moment). One of the silliest claims of universality in a "complex system" is in the "scale-free networks" literature, concerning the structure of the Internet. Walter Willinger, David Alderson and their colleagues have criticized this claim extensively in a series of careful articles, such as this one.

Finally, there are no validated theories that describe why we should expect to see power laws with certain exponents in these "complex systems". There are lots of interesting ideas, but researchers rarely do the necessary work to validate their models. Without such validation, it seems hazardous to try to generalize two systems into a universality class. Rather, similar behavior could just be a coincidence - perhaps two different mechanisms just happen to produce power law distributions with the similar exponents, but, for other reason, there is no way to put them in the same class of systems. (Or, similar behavior could be a data artifact.) Renormalization provided this deeper connection for many traditional systems, but there is not yet such a theory for any "complex systems". (At least, I'm not aware of one...) Michael Mitzenmacher has commented on some of these issues in a recent editorical for Internet Mathematics (available from his publications page). The problem is basically that people have been getting ahead of themselves (and what their data supports) in pushing into these new "complex systems" areas. I'm pretty sure that all of these issues will get sorted out eventually, but since there's already a lot of confusion in the "complex systems" scientific literature, I think we need to be careful in the wikipedia article not to repeat things as fact that are merely speculation.

Now... I wonder how we can work this perspective into the article.

I would reply that I think it depends on what you're looking at. For example, in the case of self-organized criticality I don't see any reason to doubt that well-defined universality classes exist (and indeed a lot of work has been done here, although I don't think there is a complete theory at this point).

On the other hand as regards scale-free networks I can agree, there are various ways to tune exponents with these models and there is not really a coherent point of view yet. Part of the problem may be that they are rather abstract and are not "physical" in a conventional sense. But they can emerge from physical models (see e.g. the paper by Paczuski on the self-organized criticality page) which may help to resolve some issues in the future.

The issue of coarse-grained observation I think can be dealt with simply. In some empirical cases exponents are not firmly established so it's difficult to make statements. Some may not even be power laws!

That's a very brief reply that doesn't really do justice to your detailed comments, but I think we can deal with the issue quite simply: we refer to traditional thermodynamics and SOC, and then add some comments about "whether universality can be extended beyond these cases continues to be a matter of debate, and some have said ..."

I'll try to read more deeply on the papers you suggested (any more would also be welcome) and come up with some new text. —WebDrake 09:11, 12 September 2006 (UTC)

Right, exactly. The problem here is, I think, that there are several different communities that all deal with power laws (and universality), and we need to be careful in the wikipedia article about the generality of its statements about them. That is, the article should probably note important differences between the meaning of power laws in different communities (e.g., functional versus probabilistic, or critical behavior versus other processes). In some places, such as SOC, the evidence of power laws and universality seems pretty clear (NB: I'm not an expect in SOC), while in others, like scale-free networks and terrorism, it's much less clear, and claims of universality are probably wrong (at least, I would say there's no a priori reason to believe them, and the empirical evidence seems poor). Paresnah 21:29, 12 September 2006 (UTC)

[edit] Accessability

As a layman I find this article very inaccessible. The top paragraphs do not explain at any point in plain english what power law is, it uses a mathematic sum to do the explaining. I'd rework it but I just dont understand what you are trying to explain. —The preceding unsigned comment was added by 210.49.213.27 (talk) 01:52, 16 February 2007 (UTC).