Alinghi, Bacon, & Continued Re: [NEC] 2.3: Power Laws, Weblogs,

Dave Long dl@silcom.com
Wed, 05 Mar 2003 10:14:04 -0800


[vide caveat infra]

> > Many products and many media may be entirely
> > random, without being uniformly distributed.
>
> In fact, if rankings were entirely random, they
> would have a Normal distribution thanks to the
> Central Limit Theorem

No.  Normal distributions arise when there is
some mean value from which to diffuse away, but
randomness leads to many other distributions,
including power law distributions.  Examples:

Zipf's Law
----------
Schroeder, _Fractals, Chaos, Power Laws_ (1991) : Introduction
> Zipf has endeavored to derive his law from _Human Behavior and
> the Principle of Least Effort_ (the title of his 1949 treatise).
> But Mandelbrot, in an early effort, has shown that a monkey hitting
> typewriter keys *at random* will also produce a "language" obeying
> Zipf's hyperbolic law [Man 61].  So much for lexicographic _Least
> Effort_!

The Atmosphere
--------------
? "In fact, if air molecules' heights were entirely random, they would
?  have a Normal distribution thanks to the Central Limit Theorem"

Luckily for those of us who both live near
sea level and seek inspiration on a regular
basis, air molecules are mostly found near
the bottom of the atmosphere, and are rare
in the upper parts.  If atmospheric density
was normally distributed, it'd be thin down
here and thin up below LEO and pretty thick
somewhere in the middle.

Now, this distribution (e^-mgh) is normally
presented as an equilibrium result, but it
turns out that it arises if one starts out
with any atmospheric distribution, and then
swaps energy at random (but conservatively)
between molecules.  (being an equilibrium,
it is a fixed point of the swapping)

"Re: The Big Jump"
<http://www.xent.com/pipermail/fork/2002-September/014706.html>

(this is an exponential, not a power law,
but Schroeder thinks power laws may often
result from combinations of exponentials,
parallel relaxation processes*)

possibly also of interest:
"Re: 50% of FoRK usage"
<http://www.xent.com/pipermail/fork/2001-June/000732.html>
"bell curve?"
<http://www.xent.com/FoRK-archive/oct00/0060.html>

Noises: White, Pink, Brown, and Black
-------------------------------------
(Schroeder again, Chapter V)
> Among the very many domains where self-similar power laws flourish,
> statistics ranks very high.  Especially, the power spectra [of]
> time series, often known as noises, seem addicted to simple,
> homogenous power laws in the form f^-{beta} ... Prominent among
> these is white noise, with a spectral exponent {beta}=0. ... The
> increments of Brownian motion and numerous other innovation processes,
> the learned name for a succession of surprises, belong to this class.
> Electronic and photonic shot noises, thermal noise, and many a hiss
> from man or beast aspire to membership in the white noise "sonority".
> 
> If we integrate a white noise over time, we get a "brown" noise, such
> as the projection of a Brownian motion onto one spatial dimension.
> Brown noise has a power spectrum that is proportional to f^-2 over
> an extended frequency range. ... However, white and brown noises
> are far from exhausing the spectral possibilities: between white and
> brown there is a pink noise with an f^-1 spectrum.  And beyond brown,
> black noise lurks, with a power spectrum proportional to f^-{beta}
> with {beta} > 2.
> 
> Black-noise phenomena govern natural and unnatural catastrophes
> like floods, droughts, bear markets, and various outrageous outages,
> such as those of electrical power.  Because of their black aspect,
> such disasters often come in clusters.  Indeed, "Wyse men sayth
> ... that one myshap fortuneth never alone"; so says A. Barclay in
> his translation of _The Ship of Fools_ [Bar 1509]

The english proverb is "when it rains, it pours";
a pure statmech argument notices that, out of all
possible arrangements of myshaps, there is one (up
to permutation) in which myshaps fortuneth evenly,
and many, many, more in which they are clustered.
Such clustering occurs at all scales, which might
explain why property/casualty insurers are not too
keen on Acts of God.

Schroeder : Introduction
> ... nearly 100 years ago, the Italian economist Vilfredo Pareto
> (1848-1923), working in Switzerland, found that the number of people
> whose personal incomes exceed a large value follows a simple power law
> [Par 1896][Man 63a].
and
"Re: The Rich Really Do Get Richer!"
<http://www.xent.com/pipermail/fork/2002-April/011176.html>

> That current blog rank resembles prior blog rank
> is not a prediction.

It is more of a prediction than saying "I have
no clue about rank" (simplest possible model).
Unfortunately, it depends on history.  Using a
path independent model ought to make even better
predictions, but I don't have one.  What kind of
predictions does yours make?

-Dave

:: :: ::

* he also suggests tangent bifurcations, which
are behind the processes described in:

"Re: Evolution being slow ..."
<http://www.xent.com/FoRK-archive/2001.04/0756.html>

:: :: ::

Schroeder (in relation to an earlier FoRK thread)
did his early work in figuring out what actually
made concert halls sound better or worse, and why
measuring frequency response is not so useful.

(references from [Schr 91])

[Bar 1509] A. Barclay: _The Ship of Fools_ (Translation from the
original Alsatian German of _Das Narrenschiff_, 1494, by Sebastian
Brant; an exposition of abuses within the church and precursor of
the Protestant Reformation.)

[Man 61] B.B. Mandelbrot: On the theory of word frequencies and
on related Markovian models of discourse.  In R. Jacobson (ed.):
_Structures of Language and Its Mathematical Aspects_ (AMS, NY)

[Man 63a] B. B. Mandelbrot: The stable Paretian income distribution
when the apparent exponent is near zero.  Int. Econ. Rev. 4, 111-115

[Par 1896] V. Pareto: _Oevres Completes_ (Droz, Geneva)

:: :: ::

[ I think the most important points here are:
1/ it is better to look to see how things do
   behave than to argue about how they ought
   to behave.  (the distance between theory
   and practice is greater...)
2/ models which make coarser distinctions are
   simpler (have fewer entities) than those
   which make finer ones
but those are meta-arguments, so unless we all
are interested in them, I'll try to limit this
mostly to the matter at hand ]