

In Mysterious Pattern, Math and Nature Converge (2013) - jckt
https://www.quantamagazine.org/20130205-in-mysterious-pattern-math-and-nature-converge/

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vanderZwan
Is the evolvability of complex information systems that this article speaks of
related perhaps? (was shared on here a while ago as well)

[http://nautil.us/issue/20/creativity/the-strange-
inevitabili...](http://nautil.us/issue/20/creativity/the-strange-
inevitability-of-evolution)

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westoncb
Interesting! Reading a bit more on the Wikipedia article, my understanding is
that 'Universality' is an appropriate name because it describes classes of
extremely diverse systems that can be described by the same abstract
model—which also always happens to be a scale-invarient model that resembles a
physical phase transition. Is that right, anybody? I thought this list of
systems with the same 'universality class' was interesting (from the Wikipedia
article[[http://en.wikipedia.org/wiki/Scale_invariance#Universality]](http://en.wikipedia.org/wiki/Scale_invariance#Universality\])):

"Avalanches in piles of sand. The likelihood of an avalanche is in power-law
proportion to the size of the avalanche, and avalanches are seen to occur at
all size scales. The frequency of network outages on the Internet, as a
function of size and duration. The frequency of citations of journal articles,
considered in the network of all citations amongst all papers, as a function
of the number of citations in a given paper. The formation and propagation of
cracks and tears in materials ranging from steel to rock to paper. The
variations of the direction of the tear, or the roughness of a fractured
surface, are in power-law proportion to the size scale. The electrical
breakdown of dielectrics, which resemble cracks and tears. The percolation of
fluids through disordered media, such as petroleum through fractured rock
beds, or water through filter paper, such as in chromatography. Power-law
scaling connects the rate of flow to the distribution of fractures. The
diffusion of molecules in solution, and the phenomenon of diffusion-limited
aggregation. The distribution of rocks of different sizes in an aggregate
mixture that is being shaken (with gravity acting on the rocks)."

~~~
drostie
Close but not quite. The term "universality" is different in phase transition
theory from what it means in random matrix theory (which is what's at play
here), but they've got some similarities too.

In phase transition theories, you've got two different states (like liquid
water and water vapor), and when you vary some high-level parameters (like
pressure and temperature) you can go from one of these states to the other.
This means that the only "interesting" physics (in the sense of "distinct from
the well-understood liquid/gas behavior" can only happen right as you get near
that transition, so that liquids are seamlessly becoming gases which are
seamlessly becoming liquids again. The "transition" allows a lot of behavior
not seen elsewhere, in fact all of the behavior "in the middle" between the
two regimes. Because you've got this considerable mixing of the two states,
often "zooming in" is the same as, say, adjusting the proportion of liquid to
gas -- so since it all happens in the same space, you look for these "scale
invariant" theories that are the same upon zooming in. Those theories then
can't depend on too much particulars, but just depend on various symmetries,
so they become "universal". See
[http://en.wikipedia.org/wiki/Critical_exponent](http://en.wikipedia.org/wiki/Critical_exponent)
.

Random matrix theory is similar, but for a different reason. The issue is, if
you have a very complicated system that you can represent as a huge matrix,
often the eigenvalues of the matrix tell you something concrete and physical.
The example given above was the eigenvalues of a Hamiltonian matrix in quantum
mechanics, which gives you an "energy spectrum" (discrete energies that the
system can be at, so that it can e.g. absorb a photon of energy B - A to
transition from a state of energy A to one of energy B).

Why would a Hamiltonian be random? You have to imagine a big molecule with
lots of parts, not entirely under the control of the experimentalist. Maybe
you've got a carbon nanotube hanging over a trench that you've etched under
it, but the etching has caused other atoms to be stuck to the tube in
unpredictable ways, and maybe the "islands" on either side couple to the
nanotube in complicated ways.

Wigner discovered that very often these random variations break the symmetry
between certain levels, so that where you once had 3 states at the same
energy, now it's like those "energy eigenvalues" have "repelled" each other.
He realized that the right way to start to think of the problem involved
taking a matrix and adding random elements to it; this led to a nice set of
models where you take it to the extreme and just randomize the entire matrix
and look at its eigenvalue "density" rather than the exact levels. Wigner in
particular discovered that this density function tends to look like a
semicircle.

The essential similarity between these two is, you get to some point where
"there's nothing more to say". With random matrices, when you specify how
you're building the matrix and what sorts of properties it has to have, then
you get the eigenvalue spectrum, and there's no other details to fixate upon.
Similarly when you have a phase transition, the "we're taking on all the
states in between gas and liquid" status of what you're doing means that the
particulars of those states cannot matter. So in both cases it becomes just,
"what's the configuration, what symmetries does it have" that determines how
macroscopic parameters (whether critical exponents or eigenvalue densities)
ultimately behave.

~~~
westoncb
Hey, thanks a lot! Didn't expect such a great reply. I was still a bit fuzzy
on a couple points, if you don't mind. I'm visiting another country and
haven't found reliable wifi yet, hence slow reply time.

I think I get the scale invariant theory concept from a mathematical
perspective, but I don't see why this would be the case: "Because you've got
this considerable mixing of the two states, often "zooming in" is the same as,
say, adjusting the proportion of liquid to gas"

Regarding random matrices, my understanding now is that given a random matrix
of sufficient size, under a suitable definition of "random," if we look at the
eigenvalue density function it's always going to be (roughly?) the same--we at
least know it will be semi-circular. Further, there exist physical systems
that can be modeled by random matrices; and there's a mapping between
eigenvalues of the matrix and certain physical characteristics of the system.
So, knowing the density function is always the same for these random matrices,
we can assume certain shared characteristics of any systems that can be
modeled by a random matrix.

In random matrices and phase transitions we would like to know how certain
macroscopic parameters will behave, given some data like a matrix, or state of
a phase transition. But, in both cases our starting data contains a lot of
essentially irrelevant data that these theories prescribe a method for
filtering out, since it assures us that knowledge of the symmetries involved
are all that will matter.

Am I close? :)

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kjhughes
Loved the article, but wonder why this pattern, as ubiquitous as it may be,
merits the name 'universality'. For all of the universally applicable laws in
science, it feels like naming overreach to christen this particular result
'universality'.

~~~
powertower
I kind of imagine that it's named so, because Complex systems (at least the
systems that have no closed form solutions -> they can't be solved by an
equation, they can only be modeled and simulated) end up displaying this type
of state and pattern.

These types of systems are basically anything from the 3-body problem (3
particles interacting via gravity), to the system of all the particles in this
universe. And everything that falls in between.

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dubya
This paper has 7 explicit examples of the universality phenomenon, including
the bus scheduling.

[http://arxiv.org/abs/math-ph/0603038](http://arxiv.org/abs/math-ph/0603038)

It's moderately technical, but really interesting.

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squidfood
Interesting article, but this reminds me of the whole "power laws and long
tails" that was big 10 years ago, or the perennial Golden Ratio. Seeing
commonalities of pattern can be insightful to modeling a system, but
ultimately promises that "it's all connected" are far overblown.

~~~
rndn
Here are the slides of a talk on universality that Terry Tao gave some time
ago:
[https://terrytao.files.wordpress.com/2011/01/universality.pd...](https://terrytao.files.wordpress.com/2011/01/universality.pdf)

It's basically the observation that large systems converge to exhibit rather
simple behaviors. It generalizes other reoccurring patterns such as power
laws, Euler's number, the normal distribution and the fibonacci sequence. For
some reason nature is surprisingly frugal regarding the forms and behaviors
that she allows. I think the simpest explanation for that comes from the
anthropic principle and the multiverse hypothesis which state that all
possible physical laws are realized in different universes and we happen to be
in the one that has the necessary conditions to bring us into existence
(universality being perhaps one of them).

~~~
squidfood
Well, sure. I say "oh look, when animals breed, it has this curve" and then I
say "oh look, when money is invested it grows with this curve." The insight is
that "multiplication works".

This helps, because I don't need to re-invent logarithms and e to study animal
population growth. And of course, different rules lead to different math (be
it power laws, the Normal distribution, universiality, Phi, whatever).

This is interesting philosophically. HOWEVER, these models are still
simplifications of reality. What happens with slight departures in animal
population trajectories can be very different than perturbations in interest
rates, so it would be a mistake to say "these things are truly connected"
beyond the practicality of using similar math.

~~~
rndn
I'm not too familiar with all of this (perhaps I'm even misinterpreting it),
but I think they do realize that it would be problematic to throw population
trajectories and interest rates into the same pot. I think the motivation is
rather to find reasons why things do not behave completely unpredictable at
larger scales and I cannot think of good reasons against efforts to find more
and more general descriptions of that.

~~~
squidfood
I agree with the motivation and interest! The issue comes when people try too
hard to find patterns to fit a preconceived hypothesis.

I maintain a large (publicly available) ecological dataset, and my data have
been drawn into several meta-analyses of this type. Often the idea is to
simply see if my data empirically fit the "right" distribution. And they fit
it and say "wow, it's all connected".

But then I look at the Bus data in this example (the +'s that represent actual
data). I'm guessing I could fit a lognormal distribution, a Gamma
distribution, or the Dyson distribution that they actuall use, and the data
wouldn't be enough to distinguish between them.

Now, all of these distributions result from "simple" rules, but they are three
very different sets of simple rules. For the Bus data, the "repelling" by
little slips of paper makes sense as the mechanism, so it's a good hypothesis.

But then to flip that around, and say "since this distribution fits the
ecological data, the underlying mechanism must be Bus Repelling" is wholly
unjustified (as there are other possible fits). And there's a _lot_ of junk
science that does that.

