hierarchies> thermodynamics> fractals
What is a fractal? And why does it matter? A fractal shape is
geometry with scale – a middle ground or internal
axis of scale symmetry. So it is no surprise that fractals should turn
out to model an organic world which is based on a logic of growth and
hierarchical
scale.
Ordinary geometry – like ordinary logic or ordinary science
– does not take account of scale. Differences in size are not
seen as something fundamental. A triangle is still just a triangle no
matter what
its size (or duration – temporal scale is also an issue
here). So we might say that the ordinary view is Platonic. A triangle
exists for all time and can assume any size. Or rather, for the
purposes of description, duration and extent are irrelevant to the
definition of triangle-ness.
We could say the same for a propeller, a whorl of turbulence in a
stream, or many other geometrical forms. The shape has an ideal or
Platonic existence, and real life dimensionality or scaling can be
added as local properties – extra facts needed only to define
particular propellers or whirlpools, specific triangles or hexagons.
Fractals are also Platonic in being represented as static, ready-made,
mathematical objects. But at least they
include scaling as basic to their description. And they lead on to
other mathematical models such as scale-free networks where largeness
and smallness finally is granted proper causal status. Size
becomes a mathematically meaningful attribute of a shape.
But let’s start first by considering some classic fractals
shapes such as the Koch curve, Cantor set, Sierpinski gasket and Menger
sponge. Each is created by some iterative step – you perform
the same simple act over and over to infinity - to
fracture some shape over all its scales of existence.
So with the Koch curve, we begin with a flat line segment
– the
initiating state. Next we break its middle third to make a triangular
bump. This is the generating act. Then we take each of the four smaller
flat sections so created and break them in exactly
the same way again on a new smaller scale. And we keep on going ad infinitum, or
until our pen cannot
make any smaller distinct marks.
Note several things here. We started with a line of a particular
length. And it also existed in a space, a void, with higher
dimensionality. The line was a 1D object in a 2D realm. Then the
symmetry of this flat line was broken repeatedly so that it became
buckled
into the surrounding 2D realm over all possible scales.
Originally the 1D line just sat there in the 2D void. This was a very
asymmetrical situation – a highly localised line floating
within a generalised global emptiness. But with the fractal development
of the line, it began to intrude further and further into its larger 2D
context.
This is where the term fractal comes from – the creation of a
fractional dimensionality. As the 1D line is buckled to make a Koch
curve, it takes on some fractional dimension between 1 and 2D. In fact
the Koch curve has a fractal dimension of about 1.2619. Or exactly log
4/log 3
– in other words the four unit buckled
“global” shape
divided by the three unit original “local” shape.
So the fractal dimension is derived here quite logically, and
not at all mysteriously. Note also how this fractal dimension
(in maths parlance, the Hausdorff value)
of 1.2619 is an asymptotic limit. It is a figure that the Koch curve
approaches as it becomes ever more finely broken over all possible
scales.
In theory a Koch
curve can keep approaching this limit forever. We can imagine how we
might keep magnifying sections of the curve and finding yet
smaller bumps. Like Dr Seuss’s Cat in the Hat, under every
hat will be a tinier cat still, all the way down to infinity. So while
log 4/log 3 gives some exact value, in reality we have to keep
dividing, keep magnifying, to add yet further decimal places to our
approximate Hausdorff dimension of 1.2619.
We have only just started to talk about the metaphysics of a
single fractal yet we can see just how many interesting
properties are inherent within this mathematical construct. So
let’s consider a few more fractals.
A Cantor set is created by chopping out the middle third out of a line
repeatedly. We end up with something in-between a 0D point and a 1D
line – a “Cantor dust” with a Hausdorff
dimension
of 0.63 recurring - or log 2/log 3, the
“global”
shape of a two unit outcome divided by the original
“local” or initiating shape of a three unit line.
Where the Koch curve is formed by a buckling into higher
dimensionality, penetrating a 2D plane over all scales, the Cantor dust
is formed by the disintegration of a !D line into almost point-like
segments.
The Sierpinski gasket and Menger sponge are like the Cantor set in
being iterative divisions. The initial shape is broken the same way
over and over, the division being repeated on ever smaller scales until
it asymptotically approaches some fractional dimensional limit. A
triangle has triangular voids cut out of it, a cube is riddled with
cubic voids over all scales.
So what do we have here? Each time we start with a sharply, crisply,
divided
world. We have an atomistic line floating within a void-like higher
dimensional space. Or we can go at it the other way round and start
with a solid shape out of which we begin to bite void-like chunks.
Either way, we start with two unlike extremes – asymmetric in
the strong sense of not just being different in spatiotemporal scale
but apparently
disconnected as ontological types. Then the free mixing of the two
extremes over all scales produces an emergent equilibrium balance. Line
and plane, atom and void, local and global, are muddled together over
every level of their joint existence.
In fact, fractal geometry reveals that there was this hidden extra
direction – an axis of scale – present all the
time in our geometric conceptions. Take any apparently
scaleless object like a line, a triangle, a
cube, and we can see that it was
rather exceptional that the shape was imagined as if a single scale of
representation exhausted its inherent possibilities. The
standard Platonic conception of form as, at the very
least, ambivalent about scale should now seem radically incomplete.
Indeed as
organicists we would want to turn the whole story on its head. The
standard view would be that fractals are a complexification of simpler
geometrical notions. Take a triangle or a line and then find ways to
develop it but breaking it over all scales. But now we are saying that
the fractal object is the simpler, more fundamental, case. If we have
an asymmetry of dimensionality - such as triangle floating in
a
void - then those two opposing extremes should also be mixed together
over all scales. In organic terms, it would take some extra step to
prevent such a "thermalisation" or disordering. So the disordered state
must be considered the simplest, the lowest, level of geometric
existence for the shape.
Note something else here. The standard examples of fractals have all
been created with a telling three-ness. An initial symmetry of
a
line,
triangle, or cube gets broken by something happening to its middle
third.
The reason is that this is the simplest way of representing
the
asymmetry of figure~ground, or local features within a global context.
The most extreme way of breaking a line would be to break it into the
threeness of line-point-line. But breaking it instead into three equal
line segments, or rather two
lines and a line-sized gap, preserves a balance of the local and global
features. It gives equal weight to atom and void, event and context,
and so in fact “wires in” the desired equilibrium
balance as the line is broken over all its available
scales.
Yes, fractal geometry does not create the equilibrium outcome it so
handily portrays. Instead the way it constructs a fractal shape already
presumes such a balance. However this is no problem if we understand
the metaphysical position being taken. Models make our lives simpler by
leaving out unnecessary details. We just need to be aware that it is a
simplification and not (yet) the truth of reality.
It is also worth noting how this 2-to-1 fracturing ratio is the
minimal needed to create scale within an information theoretic approach
to modelling. Elsewhere we have pointed out that information
theory is about the modern atomising of even global form. A binary code
of 1/0 gives us a representation of a smallest scale presence and then
its
absence – the located existence of a something and
then a nothing,
an atom and then a void.
But to represent a world with scale, at the very least we need to
employ one extra
bit and so create a triplet or trinary code. We need to book-end any
event
with a context. So we must have 0-1-0 to represent an atom floating
within some greater void. Or 1-0-1 to represent a void-like absence
within some a block of substance.
Just as (and for exactly the same logical reasons) the duality of a
dichotomy becomes the triality of a hierarchy with the breaking of
scale, so the minimal model of information must become triadic once the
fact of scale is admitted to the mathematical story.
self-similar worlds
So fractals are a little artificial in that they presume what
they also
seem to create. They are fabricated by the repetition of some geometric
act over a nested succession of hierarchical scale. And the result is
self-similarity, or the emergence of an axis of scale symmetry.
Fractals are called self-similar because no matter
what the
scale of observation, whether we zoom in close or step right back, we
appear to be seeing the same world. Snip off a small fragment of a Koch
curve, put it under a magnifying glass, and we will find the exact same
ragged line repeating.
Imagine some other kind of less symmetrical object such as a ball of
cotton wool. Zoom in very close to the wad and we see a tangle of
threads. If we were a bug, we could wander a near infinite maze of
paths.
But step back in scale and the tangle becomes a
fuzzy
ball. Now
we see a different world - something with a surface that can be
wandered around. Step back even further, say a hundred yards, and now
the
cotton wool ball has shrunk to a featureless grey point. It has a
location, yet no discernible structure that can be travelled.
So a
fractal is different in that changes in viewing scale make no visible
change to what is seen. This is a symmetry because moving up and down
creates no distinctions or
memories – nothing to mark the fact a change has actually
happened. It
is the same as if we were to move a step left then right, or reflect a
reflection.
Such symmetries are immensely important in physics as they virtually
define a dimension, a degree
of freedom. There underpin deep principles like Noether’s
Theorem which says for every physical symmetry there must be a
conservation law and thus an inertia. Or Goldstone’s Theorem
that states for every symmetry there must be a harmonic
reverberation and thus a type of fundamental particle. And here we are
talking about another axis of symmetry – the middle
ground axis of scale
symmetry. What can we say about its physical consequences?
Well, that is a topic for the matter pages. For the moment we want to
develop a mental image of the rightness, the logicality,
of scale-free worlds emerging from the sharp dichotomisation
of scale - how a
polarity must also create a connecting middle ground
spectrum.
A fractal starts with some stark separation of a figure from its ground
– say a “substantial” Platonic line
segment afloat in the “form” of an empty amd
unbounded Euclidean plane.
Then fractal development breaks the tension of this strong asymmetry. A
triangular buckling of the line ignites a succession of bucklings over
all possible scales until the line is infinitely buckled, or at least
buckled as far as the eye can spy.
And by this act, a symmetry is restored to the story – a
scale symmetry in
which no viewing scale is privileged and the middle ground, the
“space” between local feature and global context,
has become flat and even in its development. A fractal may look messy.
And it is indeed a complete “disordering” of scale
distinctions. Yet the result is, I would argue, a more complete
notion of geometry.
To find a line of a certain size and duration just floating in some
unlimited space begs the question why? But if line and space are
completely mixed over
every available scale, then somehow this seems more
logical. If it can happen, it will happen. And in fact with real world
systems, it does happen.
rivers, coastlines and other fractals
Fractal shapes like a Koch curve or Menger sponge exhibit some
basic
metaphysical features. For ordinary mechanical logic – which
treats scale as an extra to any description of a world
– the properties of fractal mathematics are merely a local
curiosity. Nothing to get excited about.
But for organic logic, fractals have precisely the properties we want
to explore further. Organicism takes scale as fundamental. There is
always the parts and the whole, the composing substances and the
organising form. Fractals then directly model a
“thermalised” middle ground where scale, having
been created, then gets completly disordered.
As said, Cantor dusts and Koch curves seem rather artificial or
mechanically constructed fractals. Their scale-free nature has been
artfully wired in from the start. So let’s look at some more
natural examples of fractal structures such as clouds,
coastlines, lightning strikes, river tributaries and sandpiles.
Clouds and coastlines have a self-similar geometry. Take a puff
of cloud or ragged stretch of coastline and it will look just as puffy,
or just as ragged, on any scale.
A cloud is a fine mist of water droplets. Its shape is the
result of a balance of factors. Hot air rises, gravity holds air masses
down, then the turbulence of surrounding air currents whips up around a
trapped clump of water vapour, sculpting it into various cloud
forms.
Likewise a rocky landmass is created by a complex story of forces, but
its ragged outline where it meets the sea is a simpler tale of erosion
by wave
action.
So in both cases we have a general stuff – a mass of water
vapour or rock – being eroded or otherwise acted upon by a
generalised outside force, either winds or ocean wave. This eroding
force acts
over all scales, and so no particular scale. The resulting structure of
a cloud or coastline reflects this fact. It is shaped evenly across all
scales of observation (and we can say – pansemiotically
– that the lapping of waves, the gusting of winds is a kind
of “observation” in this context).
Lightning strikes and river tributaries are a similar story, but subtly
reversed. Both lightning and rivers are a source of energy that has to
find its path through a resisting volume. Lightning has to find a path
through a block of air to dissipate itself by completing a circuit with
the ground. A river has to carve a path through a landscape to
dissipate its load of water.
Lighting forks in random fashion, breaking up into feathery
branchlets. The patterns are fractal. Zoom in on a side-frond and it
will dissolve into ever tinier branchlets. This kind of branching
dissipative pattern is modelled by fractal structures such as Cayley
trees.
Rivers also break up into a maze of tributaries when they hit a flat
plain. A weight of water coming down from the hills will slow as the
land levels and the same more gently moving water has to be
shifted in other ways.
A first response is for the river to form snaky bends. By making a
longer riverbed, the plain can absorb more of the weight. But then this
smooth response breaks down into a chaotic fracturing of the flow. Like
the branching of lightning, there will be a few broad streams and a
fractal
network of tributaries – the classic river delta.
Where once there was a sharp dichotomy between land and riverbed
– a 1D line cutting across a 2D surface – now land
and river become all jumbled up. There are tributaries and islets
distributed evenly across all scales. The actual fractal dimension for
the Nile delta is said to be 1.4; that of the Amazon basin 1.85.
Fractal organisation is to be found in all kinds of dissipative
structure – structures that “chaotically”
branch in order to dump a load of energy as fast as possible. A system
which is dissipating energy at a gentle rate, well within itself, will
show rather orderly seeming motions.
A stream in a deep enough channel will have a smooth laminar flow
– the viscosity of water molecules over-coming their thermal
jostle. But constricted or otherwise disrupted, the stream will break
into
burbling turbulence. A system pushed to its through-put limits
– one that is made to “expand” as fast as
it possibly can – will have to break up and do the job
fractally over all available spatiotemporal scales.
This same maximisation of through-put principle rules other fractally
branching systems such as lungs, blood vessels and nervous systems. A
fractal network can both absorb and excrete at a maximal rate, no
matter whether it is water and electricity, or oxygen, carbon dioxide,
bloodstream nutrients, or even neural control information.
We are making connections with a lot of different-seeming ideas here
– dissipative structures, energy flows, chaos, organ systems.
But fractal geometry models something deeply logical about the world.
It is about how the middle grounds of dynamic or self-organising
systems must
develop.
We can see that simple dissipative structures, like a thread
of river, are not really occupying the full landscape available to
them. They are excessively constrained as a geometry. They do
not really fill the rich space surrounding them. So it becomes more
fundamental once they do penetrate their worlds fully, gaining fractal
existence over all scales.
power-law distributions
The mechanical view of a system and its thermodynamics is boxed
– rigidly confined and so constrained to a single smallest
scale. The organic view of a system is unboxed and thus freely
expanding. Every scale is expressed
and the statistical picture is
very different. Therefore we need a natural yardstick for measuring
organic
systems and that yardstick is the power-law distribution or log-log
plot.
Ordinary graph are normal-normal. Each axis is divided into equal-sized
steps. A log-log plot uses logarithmic scales in which each equal-sized
step is actually a whole step up in scale. So for example, 1 to 10
becomes as big as 10 to 100, or 100 to 1000, or 1000 to 10,000.
What this does is flatten scale so that an observer is in effect flying
along an axis of scale symmetry. It models quite directly the idea that
even though every step represents an
order of magnitude change, it is as easy to move in one direction as
another. On
a log-log plot, standing at 100, you are just as close to 1000 as you
are to 10.
There is also the log-normal plot that represents exponential growth as
a straight line. Exponential or ever accelerating growth is found in
many natural
situations with “memory”. Bacterial colonies that
breed by splitting, or other kinds of doubling where what has been
created then freely recreates itself, generates a self-fueling or
geometric growth.
In log-normal graphs, the x axis is counted off in normal
units because the wider
world is taken to be moving forward in time at a steady or linear rate.
The hours and days tick off with unchanging regularity –
inertially! But the y axis is marked out with an accelerating log scale
because some system like a bacterial colony or the money earning
compound interest in a bank is building on itself, every increase in
its scale enabling even greater increases, so causing an accelerating
growth.
By plotting this disconnection on a mixed log-normal scale, a rising
exponential curve can be tamed and shown as a straight line. Every
inertial or constant-rate step made by the world, the context to the
growing system, is matched against an accelerative step being made
within the self-expanding system.
So a normal-normal graph portrays a world where inertial change rules
both figure and ground, both some local system and its more global
context. A log-normal plot shows accelerating expansion in a system
against a constant background – and it is the most
“violent” kind of change because foreground and
background are radically out of kilter. Then a log-log plot represents
a total world undergoing cohesive or generalised acceleration. Both the
local and the global,
the figure and ground, are developing together – expanding at
an accelerative rate rather than merely dawdling along at an unchanging
or inertial rate.
A log-normal world is wild. If growth of some local system like a
bacterial colony or world population goes unchecked, its doubling can
become explosive and “fill up” the larger world
within which it must exist. Exponential increase is unbalanced or
asymmetric. But a log-log world is smoothly expanding across all its
scales and it thus gives us a scale symmetric or scale-free outcome.
Both its local and global aspects develop together in harmony.
This is why a log-log plot, or what is also known as a power-law
distribution, is the natural yardstick for the middle ground
of an organic hierarchy, a world developing by dichotomisation and then
the free mixing of the dichotomised.
welcome to Extremistan
Remember chaos theory in the 1980s? Although not widely understood at
the time, power-law causality was the reason why
“deterministic chaos” in all its many guises was so
exciting and surprising. Organic logic gives us a paradigm now to
understand what chaos theory was really about.
And more recently – mainly because the internet is itself a
prime example – we have had the new kid on the block,
scale-free networks. That is networks with power-law or fractal middle
ground connectivity. Although as usual there are many names for
essentially the same phenomenon. Scale-free networks are also popularly
known as small world networks, “Kevin
Bacon’s” six degrees of separation, fat-tail
distributions, and the long tail effect.
Because of chaos theory and scale-free networks, many people have a
good mathematical grasp of power-law behaviour in systems. But the
switch we are making here is that power-laws are not the exception,
something extra to a world that is essentially normal, linear and
Gaussian. Instead a general acceleration, a general openness, is taken
as being fundamental. It is the unboxed story that comes first, the
boxed or mechanical story that is a secondary development.
A great book dramatising the difference between the two kinds of realms
is Fooled By Randomness by Nassim Nicholas
Taleb. He calls the world
ruled by normal Gaussian statistics, Mediocristan, and the one ruled by
power-laws, Extremistan.
In Mediocristan, the bell curve dominates and all outcomes are averaged
to a single bounded scale. There can be exceptions, but they are
tightly constrained. Randomness is strait-jacketed.
Take for example variations in people’s height, a typical
example of Gaussian statistics. The average height for humans is 1.67
metres
(or 5 foot 7 inches). About 1 in 6 people will be 10 cms taller than
this. But only 1 in 740 will be 30 cms taller. Variation is damped at
the extremes and dies away with exponential speed. So only 1
in a
million will be 60 cms taller than the average, or 7ft 5. And only 1 in
8,900,000,000,000,000,000 will be 90 cms taller, or 8ft 5. (Of course,
this analysis does not take account of syndromes such as pituitary
gland tumours that may distort the odds – we are talking
“normal” height variations here).
As Taleb says, in Mediocristan, every step away from the mean sees a
huge drop in probability. And it does not take many steps for the
unlikely to become the virtually impossibility.
But Extremistan which obeys power-law statistics is quite different.
Nothing really changes with distance. Larger scale events still become
relatively rarer, more sparse, however the drop-off rate is smoothly
constant rather than precipitously sudden.
Taleb uses income as an example of a scale-free realm. If personal
wealth followed a bell curve, in some model Mediocristan economy you
might have 1 in 63 people earning more than $1 million a year. Then
there would be a rapid fall-off as you moved further away from the
average. Only 1 in 127,000 would be earning more than $2 million, 1 in
14 billion more than $3 million, and to earn more than $8 million, the
odds would shoot out to 1 in 1033.
In an Extremistan economy, we again start with 1 in 63 earning over $1
million. But 1 in 125 would earn over $2 million. So instead of a $1
million step up in income creating a 2000-fold drop in probability, it
only halves the number. And every further doubling in annual income
continues only to halve the number of earners. So 1 in 500 are capable
of earning $8 million (compared to 1 in 1033)
and 1 in 1,000 of earning
$16 million (compared to some just about un-nameable number for
Gaussian-ruled Mediocristan).
In the real world, incomes indeed are more like this fractal or
power-law distribution. In the Forbes 2006 list of richest Americans,
Microsoft’s Bill Gates is top on $53 billion. Sheldon
Adelson, a casino owner, at number 3 has less than half that, or $20
billion. Then 30th placed Nike founder Philip Knight has $8 billion and
300th placed winemaker Ernest Gallo has $1.3 billion.
The US economy – which has been freely expanding for many
years – seems to support personal wealth of any scale. What
Gallo makes, or Knight makes, seems to take nothing much away from the
pool of what Gates can make. There is no gravity acting to pull
fortunes down to earth. The scale of wealth is random in this
scale-free sense. There is no constraint on scale so fortunes of every
scale can appear.
Of course in the real world, there are eventually constraints. Even the
US economy is only so big. It can only freely consume the
planet’s resources for so long. Yet for as long as the world
economy is in fact expanding, the wealth of individuals can be expected
to have the power-law randomness of Extremistan rather than the crimped
Gaussian randomness of Mediocristan.
Another way of dramatising the stark difference between the two realms
is the old 80-20 rule. A tell-tale sign that you are in power-law
Extremistan is when the few seem to have the most of anything.
Say you were told that two randomly chosen Americans jointly earned $1
million between them. What is the most likely split of their income? In
Mediocristan, it would be half a million each. But in Extremistan, it
would be more likely that the divide was $50,000 and $950,000.
Compare this to the story for heights. If you knew two people added up
to 14 foot, then it would be far more likely that they are a pair of
seven footers than a six foot and an eight foot person. Eight foot
people are just too vanishingly rare.
Or another example, if you imagine 1000 people in a hall, and the
heaviest man on the planet happens to be among them, how much of the
total does he represent? Just 0.3%. Now imagine the same hall but with
the wealthiest man on the planet, Bill Gates. How much of the assembled
wealth is he likely to represent personally? Around 99.999%.
The 80-20 rule was coined by Vilfredo Pareto who calculated that 80% of
the land in Italy was owned by 20% of the people. But in many cases of
power-law distributions, an 80-20 split can prove to be a severe
underestimate. It can end up that 1% account for 99% of the total.
The flip-side to there being far more “largeness”
than you might rightfully expect in a system with a power-law
distribution is there is also that much more
“smallness”.
So for instance, in a scale-free economy the poor are far more numerous
than any average or median income figure would suggest. This is what
has become known as the long tail phenomenon. A freely expanding system
is growing and so can support wealth and privilege of apparently any
scale. There is no brake on richness. Yet an expanding system is also
leaving an equivalent weight of smallness behind. Poverty or other
measures of small scale existence follow a flat and constant log-log
growth line as well.
This means that every standard deviation step below an
economy’s average wage finds an increasing rather than a
diminishing number of people stuck at that level. There will be many
more at the minimum wage than the average wage.
Consider the wider picture for the US economy. Over the past decade or
so, it has been expanding due to the globalisation of world markets.
All that cheap labour in China and India has been coming on-stream. All
that cheap oil and other natural resources from Russia, Brazil,
Australia, and the Middle East. So the US economy actually represents a
top slice of a new wider-base market. The wealth of Gates, Knight and
Gallo has been paid for by a matching population growth and resource
exploitation in the emerging world – the creation of an equal
weight of poor-ness everywhere else on the planet.
The world economy is dichotomising, growing evenly in both its
available directions. The super-rich are becoming the uber-rich. And
like an iceberg, nine-tenths always sinking below the line, the other 5
billion with not a lot inhabiting the planet will become the other 10
billion by 2050.
the biggest wave
Human examples are useful because they highlight the way that
power-law
statistics are surprising. We are so used to a Gaussian
“boxed average” view of events that the idea there
might have to be both Bill Gates and impoverished billions to make an
expanding economy takes us unawares. Even more disturbing is the idea
that this scale-free or long tail story is an example of a
system’s randomness. It is the pattern of disorder that has
to be created when a vagueness is dichotomised and the separated then
freely mixes over all intervening scales.
A huge number of everyday phenomena do seem ruled by power-law rather
than Gaussian statistics once you start looking. Some are well-known
examples like earthquakes and species extinctions. Others have come as
more recent discoveries.
For example, it used to be thought that ocean waves must have a
Gaussian distribution and so ships would only need to be built to
withstand waves of a certain maximum height. Oil rigs and ocean liner
have been designed to withstand waves of 15 metres (49 feet) because it
seemed safe to assume waves any larger would be vanishingly rare.
Yet a recent satellite survey of the world’s seas over a
three week period found ten rogue waves reaching 25 metres or more
– eight stories high! These waves seemed to well up for no
particular reason. So while gravity must ultimately limit the size of
monster waves, relying on the image of a bell curve to predict maximum
likely wave heights could be dangerous. Largeness does not seem so
tightly constrained as Gaussian imagery would suggest.
Coming back to suitable mathematical models of power-law or fractal
systems, scale-free networks are now the latest focus of interest. As
we have said, fractal geometry, and even other mathematical
representations of “chaos” such as strange
attractors, suffer from being overly Platonic. Their dynamism - their
dependence on history, process, memory, context, development - is
rather hidden from view. But scale-free networks are more transparently
about growth and meaningful communication or semiosis. So even as
mathematical imagery, they start out already more alive.
scale-free networks
The story of scale-free networks goes back to an experiment by the
Harvard social psychologist Stanley Milgram in the 1960s. Milgram sent
letters to individuals in Nebraska, who had been randomly picked out of
the phone book, asking them to pass on a message to a stockbroker in
Boston. The catch was they were only allowed to forward the letter to
someone they knew personally and whom they thought might be socially
"closer" to the stockbroker.
Milgram found that most of the letters that made it to their
destination did so in about six steps, leading to the now famous notion
of small world networks or the "six degrees of separation" - the idea
that we are all linked to one another through short chains of
acquaintances. The notion became popularised by the game in which you
had to link any chosen actor back to Kevin Bacon through a list of
co-stars in various movies. Mathematicians have a similar
game using co-author links to work out their collaboration distance to the
prolific (network theorist, among other things) Paul Erdös.
However it was not until the late 1990s that this kind of connectivity
phenomenon became understood as a consequence of power-law or fractal
system organisation. Various researchers such as Duncan Watts and Steve
Strogatz worked on bits of the puzzle. But the central figure has been
physicist Albert-László Barabási.
A network (also known as a “graph”) is a model of a
communicating world. There are links (or “edges”)
connecting nodes. So network theory, being about patterns of relating,
is good for modelling knowledge structures, social structures and
ecological structures. Complex worlds. But if the universe itself is
considered as a pattern of interactions, of connecting events, then
there is no reason why network theory cannot also be a powerful model
of apparent simplicity, of physical reality itself.
Certainly networks are congenial to our organic approach to causality.
A network models the local and the global in interaction. There are the
particular or individual connections and then the holistic causal
organisation of the network, its behaviour as a whole.
Of course, many models of networks are mechanical. High level
behaviour only “emerges” as a consequence of
bottom-up or “feed-forward” construction. However
once cybernetic feedback is allowed into the story - or better yet,
long term memories and anticipation, as with the neural networks of
Stephen Grossberg and others
– we do indeed get a properly
organic causality. There is a dichotomisation of the causality and thus
a
local~global interaction.
Anyway, network theory has been around a long time and as a
“geometry of connectivity” is obviously of great
ontological importance. So it is surprising that it took quite so long
for scale-free networks to be fully described.
Scale-free networks are created by random connections – the
randomness of an ever-expanding Extremistan, that is. In the 1950s,
Erdös and Rényi worked out the maths of networks
with Gaussian randomness. They set up a circle of nodes and then
assigned an average number of connections to each of them. Some nodes
had
more, some less, but overall the variation was constrained to a bell
curve distribution.
However as Barabási notes, this is a topographic or static
approach to randomising networks. It starts with a fixed number of
nodes and wires them up in one go. He instead decided to explore a more
active, semiotic and growth-based approach.
His
inspiration came from analysing the connectivity patterns of the
internet - the way that pages link to pages to form like-minded
communities. As any surfer knows, there are many pockets of
common interest on the web. One study of 100 million
web pages by Jon Kleinberg claimed they could be parsed by subject
matter into about 50,000
identifiable clusters. Although some of the groupings
were fairly esoteric, such as a cluster of pages devoted to tracking
oil spills off the coast of Japan.
The groupings were of course loosely bordered. A web-site builder
worried about ecological disasters might also have interests in
rock-climbing and Roman history. So "randomly" there would be
some surprising connections made across
the breadth of the web. And within any community, some sites were
always more
prominent than others. These acted like hubs, with many other pages
hanging off them, while other sites were isolated and rarely
visited.
The pattern looked much like that for the world’s air
transport network. A few airports, such as Heathrow, Chicago and
Frankfurt, are really big. Nearly everyone passes through them at some
time even if they are heading on to other places. But still more are
remote
airfields which only get visited by people who really want to get there.
This kind of network is multi-level - a nested hierarchy. There are
top-level airports that connect far and wide, then the regional
airports that connect in a more locally selective fashion.
It seems logical that an air transport or information network might be
organised in this way. But what Barabási and others managed
to show was how such networks would spontaneously self-assemble through
“randomness”.
That is, in a network with scale – in which the global and
local levels are in free interaction and thus able to arrive at their
natural equilibrium balance – then a scale-symmetric or
fractal pattern of connectivity must emerge. It does not have to be
designed in but self-organises
through its own devices.
assuming free growth and preference
The modelling of scale-free networks has to make a few assumptions.
First
there must be free additive growth. The number of available nodes must
always be increasing. That is, in our terms, the local bound - the
universe of distinct locales - is always
expanding.
This is what we see with the internet or global air travel. New pages
and new flight destinations, are being added all the time (though the
internet
has grown up over a decade and air travel over the course of nearer a
century). So a scale-free network is a thermodynamically open rather
than closed system. If the free addition of further nodes halted, the
system would silt up. It would in effect become boxed and so would
eventually be expected to turn Gaussian.
The second key assumption is semiotic. New nodes will show a preference
in where they attach. Even “just by chance”, they
will be more likely to attach to large hubs than small isolated nodes.
This has been dubbed the "Matthew Effect" (Matt. 13:12), because the
rich get richer. Or as they also say, them that has, gets.
It makes intuitive sense. When a new node strays into the network
looking for a place to attach, it is just more likely to strike
something large and obvious than something small and isolated.
And so the rich – the hubs of a network like a Bill Gates or
Heathrow Airport – just keep getting richer with no upward
boundary on their scale. The larger a hub becomes, the more likely it
will be that it takes the lion’s share of the rain of new
nodes.
This logical principle also explains what is know as the prime mover
advantage. Being the first nodes in a network make it more likely you
will gather the early weight of connections and so evolve to become a
future hub, whereas latecomers to the party will struggle to make an
impact (though accidents do happen and it is possible for some new node
to luck out and quickly gather enough links to grow away and become a
hub).
So a scale-free network is based on just a few minimal assumptions
about an open flow of new nodes and their natural habits of attachment.
Yet the model is useful because it delivers some mathematically precise
characterisations of ideal network behaviour.
For a start, the pattern of connectivity is indeed power-law or
fractal. In a scale-free network, there are hubs of every scale, of
every size. Of course, there are only a few really large ones. But in
power-law fashion, the fall-off in largeness is constant not
exponential. So there is no actual limit on largeness (except
obviously no hub can be larger than the system that contains
it).
The balance between bottom-up and top-down connectivity can also be
precisely measured. In a scale-free network, the probability of a node
being connected to k other nodes is 1/kn. In real-life examples like
the internet, n is usually between 2 and 3. Which means that any
particular node chosen at random is four times as likely to have just
half the number of incoming links as any other node.
That is, following the flip side of power-law statistics, it is far
more likely you will find yourself a small player than a large hub.
Like the chances of being born one of the world’s
impoverished billions or one of the Forbes 400 list of billionaires, in
a scale-free realm you are more likely to find yourself looking up from
the bottom of the pile than down from the top.
The chaos-savvy will recognise that this kind of connectivity rule is
like you find in “edge of chaos” models such as
Kauffman’s boolean nets. Scale-free networks also have to be
wired with the right balance of input to output to prevent them
becoming either over-connected or under-connected.
If connectivity falls too low in a scale-free network, the whole system
would simply break up into isolated pockets of activity. You would get
the equivalent of steam - the vapour phase of water - where everything
has become too hot and fallen apart into localised bits. Or indeed the
equivalent of hyperbolic geometry, where every path leads away from
every other, causing a fragmenting divergence.
And if connectivity is tuned too high, then like water freezing to ice
or the closed paths of hyperspheric geometry, the network will close in
on itself and lock up solid. Instead of the rich getting richer, you
would now get the “winner takes all story” that is
a star network topology, with every node attached to the one central
hub.
Like Goldilocks, scale-free networks self-organise to find the flat
middle ground where their connectivity is neither too hot, nor too
cold, but just right. Where their opposing local and global tendencies
– the hyperbolic tendency to diverge and hyperspheric
tendency to converge – are properly mixed and so cancel to
create a flat equilibrium balance over all scales.
slithering sand piles
Power-law statistics and scale-free behaviour explain the organisation
of many complex natural systems from the internet and airports to the
spread of disease epidemics, the branching of rivers, the frequencies
of commonly used word and the growth of towns.
As should by now be obvious, these are all examples of unboxed or
freely expanding systems. They are freely expanding in scale. And
because their scale is always being “thermalised”
– allowed to mix and reach equilibrium – they will
naturally become scale symmetric. Their middle ground is being
disordered – made random. But it is the randomness of organic
Extremistan not the more familiar mechanical Mediocristan.
The rightful maths of Extremistan now has many faces. As well
as fractals, power-laws and scale-free
networks, there are dissipative structures, strange attractors, phase
transitions, criticality, autocatalytic nets, cellular automata,
renormalisation theory, universality, percolation, fluid dynamics, the
edge of chaos, synchonicity – dozens of names for essentially
the same thing. But often quite the wrong causal messages get read into
these chaotic or non-linear models because they
are interpreted from a mechanical perspective
rather than an organic one.
So let’s briefly consider another couple of
standard examples of deterministic chaos - middle ground
disorder - to give a feel for the deeper logic at play.
A favourite of chaos studies is Per Bak’s sandpile. No matter
that real sandpiles do not act in quite the scale-free way modelled
because grains of sand have too much friction. To make a sandpile work
the way it should, it has to be constructed using rice grains trapped
between upright glass plates. Yet still the sandpile is a memorable
image of fractal behaviour.
The idea is that there is a heap of sand being steadily fed by a
trickle of new grains from above (so it is set up as an open or
constantly expanding system). As the pile builds up, its sides
become too steep to be stable and there are sand avalanches that bring
the slope back to an equilibrium balance – a local~global
balance between the global force of gravity and the local forces of
particle friction.
Anyway the model says these avalanches can be of any scale and so will
follow a power-law distribution. Mostly each freshly falling grain will
only
trigger small micro avalanches. But randomly, a really large landslip
can occur.
The “marvel” of the system is that small
causes can have unpredictable, or at least non-linear, results. This
was also what made people goggle about the famous butterfly
wing effect – where due to chaos, the single flick of a
butterfly’s wing in Brazil could some time later be
considered the cause of a tornado in Texas.
However this kind of “from small acorns, mighty oak trees can
grow” take-home message was quite misleading. It was the view
from mechanical causality – from RAMML - where it is
axiomatic that effects are in direct or linear proportion to their
causes. For the mechanist, it seems quite
“random”, indeed alarmingly chaotic, that
small triggering events can sometimes have small effects, yet at other
times
produce cataclysmic convulsions.
Organic logic changes the expectations around because we now take a
holistic view of system’s causality. The middle ground
behaviour of any system is axiomatically the result of local and global
forces in interaction. The lower bound constructs and the upper bound
constrains. So no longer can the whole burden of causal explanation
come to rest on just one grain of sand, or one random butterfly flap.
The causality is shared between the particular events and their general
contexts.
There is still the “wonder” that some very small
and localised event can appear to be the trigger for a monstrously
out-sized result. But the total causal picture now includes all the
other sand grains, all the other flapping butterflies, that did not
seem to tip the balance. Every scale-free system will have its hubs.
But theory also predicts an equal weight of largely isolated activity.
bubbling Bénard cells
Another icon of chaos theory that typically has quite the wrong
take-home message is the Bénard cell.
Take a shallow pan of oil and heat it. Soon it will develop rolling
convection currents. Squeezed together by the sides of the pan, these
ring-like vortices will then get squished to make a vaguely hexagonal
pattern of cells.
Mechanical logic paints this as an example of “order for
free”. Out of the disorderly jostlings of heated oil
molecules emerges magically a high-level pattern of orderly flow. This
hexagonal order pays for itself because it dissipates the heat faster.
What is happening is that when the pan is only gently warmed, it is
each heated molecule for
itself. The molecules chaotically barge about and only eventually
arrive at the surface where they can shed their excess heat. But at a
certain critical temperature there is a phase transition. A symmetry
breaking. The jostling molecules are entrained into a coherent flow,
the rolling convection current which moves them smoothly from the
heated bottom to the cooler surface.
This mechanism is a more efficient way of disposing of the
system’s heat and so obeys the second law of thermodynamics.
The law says disorder must generally prevail, but you can still have
local examples of order, such as the hexagonal
Bénard cells, so long as overall they contribute to the
wider disordering of the world, the dissipation of energy as entropy or
waste heat.
This is again a causal surprise apparently explained. It is a
puzzle
that order in the shape of life and mind can appear in the midst of a
universe ruled generally by a disintegration into thermal
disorder. A Bénard cell shows
how order can arise spontaneously within wider or more global disorder
as a
localised dissipative structure.
This is all certainly true. But heat a Bénard cell just a
little more
and its orderly hexagonal cells soon break up and you get instead
turbulent – or chaotic – disorder again. The order
turns out to
exist only for a very finely controlled balance of heat dissipation.
Organic logic would prompt us to look at a Bénard cell
in different light. What is going on now? Well, we start with
the gentle heating and
localised thermal motions of oil molecules. We reach the critical point
and suddenly the oil molecules first start to feel “the shape
of the container”. They are moving about enough for the walls
of the pan to begin to exert a constraining or boxing effect. The
result is the lining up of the molecules into rolling convection
currents. We have the first inklings of a local~global system in place.
But it is still only the start of the story as we have simply jumped
from a locally-dominated system – one where the only pattern
is the local jostlings – to a globally-dominated one, where
the molecules are now entrained to some single global scale of orderly
behaviour.
We have a dichotomy being created in stages in other words. First the
local boundary, then the global boundary. Next must come the free
mixing of the dichotomised. We must see the eruption of convection
currents on every scale – a power-law distribution of
turbulent flows - to complete the natural causal trajectory to full
blown "systems-hood".
So when the hexagonal cells break down into chaotic boiling motion with
a little extra heating, this third stage is actually the take-home
message. Out of dichotomisation erupts a flat, scale-symmetric, middle
ground. The system has completed its transition when some new
feature – in this case cooling convection patterns
– has made its power-law appearance over all possible
scales.
And this result obeys the second law of thermodynamics. The
hexagonal cells – the simple global order – is more
efficient at dissipating heat than the simple local motions. But a
complexly developed or fractal pattern of turbulence dissipates the
maximum possible degree of heat. It “expands” the
system the fastest (throughput being equivalent to expansion in open
systems ontology).
and back to organic hierarchies
Well we have spent a lot of time here on the possible mathematical
representations of the middle ground of organic hierarchies. It is good
news that there are so many ready-made models lying about because a
logic needs to be fully fleshed out.
We can think of organic
logic as itself a hierarchy. At the most
general (and indeed vague) level we start with some broad notions about
organicism, holism or systems science. We have a few slogans and mental
images to do with a complex causality, such as the conviction that the
whole is more than the sum of its parts, that feedback, mutuality and
process are important, that purpose and meaning must be modelled as
well as information and substance.
Then the purpose of this site is to distil this general organic
understanding down to a single crisp causal mechanism – a
particular logic that describes how any event, any development, must
happen.
But in-between the general idea and the particular mechanism there must
be a variety of concrete models. We must have a middle strata
of
useful mathematics that give various glimpses into real developed
systems behaviour. For instance, some models like scale-free networks
will be good for accounting for steadily expanding worlds. Others like
Ising spin glasses are better at modelling sudden shifts in a system
such as a phase transition.
So now we can get back to the logic of hierarchies as a whole
–
talking about not just their middle ground thermodynamics but also
their constructing and constraining upper and lower bounds. Thus to
resume
the organic hierarchy thread........[under construction]