A system that is in a state of chaos, high entropy, or far-from-equilibrium
conditions would exhibit high-dimensional changes in behavior patterns
over time, but not indefinitely so. Systems in that state tend to adopt new
structures that produce Self-organization is sometimes known as "order for
free" because systems acquire their patterns of behavior without any input
from outside sources.
There are four commonly acknowledged models of self-organization:
synergetics, introduced by Herman Haken; the rugged landscape, which
was introduced by Stuart Kauffman; the sandpile, introduced by Per Bak;
and multiple basin dynamics, introduced by James Crutchfield. What they
all have in common is that the system self-organizes in response to the
flow of information from one subsystem to another. In this regard the
principles build on the concepts of cybernetics that were introduced in the
early 1960s, and John von Neumann's principle of artificial life: all life can
be expressed as the flow of information.
The basic synergetic building block is the driver-slave relationship, which
can be portrayed with simple circles and arrows. The driver behaves
over time (produces output or information) according to some temporal
dynamic such as an oscillation or chaos. The driver's output acts a control
parameter for to an adjacent subsystem, which one the one hand responds
to the temporal dynamics from the driver and produces its own temporal
output. In the simple case, the driver-slave relationship is unidirectional. In
other cases, such as when effective communication and coordination occur
between two people, the relationships are bidirectional.
A larger system would contain more circles and arrows. What we want to
know, however, is what do the arrows mean? This is where the dynamics
are of great importance.
Once patterns form and reduce internal entropy, the structures maintain
for a while until a perturbation of sufficient strength occurs that disrupts
the flow. The system adapts again to accommodate the nuances in some
fashion, either through small-scale and gradual change or a marked
reorganization. The latter is a phase shift
. For instance, a person might be
experiencing a medical or psychological pathology that is unfortunately
stable, and thus prone to continue, until there is an intervention. The intervention takes some time to be effective but the system eventually breaks up
its old form of organization and adopts a new one.
The phase shift in the
system is akin to water turning to ice or to vapor, or vice versa.
challenge is to predict when the change will occur. There is a sudden
burst of entropy in the system just before the change takes place, which
the researcher (therapist, manager) would want to measure and monitor.
A concise intervention at the critical point could have a large impact on what happens to the system next.
An important connection here is that the phase shift that occurs in
self-organizing phenomena is a cusp catastrophe function. Researchers
do not always describe it as such, but the equation they generally use
to depict the process is the potential function for the cusp; the only
difference is that sometimes the researchers hold the bifurcation variable
constant rather than a variable that is manipulated or measured.
The red ball in the phase shift diagram indicates the state the system is
in. In the top portion of the diagram it is stuck in a well that represents
an attractor. When sufficient energy or force is applied, the ball comes
out of the well and with just enough of a push moved into the second
well. In some situations we know what well we're stuck in, but not
necessarily what well we want to visit next. The question of how to form
a new attractor state is a challenge in its own right.
For the rugged landscape scenario, imagine that a species of organism
is located on the top of a mountain in a comfortable ecological niche.
The organisms have numerous individual differences in traits that are
not relevant to survival. Then one day something happens and the
organisms need to leave their old niche and find new ones on the
rugged landscape, so they do. In some niches, they only need one or
two traits to function effectively. For other possible niches, they need
several traits. As one might guess, there will be more organism living in a new 1-trait environment, not as many in a 2-trait environment, and so on. Figure 9 is a distribution of K, the number of traits required, and N the number of organisms exhibiting that many traits in the new environment. It is also interesting that there is a niche at the high-K end of the graph that seems to contain a large number of new inhabitants.
The niches in the landscape can also be depicted as having higher and lower elevation levels, where the highest elevation reflects high fitness for the inhabiting organism, and lower elevations for less fit locations. Organisms thus engage in some exploration strategies to search out better niches. Niches have higher elevations to the extent that there are many forms of interaction taking place among the organisms in the niche. The rugged landscape idea became a popular metaphor for business strategies in the 1990s. For further elaboration, see Kevin Dooley's linked contribution on rugged landscapes.
For the avalanche model, imagine that you have a pile of sand, and new sand is slowly drizzled on top the pile. At first nothing appears to be happening, but each grain of sand is interacting with adjacent grains of sand as new sands falls. There is a critical point at which the pile avalanches into a distribution large and small piles. The frequency distribution of large and small piles follow a power law distribution.
A power law distribution is defined as FREQ[X] = aXb
, where X
is the variable of interest (pile size), a
is a scaling parameter, and b
is a shape parameter. Two examples of power law distributions are shown in the diagram. Note the different shapes that are produced when b
is negative compared to when b
is positive. When b
becomes more severely negative, the long tail of the distribution drops more sharply to the X
axis. All the self-organizing phenomena of interest contain negative values of b
. The |b
| is the fractal dimension for the process that presumably produced them. The widespread nature of the 1/f/span> relationships led to the interpretation of fractal dimensions between 1.0 and 2.0 as being the range of self-organized criticality.
An easy way to determine the fractal structure of a self-organized process is to take the log of the frequency and plot it against the log of the object size. Then calculate a correlation between the two logs. The regression coefficient is the slope of the line, which is negative. The absolute value of the slope is the fractal dimension.
The multiple basin concept of self-organization also builds
on a biological niche metaphor and attempts to explain how
biological species could cross a species barrier. Imagine there
are several basins, each containing a population of some
sort. The populations stay in their niches while they interact,
change, and do whatever else they do. But the niches are
connected, so that once enough entropy builds up within a
basin, a few of the members bounce out into the adjacent
Multiple basin dynamics can also be found in economics
where, for instance, product designs and product prices
combine to meet distinct market needs. Sometimes, however,
a product producer will jump into another basin. It is an open
question as to how similar the process of jumping basins is to
jumping fitness peaks in the N|K model. Arguably, the multiple
basin scenario is a continuation of the N|K story.