>Reflectorites interested in chaos, complexity and how they may be related
>to punctuated equilibrium may want to take a look at
>
>"Criticality and Parallelism in Combinatorial Optimization" by William G.
>Macready, Athanasios G. Siapas and Stuart A. Kauffman, ppp 56-59 of the 5
>January 1996 issue of Science.
>
Thanks for the reference Bill.
This is an example of the "new paradigm" that I referred to earlier, i.e.
self-organized criticality (SOC). It is sometimes (especially in the
Physics lit.) referred to as the BTW model, after its originators,
Bak, Tang and Wiesenfeld. The model was introduced fairly recently:
===================================================================
Bak, P., C. Tang and K. Wiesenfeld (1987). "Self-Organized
Criticality: An Explanation of 1/f Noise," Physical Review
Letters 59(4):381-384.
ABSTRACT: We show that dynamical systems with spatial degrees
of freedom naturally evolve into a self-organized critical
point. Flicker noise, or 1/f noise, can be identified with
the dynamics of the critical state. This picture also yields
insight into the origin of fractal objects.
Bak, P., C. Tang and K. Wiesenfeld (1988). "Self-Organized
Criticality," Physical Review A 38(1):364-374.
ABSTRACT: We show that certain extended dissipative dynamical
systems naturally evolve into a critical state, with no
characteristic time or length scales. The temporal "fingerprint"
of the self-organized critical state is the presence of flicker
noise or 1/f noise; its spatial signature is the emergence of
scale-invariant (fractal) structure.
=====================================================
Some recent papers by Bak and his colleagues related to Punk Eek and
macroevolution follow (abstracts supplied so you can get some idea
of what they are up to):
===================
Bak, P. and K. Sneppon (1993). "Punctuated Equilibrium and
Criticality in a Simple Model of Evolution," Physical Review
Letters 71(24):4083-4086.
ABSTRACT: A simple and robust model of biological evolution of
an ecology of interacting species is introduced. The model
self-organizes into a critical steady state with intermittent
coevolutionary avalanches of all sizes; i.e., it exhibits
"punctuated equilibrium" behavior. This collaborative evolution
is much faster than noncooperative scenarios since no large
and coordinated, and hence prohibitively unlikely, mutations
are involved.
Flyvbjerg, H., K. Sneppen, and P. Bak (1993). "Mean Field Theory for
a Simple Model of Evolution," Physical Review Letters,
71(24):4087-4090.
ABSTRACT: A simple dynamical model for Darwinian evolution on
its slowest time scale is analyzed. Its mean field theory is
formulated and solved. A random neighbor version of the model
is simulated, as is a one-dimensional version. In one dimension,
the dynamics can he described in terms of a "repetitious random
walker" and anomalous diffusion with exponent 0.4. In all cases
the model self-organizes to a robust critical attractor.
Bak, P., H. Flyvbjerg, and K. Sneppen (1994). "Can we Model Darwin?,"
New Scientist 141(12 March 1994):36-39.
ABSTRACT: Reducing Darwin to a set of equations may never be
possible. But a promising computer model shows that mass
extinctions could have happened naturally as a consequence of
the simple principles of evolution.
Bak,P. and Paczuski,M. (1995). "Complexity, Contingency and
Criticality", Proceeding of the National Academy of Science
USA, 92:6689-6696.
ABSTRACT: Complexity originates from tbe tendency of large
dynamical systems to organize themselves into a critical state,
with avalanches or "punctuations" of all sizes. In the critical
state, events which would otherwise be uncoupled become
correlated. The apparent, historical contingency in many
sciences, including geology, biology, and economics, finds a
natural interpretation as a self-organized critical phenomenon.
These ideas are discussed in tbe context of simple mathematical
models of sandpiles and biological evolution. Insights are gained
not only from numerical simulations but also from rigorous
mathematical analysis.
Sneppen, K., P. Bak, H. Flyvbjerg, and M.H. Jenson (1995).
"Evolution as a Self-Organized Critical Phenomenon", Proceedings
of the National Academy of Sciences USA, 92:5209-5213.
ABSTRACT: We present a simple mathematical model of biological
macroevolution. The model describes an ecology of adapting,
interacting species. The environment of any given species is
affected by other evolving species; hence, it is not constant
in time. The ecology as a whole evolves to a "self-organized
critical" state where periods of stasis alternate with
avalanches of causally connected evolutionary changes. This
characteristic behavior of natural history, known as
"punctuated equilibrium," thus finds a theoretical explanation
as a self-organized critical phenomenon. The evolutionary
behavior of single species is intermittent. Also, large bursts
of apparently simultaneous evolutionary activity require no
external cause. Extinctions of all sizes, including mass
extinctions, may be a simple consequence of ecosystem dynamics.
Our results are compared with data from the fossil record.
===================================================================
Previously, I indicated my intent to critique SOC from the point of
view of algorithmic info-theory. Lest this be misunderstood let me
say first that I think that SOC seems very promising in a number of
respects. For example, that the basic features of punk eek arise
naturally from nonlinear dynamics seems promising in gaining some
understanding of evolution. I believe that Phil has criticized Punk
Eek as being simply an acknowledgement of the empirical results but
lacking any explanation. Perhaps this must be modified somewhat in
light of Bak's work.
Another positive aspect of SOC is that it apparently turns up in a
wide variety of physical phenomena. I was delighted to find an
application even in my own particularly specialized field of
polymer mechanics where SOC has been used to model several things
including the behavior of polymers at the so-called glass transition.
So, I have a professional interest now in SOC and am curious to see
whether this will offer me any new insights.
My objection to SOC is the proposal that this behavior be considered
as a paradigm for complex systems is general. What clued me in on this
is exactly what I mentioned above wrt polymers. The glass transition
in polymers is an example of what is more generally referred to as a
second order phase transition. Bill also mentioned this in his summary
of the Kauffman paper "They compare it to a phase change phenomenon".
The problem is that phase changes are processes of simplification
rather than complexification, i.e. processes with increasing order.
This is "looking through the wrong end of the telescope" as Yockey
likes to say :). These processes may play an important role in
understanding some types of behavior of complex systems but I think
it will prove a dead end for the grand scheme of trying to find the
underlying principles of complexification.
Another way of looking at this is to note from the above abstracts the
link between SOC and fractal structures. From the point of view of
algorithmic complexity, fractals are exceedingly simple.