10.5 wrapping up
We started off this course
with the question
What are Complex Systems?
Our informal answer was
they are
large networks of
simple interacting elements,
which, following simple rules,
produce emergent, collective,
complex behavior.
and I told you that this course
would provide some
insight into what this all means.
The purpose of this sub-unit
is to go over what we covered
and see how much insight
we've gained.
So remember that I mentioned
four core disciplines of the
sciences of complexity.
The discipline of Dynamics,
Information, Computation,
and Evolution and Learning.
and the goals of this course,
were, to give you an overview of what the topics are about,
and to give you a sense of how these
topics are integrated
into the study of Complex Systems.
and while doing so, to give you a sense of
how we can use idealized models
to study these topics.
and we've looked at a lot of idealized models,
in NetLogo.
So if you've gotten this far
in the course,
you should be very proud of yourself,
because we've covered quite a lot
and hopefully,
you've learned quite a lot about complex systems.
So let's review what we've done,
very quickly,
We looked at Dynamics and Chaos,
and learned how it can provide
a vocabulary.
for describing how Complex Systems change over time.
The vocabulary included ideas, such as
fixed points, periodic attractors, chaos,
sensitive dependence on initial conditions,and other terms.
Dynamics showed us
how Complex behavior can arise from iteration
and iteration of simple rules, such as
the logistic map
and we were able to characterize the complexity of behavior,
in terms of the particular kinds of dynamics we saw,
whether they be fixed points, cycles or chaos,
Also the field of Dynamics showed a contrast,
between intrinsic unpredictability
which we saw in chaotic systems.
and universal properties such as,
the period doubling route to chaos, and Figenbaum's constant.
Our next topic was fractals.
Fractals showed us how a new kind of geometry can be developed.
that characterizes real world patterns,
in a more realistic way than euclidean geometry.
Like Dynamics,
the study of Fractals shows us how complex patterns can arise,
from the iteration of simple rules.
and we are able to characterize complexity,
in a different way here,
in terms of fractal dimension.
Information Theory was next and we learned how
Information Theory makes an analogy
between information and physical entropy.
and also a different way of characterizing complexity
that is in terms of Information Content.
So now we've seen several different ways
in which complexity can be characterized.
Genetic Algorithms was next,
and that showed us how
idealized models of evolution and
adaptation can be constructed.
and it also demonstrated how complex behavior
or complex shapes can emerge
from the simple rules of evolution.
which themselves can be thought of as iterative,
Cellular automata...
again we saw, how
cellular automata were
idealized models of complex systems.
This was another way in which
complex patterns emerged from
the iteration of simple rules.
and we learned about the idea of
Wolfram classes which characterize
the complexity of cellular automata behavior,
in terms of these classes of patterns.
We looked at some models of self-organization in biology,
like firefly, synchronization,
bird flocking, fish schooling, ant foraging,
ant task allocation,
and there are many other possible models we didn't cover,
We saw how we can build idealized models such as these
NetLogo models of self-organizing behavior,
and we made an attempt to isolate some of
the common principles of these systems,
in terms of their dynamics,
the information that they process,
the computation they do, and their adaptation,
We looked at models of cooperation,
in particularly the Prisoner's Dilemma model and El Farol Problem model,
This gave us a sense of how idealized models can explain,
self-organized cooperation and social systems,
and in general how idealized models can be
used to study very complex phenomenon.
Then we looked at networks,
Networks gave us a vocabulary for describing
the structure and dynamics of networks in real-world
in terms of concepts such as small-world, scale-free,
degree distribution, clustering, path length, etc.
The models that we explored,
captured some aspects of real world network structure.
such as preferential attachment showed us how we can get
scale-free structure in a network,
This captured the idea of a power law in the degree distribution of networks,
After networks, we covered scaling,
in which we looked at some theories of metabolic scaling in biology,
and the very new area of urban scaling.
We saw that looking at how complex systems scale,
as the sizes increased
can give clues to the underlying structure and dynamics of these systems,
such as fractal distribution networks.
At the beginning of this course,
I said there were two goals of the Sciences of Complexity.
The first is
to provide cross-disciplinary insights into complex systems,
and the second one is
is to develop a general theory of complex systems.
Well clearly we've managed to accomplish the first,
We've seen a lot of cross-disciplinary insights,
that we get from studying different complex systems,
but in this class we haven't talked about
what it might mean to have a general theory of complex systems.
Many people are asking,
Can we develop some kind of general unified theory of complex systems?
that is,
Can we develop a mathematical language that is going to unify
the core disciplines of dynamics,
information processing, computation, and evolution in these systems
Some people have referred to this,
hypothetical language as
a "Calculus of Complexity",
In my book, "Complexity, a Guided Tour,"
I make an analogy with some history,
In the late 1600's,
Issac Newton, along with Gottfried Liebniz
was developing The Calculus.
As James Gleick said in his biography
of Isaac Newton,
He was hampered by the chaos of language,
words still vaguely defined,
and words not quite existing...
Newton believed he could marshal a complete science of motion,
if only he could find the appropriate lexicon...
When I read this,this sounded very much like the state of
complex systems today,
Newton had a lot of concepts
floating around in his head,
that had been developed by previous mathematicians,
notions that he was trying to bring together in a unified whole
such as infinitesimal,
derivative, integral, limit,
and he was able, finally,
to unify these different concepts to develop
the appropriate lexicon
for understanding motion,
that is, the mathematics that we call calculus.
Well, the state of complex systems today,
is very much like the state of mathematics
back before calculus was invented.
We have all these notions
that seem separate in many ways,
I've listed a few of them here,
kind of floating word cloud,
and the idea is
that we need to unify them in some way
to have a mathematics that unifies all these disparate notions,
I am not sure that is going to happen,
It is a very exciting prospect,
but I will leave you with the question
whether complex systems is going to find its own Issac Newton,
either in near or far future perhaps,
To conclude, I will give you a quotation that I like very much,
attributed to Oliver Wendell Holmes,
I do not give a fig for the simplicity on this side of complexity,
but I would give my life,
for the simplicity, on the other side of complexity.