So my one sentence definition is

systems that don't yield to

compact forms of representation

or description, and I should explain that.

In the systems that physicists study,

you can often write down on one page

a few very beautiful, elegant equations, like Newton's

laws for the conservation of momentum, or the Maxwell field

equations for electromagnetism, and so forth.

And you can explain a huge amount of empirical data

when it comes to the genome, or the brain, or properties

of society or literary history, as far as we know,

there are no such beautiful, elegant, compact descriptions.

And that, for me, is evidence

that we're dealing with a complex system.

Now, why is that? So, the reason why I think it's difficult

to do is because those are systems that encode long histories.

One of the characteristics, for me,

of a complex system, is that it has found a means,

or a mechanism, for extracting from its environment

some information, in order to use it to behave

adaptively. To predict and control.

And consequently, it needs to be described using models

that have a slightly different flavor to the ones that

we have been traditionally familiar with in the mathematical natural sciences.

And typically, those models will be computation.

So, I'm going to ask you

the same question that I'm asking everybody, which is

what is your definition of a complex system. Oh no!

That's what everybody says.

In theoretical computer science, we don't say that

systems are complex or simple, per se,

we more typically say that questions are

complex if those questions require a lot of computational

resources to solve. A lot of time, a lot of memory,

a lot of communication between people.

Some limited resource.

Different questions might have different levels of computational complexity. So for instance,

if what you want to know is what will the system

look like t time steps from now,

you can answer that question with about t time

by simulating it forward, but an interesting

question might be, well, maybe there is no algorithm that works much

faster than that. Maybe there's no way to leapfrog over

the history. Maybe like a chaotic dynamical system that

has no closed-form solution, maybe there is no shortcut to doing

that laborious step-by-step simulation.

So, for me, I find it helpful to, rather than

saying, is this system simple or complex? I mean, I don't deny

that we often have clear ideas about that, but I find it helpful

to change the question a little bit to, give me

a yes or a no question you want to answer about this sytem,

or a quantity you want to compute about this system, and then let's talk

about how hard it is computationally to answer that question

or compute that quantity.

Well, it's a complicated

concept. I didn't say complex, just kind of

complicated.

So, this actually ties in with a discussion I'm sure we'll

have on information, so I have a rather precise notion

of what I mean when I refer to a natural or artificial

system as complex, and what I mean in particular is that it has a very sophisticated

internal causal architecture that stores and processes information.

So, the technical things that we'll talk about shortly have to do

with how we measure stored information and the amount of structure.

So, information in many ways stands in for trying to describe

how complex a complex system is, and various

kinds of information processing and storage can be associated with how a system is organized.

So it's a key concept, certainly

Shannon's original notion of information as degree of surprise,

degree of unpredictability in a system, or how random a system is

needs to be augmented. So that's certainly the focus of a lot

of my work is trying to delineate that there are many different kinds of information,

not just Shannonian information.

So, my definition of complex systems

is a system that has

interactions. It has nonlinear elements in it.

I tend to work on high dimensional systems not low dimensional systems.

And I like to use the methods of statistical mechanics

from physics to understand problems in these systems.

Most of the time, the interesting features in these systems have

scaling properties, that is to say they have power laws

or fractal objects in them, embedded in them someplace,

either in the actual physical arrangement of them

or in terms of the statistics that you see.

So my basic definition is that a complex system consists of a bunch of entities

that may not start out diverse, but end up being diverse. They are connected

in some way, usually through some sort of network structure or some spatial

structure, and they get information and signals through that network

or local structure, but they also sometimes get some global signals or global information,

which could be prices in a market, or temperature in a system, so that

in addition to be sort of diverse and interconnected, they are also interdependent, so the

actions of one agent in the system will sort of

influence or have implications for another agent. So in the context

of a social system, like in economics, I'll say that if I go in and buy bread in the grocery store

whether I buy whole wheat bread or white bread, you really don't

care. It's not interdependent. There's no real strong interdependence, other than to

the prices. But if I decide to drive my car on the road

or drive my car really fast down the road, those sort of things, then that actually can affect you

in a big way. So they're interdependent. And the last thing

is in addition to having these

interdependent behaviors and networks and diverse agents, that the agents adapt

and respond to the environment which that they're in. So it's not just a case of them following

simple rules, but that they sort of adapt. Now this last part

gets a little bit tricky philosophically, because adaptation is really just a higher-order

rule, so you can have a first-level rule and then a meta rule

and so, you could say that they are rule based, but they are sort of meta rule based,

that they allow for behavior that

can respond to the signals that they are getting both globally and locally. Now the last thing

will mention is another sort of paradox in the definition of complex adaptive systems

is that a system like that can be complex, but it need not be.

So a system can have those components to it, but it can end up producing

an equilibrium. Especially if I look at an economic system

some parts of economic systems really equilibriate quite well, but then others

end up being really complex. So if you look at oil consumption

over time at the global level, that's pretty predictable, it's a pretty stable pattern

but if I look at oil prices over time, that's complex, because

there's much more interdependencies and all those things come into play.

So John Holland and I sometimes joke that we should call them systems capable of producing

complexity. That doesn't sound as remarkable.

Okay, well, that is a question that people have debated a lot.

I guess

most people would agree that a complex system is a system of many interacting parts

where the

system is more than just the sum of its parts.

It shows emergent behaviors which are not just the sum of the individual behaviors

of the parts. Other people add

extra elements to that, but that's probably what my definition is

pretty much. It's a system of interacting parts which shows emergent behaviors.

Okay, that's pretty simple. Sort of unpacking that

takes a little more time.

So I'm going to give you a definition of complex systems, but

I will remind you that many important concepts

like virtue and life are very hard to define

and I think complex systems are somewhat in that category.

Nonetheless, the kinds of systems that I call complex

have many interacting active components

and the interactions between the components have

nontrivial or nonlinear interactions

and that leads to the system having unpredictable behavior. You may have

heard those things before. But importantly,

all of the components are either learning or

modifying their behavior in some way while the system

is behaving.

And so that leads to all kinds of interesting dynamics.

So that's roughly what I think of when I think

of a complex system. So do you think that adaptation is

essential for complex behavior?

Well, it's essential for the kind of complex behavior that I'm most interested in

Okay, fair enough.

I think my definition is probably like a lot of other people's

definition in that a complex system is something

with a lot of interacting parts where something

about the way those parts behave when they interact is qualitatively different

than the way they behave if you look at them individually. So

it's something with emergent phenomena. And I think we can then quibble

about what exactly all those words really mean

so for example I might mean something a little bit different than some other people, but I

don't think I have anything particularly unique in my definition.

Complex systems tend to be things that

are different from simpler, usually

physical, systems in that they tend to be heterogenous, they tend to

be made up of parts that are the not the same kind of parts. For example,

people and firms in a city are all different. They are not all the same.

They tend to be, many of them are open ended, although not all of them.

So a city or an ecosystem can keep on evolving

Often the thing that makes them hardest to study in terms of making predictions