Spring School in Complexity Science

Introduction toComplexity Science

Complexity: Scale and Connectivity

Seth Bullock, 2006

Conceptual Landscape

In this lecture we will explore two things:

some of the conceptual diversity running through the complexity literature

some key issues for understanding how complexity can be applied across domains

Seth Bullock, 2006

Defining Complexity

It is widely acknowledged that "complexity" is:poorly definedmultiply definedCan mean:challenginginterestingcomplicatedor just large

Seth Bullock, 2006

Definitions

A plethora of attempted definitions (36+!).

Approaches to defining complexity: computational vs. statistical structural vs. functional sequential, hierarchical, etc.

Particular definitions include: algorithmic c. Kolomogorov c. minimum description length effective measure c., effective c., physical

c.

Seth Bullock, 2006

Motivations

Each definition attempts to formalise an intuition.Systems can be placed on a continuum:

Both regular and random systems are simple - their aggregate behaviour is straightforward to explain (e.g., pendulum, ideal gas)

Complex systems are more difficult to understand due to the “entwined” nature of their parts.Standard “divide-and-conquer” approaches to explanation are limited, here.

Com

ple

xit

y

Regularity

Seth Bullock, 2006

Beyond Intuition

Much hinges on unpacking what we mean by intuitive terms: “straightforward” or “difficult”.

If we cannot formalise them, then to claim that one system is more complex than another is just to claim that we currently find it harder to understand.

Seth Bullock, 2006

Problems

Some of the formalisms have obvious problems:Kolomogorov complexity measures predictability in a system. Homogeneous Systems → Low KC Regular, Periodic Systems → Higher KC Complex, Chaotic Systems → Even Higher

KC Totally Random Systems → Highest KC

It is the intermediate systems that we want to single out.

Seth Bullock, 2006

Emergence

Low-level interactions bring about systemic organization in complex systems. How?

System-level behaviour “emerges” from the low-level interactions of individual system components in a non-trivial manner.But again, much hinges on what we mean by “non-trivial”.

Seth Bullock, 2006

Emergent = Mysterious?

Andy Clark points out that when a number of small children tip a see-saw, we gain little by tagging this as “emergent behaviour”.

But reserving “emergent” for systems that are currently unexplained (or perhaps inherently inexplicable)…

“robs the notion of immediate scientific interest”

Seth Bullock, 2006

Four Kinds of Emergence

Clark again:1.collective self-organization2.un-programmed functionality3.interactive complexity4.incompressible unfoldingNo time to deal with all four.

Each drives at an account of the opacity in the relationship between a system’s levels of description that is not subjective.

Seth Bullock, 2006

Non-linearity

Simplifying:To the extent that a system’s interactions are non-linear, an account of their impact on global behaviour will be increasingly involved.

For “non-linear” read: multiple, ecologically embedded non-additive, inseparable, heterogeneous interactive, asynchronous, lagged, or

delayed.

Seth Bullock, 2006

Naturalising Emergence

A continuum: non-linearity in a system’s interactions corresponds with a notion of complexity and emergence.

Between simple (weight) and irreducibly complex (protein folding) sit moderately complex systems with challenging but tractable.

Seth Bullock, 2006

Issues

These ideas are not new. People have been fretting about these questions for a long time.Given this, can we expect significant progress any time soon?First, let's look at some stumbling blocks that have prevented complexity ideas from entering the mainstream of science and particularly engineering...

Seth Bullock, 2006

Plurality

Lack of consensus on defining complexity is sometimes taken to reflect poorly on the field. diverse communities → multiple definitionsA single tightly-defined concept may be impossible/undesirable.We might expect a cluster of ideas to share a common centre of gravity.Increased interdisciplinarity could accelerate this? Some evidence that this is happening already.

Seth Bullock, 2006

Subjectivity

“behaviour is emergent if it surprises us ”“a system is complex when we find it hard to understand”

Limits scientific utility.

Non-linearity is not subjective. Can it be made core to notions of complexity and emergence?N.b. Complex systems may of course remain counter-intuitive even when we have a full theory in place.

Seth Bullock, 2006

Complicated vs. Complex

“Well, you are talking about complexity, but a car is not complex it's just complicated.”Complicated systems: difficult, but succumb to divide-and-conquer approaches. a car’s turning circle

Complex systems are different: Hofstadter’s “thrashing” e.g.

“Why can't you just open up the computer, find the number ‘35’ and change it to ‘50’?”Complicated is easier to cope with than complex?

Seth Bullock, 2006

Complications

But complicated systems are often complex:

Cars do exhibit “unwanted functionality”. Software does suffer from “emergent” bugs

And complex systems do exhibit complicatedness:

the body’s many modular sub-systemsIf this were not so (i) engineering would be much easier, (ii) science would be much much harder.

complexity arises from complication complication evolves in complex systems

The distinction is not clear-cut.

Seth Bullock, 2006

PredictabilityComplexity = Unpredictability = Untrustworthy?

← Simple Gas Complex Pigs →

Low-level behaviour is unpredictable (gas molecules bouncing around, pigs pigging about).Yet, some high-level behaviours are predictable.

E.g., Stock control must be reliable, therefore we cannot use a complex systems approach, and must eradicate complexity from our systems!

Seth Bullock, 2006

Explicability not Predictability

It is relatively easy to explain how more gas increases temperature (ideal gas law) but not easy to explain how more pigs brings about an abrupt phase transition in pig violence. For simple (linear) systems:

a small change to a system’s components → a small change at the system level

For complex (non-linear) systems:a small change to a system’s

components → large/small/no change at the system level

Seth Bullock, 2006

Example

The periodic table organises and labels these transitions. But it does not explain them.Complexity science is in the process of building it's own periodic table, but we are not there yet.

If we add a proton to each atom of a bar of gold, radical but predictable change occurs.

Seth Bullock, 2006

Finally…Braehe’s

Observations

Kepler’s Patterns

Newton’s Laws

Complexity Science