Introduction to Complexity Science

1. Introduction toComplexity Science Complexity: Scale and Connectivity

2. 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

3. Defining Complexity It is widely acknowledged that "complexity" is: poorly defined multiply defined Can mean: challenging interesting complicated or just large

4. 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.

5. Complexity Regularity 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.

6. 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 wecurrently find it harder to understand.

7. 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.

8. Emergence Low-level interactions bring about systemic organization in complex systems. How? System-level behaviour “emerges” from the low-level interactions of individual system componentsin a non-trivial manner. But again, much hinges on what we mean by “non-trivial”.

9. Emergent = Mysterious? Andy Clarkpoints 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”

10. Four Kinds of Emergence Clark again: collective self-organization un-programmed functionality interactive complexity incompressible unfolding No 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.

11. 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.

12. 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.

13. 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...

14. Plurality Lack of consensus on defining complexity is sometimes taken to reflect poorly on the field. diverse communities → multiple definitions A 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.

15. 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 bemade 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.

16. 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?

17. Complications But complicated systems are often complex: • Cars do exhibit “unwanted functionality”. • Software does suffer from “emergent” bugs Andcomplex systems do exhibit complicatedness: • the body’s many modular sub-systems If 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.

18. ← Simple Gas Complex Pigs → Predictability Complexity = Unpredictability = Untrustworthy? E.g., Stock control must be reliable, therefore we cannot use a complex systems approach, and must eradicate complexity from our systems! Low-level behaviour is unpredictable (gas molecules bouncing around, pigs pigging about). Yet, some high-level behavioursare predictable.

19. 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

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

21. Braehe’s Observations Complexity Science Kepler’s Patterns Newton’s Laws Finally…