New science of complexity?

1. New science of complexity? • Much attention has been given to “complex adaptive systems” in the last decade. • Popularization of not just information and entropy, but phase transitions, criticality, fractals, self-similarity, power laws, chaos, emergence, self-organization, etc. • Aim is to find universal characteristics of complexity. • From Physics: an emphasis on emergent complexity via self-organization of a homogeneous substrate near a critical or bifurcation point (SOC/EOC).

2. Highly Optimized Tolerance (HOT) • Complex systems in biology, ecology, technology, sociology, economics, … • are driven by design or evolution to high-performance states which are also tolerant to uncertainty in the environment and components. • This leads to specialized, modular, hierarchical structures, often with enormous “hidden” complexity, • with new sensitivities to unknown or neglected perturbations and design flaws. • “Robust, yet fragile!”

3. “Robust, yet fragile” • Robust to uncertainties • that are common, • the system was designed for, or • has evolved to handle, • …yet fragile otherwise • This is the most important feature of complex systems (the essence of HOT).

4. HOT features of ecosystems • Organisms are constantly challenged by environmental uncertainties, • And have evolved a diversity of mechanisms to minimize the consequences by exploiting the regularities in the uncertainty. • The resulting specialization, modularity, structure, and redundancy leads to high densities and high throughputs, • But increased sensitivity to novel perturbations not included in evolutionary history. • Robust, yet fragile! • Complex engineering systems are similar.

5. Reduces risk in high-speed collisions Increases risk otherwise Increases risk to small occupants Mitigated by new designs with greater complexity Could just get a heavier vehicle Reduces risk without the increase! But shifts it elsewhere: occupants of other vehicles, pollution Example: Auto airbags

6. Robustness Complexity

7. The simplest possible spatial model of HOT. Square site percolation or simplified “forest fire” model. Carlson and Doyle, PRE, Aug. 1999

8. empty square lattice occupied sites

9. not connected connected clusters

10. Assume one “spark” hits the lattice at a single site. A “spark” that hits an empty site does nothing.

11. A “spark” that hits a cluster causes loss of that cluster.

12. yield density loss Yield = the density after one spark

13. no sparks sparks 1 0.9 “critical point” 0.8 Y= (avg.) yield 0.7 0.6 0.5 N=100 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1  = density

14. 1 0.9 “critical point” Y= (avg.) yield 0.8 limit N   0.7 0.6 0.5 0.4 0.3 0.2 c = .5927 0.1 0 0 0.2 0.4 0.6 0.8 1  = density

15. Y Fires don’t matter. Cold 

16. Everything burns. Y Burned 

17. Critical point Y 

18. all sites occupied 1 critical point c 0 1 no  cluster Percolation P( ) = probability a site is on the  cluster P( ) 

19. miss  cluster full density hit  cluster lose  cluster Y  Y =( 1-P() ) + P() (  -P() ) =  - P()2 P() c

20. Power laws Criticality cumulative frequency cluster size

21. Average cumulative distributions fires clusters size

22. high density low density cumulative frequency Power laws: only at the critical point cluster size

23. Tremendous attention has been given to this point. yield density

24. Edge-of-chaos, criticality, self-organized criticality (EOC/SOC) yield Claim: Life, networks, the brain, the universe and everything are at “criticality” or the “edge of chaos.” density

25. yield density Edge-of-chaos, criticality, self-organized criticality (EOC/SOC) • Essential claims: • Nature is adequately described by generic configurations (with generic sensitivity). yield • Interesting phenomena is at criticality (or near a bifurcation).

26. 18 Sep 1998 Forest Fires: An Example of Self-Organized Critical Behavior Bruce D. Malamud, Gleb Morein, Donald L. Turcotte 4 data sets

27. 6 10 -1 -1/2 4 10 2 10 0 10 -6 -4 -2 0 2 10 10 10 10 10 WWW files Mbytes (Crovella) Cumulative Frequency Forest fires 1000 km2 (Malamud) Size of events

28. 6 10 4 10 -1/2 2 10 SOC FF 0 10 -6 -4 -2 0 2 10 10 10 10 10 Exponents are way off. WWW Cumulative FF Frequency .15 Size of events

29. 3 10 SOC FF 2 10 -1/2 1 10 0 10 -2 -1 0 1 2 3 4 10 10 10 10 10 10 10 Additional 3 data sets

30. ? Forget random, generic configurations. What about high yield configurations? Would you design a system this way?

31. Barriers What about high yield configurations? Barriers

32. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1

33. Rare, nongeneric, measure zero. • Structured, stylized configurations. • Essentially ignored in stat. physics. • Ubiquitous in • engineering • biology • geophysical phenomena? What about high yield configurations?

34. Highly Optimized Tolerance (HOT) critical Cold Burned

35. both analytic and numerical results. Why power laws? Optimize Yield Almost any distribution of sparks Power law distribution of events

36. Optimize Yield No fires Optimize Yield Uniform grid Special cases Singleton (a priori known spark) Uniform spark

37. Special cases No fires In both cases, yields 1 as N . Uniform grid

38. Generally…. Optimize Yield • Gaussian • Exponential • Power law • …. Power law distribution of events

39. “optimized” Numerical Example 32x32 grid 1 0.9 0.8 0.7 0.6 yield 0.5 0.4 random 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 density

40. Probability distribution (tail of normal) x1=2.^(- ((1+ (1:n)/n)/.3 ).^2 ); x2=2.^(- ((.5+ (1:n)/n)/.2 ).^2 );

41. High probability region Probability distribution (tail of normal) 2.9529e-016 0.1902 5 10 15 20 25 30 5 10 15 20 25 30 2.8655e-011 4.4486e-026

42. We expect barriers concentrated in the upper left. • Computing the global optimal is difficult, since the number of configurations is • But practically anything you do… • …gives high yields and power laws… • …at almost all densities.

43. Grid design: optimize the position of “cuts.” cuts = empty sites in an otherwise fully occupied lattice. Compute the global optimum for this constraint.

44. large events are unlikely Optimized grid Small events likely density = 0.8496 yield = 0.7752

45. Optimized grid density = 0.8496 yield = 0.7752 1 0.9 High yields. 0.8 0.7 0.6 grid 0.5 0.4 random 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1

46. “Evolutionary” algorithm “grow” one site at a time to maximize incremental (local) yield

47. density= 0.8 yield = 0.8 “grow” one site at a time to maximize incremental (local) yield

48. density= 0.9 yield = 0.9 “grow” one site at a time to maximize incremental (local) yield

49. Optimal “evolved” density= 0.97 yield = 0.96 “grow” one site at a time to maximize incremental (local) yield

50. Optimal “evolved” density= 0.9678 yield = 0.9625 Several small events A very large event.