1. Are Forest Fires HOT? A mostly critical look at physics, phase transitions, and universality Jean Carlson, UCSB

2. Background Much attention has been given to “complex adaptive systems” in the last decade. Popularization of information, entropy, phase transitions, criticality, fractals, self-similarity, power laws, chaos, emergence, self- organization, etc. Physicists emphasize emergent complexity via self-organization of a homogeneous substrate near a critical or bifurcation point (SOC/EOC)

3. Criticality and power laws Tuning 1-2 parameters  critical point In certain model systems (percolation, Ising, …) power laws and universality iff at criticality. Physics: power laws are suggestive of criticality Engineers/mathematicians have opposite interpretation: –Power laws arise from tuning and optimization. –Criticality is a very rare and extreme special case. –What if many parameters are optimized? –Are evolution and engineering design different? How? Which perspective has greater explanatory power for power laws in natural and man-made systems?

4. 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!”

5. “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).

6. Robustness of HOT systems Robust Fragile Robust (to known and designed-for uncertainties) Fragile (to unknown or rare perturbations) Uncertainties

7. Complexity Robustness Aim: simplest possible story Interconnection

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

9. empty square latticeoccupied sites

10. connected not connected clusters

11. Draw lattices without lines.

12. 20x20 lattice

13. Density = fraction of occupied sites.2.4.6.8

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

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

16. Think of (toy) forest fires.

17. yielddensityloss Think of (toy) forest fires.

18. Yield = the density after one spark yielddensityloss

19. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Y = yield  = density no sparks

20. density=.5 yield = Average over configurations.

21. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Y = (avg.) yield  = density “critical point” N=100 no sparks sparks

22. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 limit N   “critical point” Y = (avg.) yield  = density  c =.5927

23. Cold Fires don’t matter. Y 

24. Y Burned Everything burns. 

25. Critical point  Y

26. critical phase transition This picture is very generic and “universal.” Y 

27. Statistical physics: Phase transitions, criticality, and power laws

28. Thermodynamics and statistical mechanics Mean field theory Renormalization group  Universality classes Power laws Fractals Self-similarity “hallmarks”or “signatures” of criticality

29. Statistical Mechanics Microscopic models Ensemble Averages (all configurations are equally likely) Distributions log(size) log(prob) clusters     = correlation length and correlations

30. Renormalization group low density high densitycriticality

31. Fractal and self-similar Criticality

34. Percolation   cluster

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

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

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

38. Power laws Criticality cluster size cumulative frequency

39. Average cumulative distributions clusters fires size

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

41. Other lattices Percolation has been studied in many settings. Higher dimensions Connected clusters are abstractions of cascading events.

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

43. Self-organized criticality (SOC) Create a dynamical system around the critical point yield density

44. Self-organized criticality (SOC) Iterate on: 1.Pick n sites at random, and grow new trees on any which are empty. 2.Spark 1 site at random. If occupied, burn connected cluster.

45. lattice fire distribution density yield fires

46. -.15

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

48. 10 -2 10 10 0 1 2 3 4 0 1 2 SOC FF Exponents are way off -1/2

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

50. Qualitatively appealing. Power laws. Yield/density curve. “order for free” “self-organization” “emergence” Lack of alternatives? (Bak, Kauffman, SFI, …) But... This is a testable hypothesis (in biology and engineering). In fact, SOC/EOC is very rare.

51. Self-similarity?

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

53. Barriers What about high yield configurations?

54. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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

56. critical Cold H ighly O ptimized T olerance (HOT) Burned

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

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

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

60. Generally…. 1.Gaussian 2.Exponential 3.Power law 4.…. Optimize Yield Power law distribution of events

61. 51015202530 5 10 15 20 25 30 0.1902 2.9529e-016 2.8655e-0114.4486e-026 Probability distribution (tail of normal) High probability region

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

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

64. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 random grid High yields. Optimized grid density = 0.8496 yield = 0.7752

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

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

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

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

69. Optimal density= 0.9678 yield = 0.9625 “grow” one site at a time to maximize incremental (local) yield At density  explores only choices out of a possible Very local and limited optimization, yet still gives very high yields. Several small events A very large event.

70. Optimal density= 0.9678 yield = 0.9625 choices out of a possible Small events likely large events are unlikely Very local and limited optimization, yet still gives very high yields. “grow” one site at a time to maximize incremental (local) yield At density  explores only

71. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 random “optimized” density High yields. At almost all densities.

72. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 random density Very sharp “phase transition.” optimized

73. 10 0 1 2 3 -4 10 -3 10 -2 10 10 0 grid “grown” “critical” Cumulative distributions

74. 10 0 1 2 3 -4 10 -3 10 -2 10 10 0 grid “grown” “critical” size Cum. Prob. All produce Power laws

75. 10 0 1 2 3 -12 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 Local Incremental Algorithm This shows various stages on the way to the “optimal.” Density is shown..9.8.7 optimal density= 0.9678 yield = 0.9625

76. Gaussian log(size) log(prob>size) Power laws are inevitable. Improved design, more resources

77. HOT SOC d=1 dd dd HOT  decreases with dimension. SOC  increases with dimension. SOC and HOT have very different power laws.

78. HOT yields compact events of nontrivial size. SOC has infinitesimal, fractal events. HOT SOC size infinitesimal large

79. A HOT forest fire abstraction… Burnt regions are 2-d Fire suppression mechanisms must stop a 1-d front. Optimal strategies must tradeoff resources with risk.

80. Generalized “coding” problems Fires Web Data compression Optimizing d-1 dimensional cuts in d dimensional spaces.

81. -6-5-4-3-2012 0 1 2 3 4 5 6 Size of events Cumulative Frequency Decimated data Log (base 10) Forest fires 1000 km 2 (Malamud) WWW files Mbytes (Crovella) Data compression (Huffman) (codewords, files, fires) Los Alamos fire d=0d=1 d=2

82. -6-5-4-3-2012 0 1 2 3 4 5 6 Size of events Frequency Fires Web files Codewords Cumulative Log (base 10) -1/2

83. -6-5-4-3-2012 0 1 2 3 4 5 6 WWW DC Data + Model/Theory Forest fire SOC  =.15

84. -6-5-4-3-2012 0 1 2 3 4 5 6 FF WWW DC Data + PLR HOT Model

85. HOT SOC HOTData Max event sizeInfinitesimalLarge Large event shapeFractalCompact Slope  SmallLarge Dimension d  d-1  1/d SOC and HOT are extremely different.

86. SOC HOT & Data Max event sizeInfinitesimal Large Large event shapeFractal Compact Slope  Small Large Dimension d  d-1  1/d SOC and HOT are extremely different. HOT SOC

87. HOT: many mechanisms gridgrown or evolvedDDOF All produce: High densities Modular structures reflecting external disturbance patterns Efficient barriers, limiting losses in cascading failure Power laws

88. Robust, yet fragile?

89. Extreme robustness and extreme hypersensitivity. Small flaws

90. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

91. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

92. If probability of sparks changes. disaster

93. Conserved? Sensitivity to: sparks flaws assumed p(i,j)

94. Can eliminate sensitivity to: assumed p(i,j) Uniform grid Eliminates power laws as well. Design for worst case scenario: no knowledge of sparks distribution

95. Can reduce impact of: flaws Thick open barriers. There is a yield penalty.

96. Critical percolation and SOC forest fire models HOT forest fire models Optimized SOC & HOT have completely different characteristics. SOC vs HOT story is consistent across different models.

97. Characteristic Critical HOT Densities Low High Yields LowHigh Robustness GenericRobust, yet fragile Events/structureGeneric, fractal Structured, stylized self-similarself-dissimilar External behavior Complex Nominally simple Internally Simple Complex Statistics Power laws Power laws only at criticalityat all densities

98. Characteristic Critical HOT Densities Low High Yields LowHigh Robustness GenericRobust, yet fragile. Events/structure Generic, fractal Structured, stylized self-similarself-dissimilar External behavior Complex Nominally simple Internally Simple Complex Statistics Power laws Power laws only at criticalityat all densities Characteristics Toy models ? Power systems Computers Internet Software Ecosystems Extinction Turbulence Examples/ Applications

99. Power systems Computers Internet Software Ecosystems Extinction Turbulence But when we look in detail at any of these examples... …they have all the HOT features... Characteristic Critical HOT Densities Low High Yields LowHigh Robustness GenericRobust, yet fragile. Events/structure Generic, fractal Structured, stylized self-similarself-dissimilar External behavior Complex Nominally simple Internally Simple Complex Statistics Power laws Power laws only at criticalityat all densities

100. Summary Power laws are ubiquitous, but not surprising HOT may be a unifying perspective for many Criticality & SOC is an interesting and extreme special case… … but very rare in the lab, and even much rarer still outside it. Viewing a system as HOT is just the beginning.

101. The real work is… New Internet protocol design Forest fire suppression, ecosystem management Analysis of biological regulatory networks Convergent networking protocols etc

102. Forest fires dynamics Intensity Frequency Extent Weather Spark sources Flora and fauna Topography Soil type Climate/season

103. Los Padres National Forest Max Moritz

104. Yellow: lightning (at high altitudes in ponderosa pines) Red: human ignitions (near roads) Ignition and vegetation patterns in Los Padres National Forest Brown: chaperal Pink: Pinon Juniper

105. FF  = 2  = 1 California brushfires

106. Santa Monica Mountains Max Moritz and Marco Morais

107. SAMO Fire History

108. Fires 1991-1995 Fires 1930-1990 Fires are compact regions of nontrivial area.

109. We are developing realistic fire spread models GIS data for Landscape images

110. Models for Fuel Succession

111. 1996 Calabasas Fire Historical fire spread Simulated fire spread Suppression?

112. Preliminary Results from the HFIRES simulations (no Extreme Weather conditions included)

113. Data: typical five year periodHFIREs Simulations: typical five year period Fire scar shapes are compact

114. Excellent agreement with data for realistic values of the parameters

115. Agreement with the PLR HOT theory based on optimal allocation of resources α=1 α=1/2

116. What is the optimization problem? Fire is a dominant disturbance which shapes terrestrial ecosystems Vegetation adapts to the local fire regime Natural and human suppression plays an important role Over time, ecosystems evolve resilience to common variations But may be vulnerable to rare events Regardless of whether the initial trigger for the event is large or small (we have not answered this question for fires today) We assume forests have evolved this resiliency (GIS topography and fuel models) For the disturbance patterns in California (ignitions, weather models) And study the more recent effect of human suppression Find consistency with HOT theory But it remains to be seen whether a model which is optimized or evolves on geological times scales will produce similar results Plausibility Argument: HFIREs Simulations:

117. The shape of trees by Karl Niklas L: Light from the sun (no overlapping branches) R: Reproductive success (tall to spread seeds) M: Mechanical stability (few horizontal branches) L,R,M: All three look like real trees Simulations of selective pressure shaping early plants Our hypothesis is that robustness in an uncertain environment is the dominant force shaping complexity in most biological, ecological, and technological systems