Overview

1
Are Forest Fires HOT?
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
18 Sep 1998
Forest Fires An Example of Self-Organized
Critical Behavior Bruce D. Malamud, Gleb Morein,
Donald L. Turcotte
4 data sets
4
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?

5
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!

6
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).

7
Robustness of HOT systems
Fragile
Fragile (to unknown or rare perturbations)
Robust (to known and designed-for uncertainties)
Uncertainties
Robust
8
Robustness
Complexity
Interconnection
Aim simplest possible story
9
The simplest possible spatial model of HOT.
Square site percolation or simplified forest
fire model.
Carlson and Doyle, PRE, Aug. 1999
10
empty square lattice
occupied sites
11
connected
not
connected clusters
12
20x20 lattice
13
Assume one spark hits the lattice at a single
site.
A spark that hits an empty site does nothing.
14
A spark that hits a cluster causes loss of that
cluster.
15
Yield the density after one spark
16
Average over configurations.
density.5
17
1
0.9
critical point
0.8
Y (avg.) yield
0.7
0.6
0.5
N100
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
? density
18
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
19
Y
Fires dont matter.
Cold
?
20
Everything burns.
Y
Burned
?
21
Critical point
Y
?
22
This picture is very generic and universal.
Y
critical phase transition
?
23
Statistical physics Phase transitions,
criticality, and power laws
24
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25
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26
Power laws
Criticality
cumulative frequency
cluster size
27
Average cumulative distributions
fires
clusters
size
28
cumulative frequency
Power laws only at the critical point
cluster size
29
Self-organized criticality (SOC)
yield
Create a dynamical system around the critical
point
density
30
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.

31
lattice
distribution
fire
density
yield
fires
32
-.15
33
18 Sep 1998
Forest Fires An Example of Self-Organized
Critical Behavior Bruce D. Malamud, Gleb Morein,
Donald L. Turcotte
4 data sets
34
2
10
-1/2
1
10
0
10
-2
-1
0
1
2
3
4
10
10
10
10
10
10
10
Exponents are way off
35
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).

36

• 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.

37
Self-similarity?
38
?
Forget random, generic configurations.
What about high yield configurations?
Would you design a system this way?
39
Barriers
What about high yield configurations?
Barriers
40
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
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• Rare, nongeneric, measure zero.
• Structured, stylized configurations.
• Essentially ignored in stat. physics.
• Ubiquitous in
• engineering
• biology
• geophysical phenomena?

What about high yield configurations?
42
Highly Optimized Tolerance (HOT)
critical
Cold
Burned
43
Why power laws?
Optimize Yield
Almost any distribution of sparks
Power law distribution of events
44
Special cases
Singleton (a priori known spark)
Uniform spark
45
Special cases
No fires
In both cases, yields ?1 as N ??.
Uniform grid
46
Generally.
Optimize Yield

1. Gaussian
2. Exponential
3. Power law
4. .

Power law distribution of events
47
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
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Grid design optimize the position of cuts.
cuts empty sites in an otherwise fully
occupied lattice.
Compute the global optimum for this constraint.
49
Optimized grid
Small events likely
density 0.8496 yield 0.7752
50
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
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Local incremental algorithm
grow one site at a time to maximize incremental
(local) yield
52
density 0.8 yield 0.8
grow one site at a time to maximize incremental
(local) yield
53
density 0.9 yield 0.9
grow one site at a time to maximize incremental
(local) yield
54
Optimal density 0.97 yield 0.96
grow one site at a time to maximize incremental
(local) yield
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Very sharp phase transition.
1
0.9
optimized
0.8
0.7
0.6
0.5
0.4
random
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
density
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0
10
-1
10
Cum. Prob.
grown
-2
10
All produce Power laws
-3
10
grid
-4
10
0
1
2
3
10
10
10
10
size
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HOT
SOC and HOT have very different power laws.
d1
SOC
d1

• HOT ? decreases with dimension.
• SOC?? increases with dimension.

58

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

HOT
SOC
large
infinitesimal
size
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A HOT forest fire abstraction
Fire suppression mechanisms must stop a 1-d front.
Optimal strategies must tradeoff resources with
risk.
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Generalized coding problems
Data compression
Optimizing d-1 dimensional cuts in d dimensional
spaces.
Web
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6
Data compression (Huffman)
WWW files Mbytes (Crovella)
5
Cumulative
d0
d1
4
3
Frequency
Forest fires 1000 km2 (Malamud)
2
1
d2
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
Decimated data Log (base 10)
Size of events
(codewords, files, fires)
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6
Web files
5
Codewords
4
Cumulative
3
Frequency
Fires
2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
Size of events
Log (base 10)
63
Data Model/Theory
6
DC
5
WWW
4
3
2
1
Forest fire
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
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Data PLR HOT Model
6
DC
5
WWW
4
3
FF
2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
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SOC and HOT are extremely different.
SOC HOT Data
Max event size Infinitesimal Large Large
Large event shape Fractal Compact Compact
Slope ? Small Large Large
Dimension d ??d-1 ??1/d ??1/d
HOT
SOC
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SOC and HOT are extremely different.
SOC HOT Data
Max event size Infinitesimal Large
Large event shape Fractal Compact
Slope ? Small Large
Dimension d ??d-1 ??1/d
HOT
SOC
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HOT many mechanisms
grid
grown or evolved
DDOF
All produce

• High densities
• Modular structures reflecting external
  disturbance patterns
• Efficient barriers, limiting losses in cascading
  failure
• Power laws

68
Robust, yet fragile?
69
Extreme robustness and extreme hypersensitivity.
Small flaws
Robust, yet fragile?
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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
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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
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If probability of sparks changes.
disaster
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flaws
assumed p(i,j)
Sensitivity to
74
Critical percolation and SOC forest fire models

• SOC HOT have completely different
  characteristics.
• SOC vs HOT story is consistent across different
  models.

75
Characteristic Critical HOT
Densities Low High Yields Low High Robust
ness Generic Robust, yet fragile Events/struc
ture Generic, fractal Structured,
stylized self-similar self-dissimilar Externa
l behavior Complex Nominally
simple Internally Simple Complex Statistics
Power laws Power laws only at
criticality at all densities
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Characteristic Critical HOT Densities Low
High Yields Low High Robustness
Generic Robust, yet fragile. Events/structure
Generic, fractal Structured, stylized self-sim
ilar self-dissimilar External behavior Complex
Nominally simple Internally Simple
Complex Statistics Power laws Power
laws only at criticality at all densities
Characteristics
Examples/ Applications
Toy models
?
77
Characteristic Critical HOT Densities Low
High Yields Low High Robustness
Generic Robust, yet fragile. Events/structure
Generic, fractal Structured, stylized self-sim
ilar self-dissimilar External behavior Complex
Nominally simple Internally Simple
Complex Statistics Power laws Power
laws only at criticality at all densities

• 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...
78
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!

79
HOT and evolution mutation and natural
selection in a community

• Begin with 1000 random lattices, equally divided
  
• between tortoise and hare families
• Each parent gives rise to two offspring
• Small probability of mutation per site
• Sparks are drawn from P(i,j)
• Fitness Yield (1 spark for hares, full P(i,j)
  for tortoises)
• Death if Fitnesslt0.4
• Natural selection acts on remaining lattices
• Competition for space in a community of bounded
  size

Barriers to cascading failure an abstraction of
biological mechanisms
for robustness
Tong Zhou
80
Fast mutators (hares)
Slow mutators (tortoises)
Hares -noisy patterns -lack protection for
rare events
Genotype (heritable traits) lattice
layout Phenotype (characteristics which can be
observed in the environment)
cell sizes and probabilities Fitness (based on
performance in the organisms lifetime) Yield
81
(Primitive) Punctuated Equilibrium
Hares win in the short run. But face
episodic extinction due to rare events
(niche protects 50). Tortoises take over, and
diversity increases. Until hares win again.
tortoises
hares
82
Tortoise population exhibits power laws Hares
have excess large events
83
Convergent Evolution Species which evolve in
spatially separate, but otherwise similar
habitats develop similar phenotypic traits. They
are not genetically close, but have developed
similar adaptations to their environmental niches.
Our analogy different runs with the same P(i,j)
evolve towards phenotypically similar,
genotypically dissimilar lattice populations
84
The five great extinctions are associated with a
rate of disappearance of species well in excess
of the background, as deduced from the fossil
record.
Paleontologists attribute these to rare
disturbances, such as meteor impacts.

Robust, yet Fragile!
85
Punctuated Equilibrium (left) vs. Gradualism
(right)
PE rapid, bursts of change (horizontal lines),
followed by extended periods of relative
stability (vertical lines), followed by
extinction.
Our analogy after a transient period of rapid
evolution lattices have barrier patterns, which
are relatively stable until extinction
86
Large extinction events are typically followed by
increased diversity. The recovery period is the
time lapse between the peak extinction rate,
and the maximum rate of origination of new
species.
Our analogy extinction of the hares is
are followed by diversification of both families
87
The current mass extinction is frequently
attributed to overpopulation and causes which
can be attributed to humans, such as
deforestation
Our analogy large events can be due to rare
disturbances, especially if they are not not
part of the evolutionary history of the
(vulnerable) species.
Robust, yet Fragile!
88
Evolution by natural selection in coupled
communities with different environments
Uniform Sparks
Skewed Sparks
Fitness based on a single spark. Eliminate
protective niches. Fixed maximum capacity for
each habitat. Fast and slow mutation rates (rate
subject to mutation).
89
Coupled Habitats Fast and slow mutators compete
with each other in each habitat, with a small
chance of migration from one habitat to the other.
Efficient barrier patterns develop in the uniform
habitat. After an extinction in the skewed
habitat, uniform lattices invade,
and subsequently lose their lower right barriers
a successful strategy in the short term, but
leads to vulnerability on longer time scales
90
Patterns of extinction, invasion, evolution
Over an extended time window, spanning the two
previous extinctions, we see the long term
fitness ltYgt initially increases as the
invading lattices adapt to their new environment.
This is followed by a sudden decline when the
lattices lose a barrier. This adaptation is
beneficial for common events, but fatal for rare
events.
91
Evolution and extinction
fitness
density
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HOT and Evolution

• Robustness in an uncertain environment
• provides a mechanism which leads to a
• variety of phenomena consistent with
• observations in the fossil record (large
• extinctions associated with rare disturbance,
• punctuated equilibrium, genotypic divergence,
• phenotypic convergence).

• In a model which retains abstract notions of
  genotype, phenotype,
• and fitness, highly evolved lattices develop
  efficient barriers to
• cascading failure, similar to those obtained by
  deliberate design.

• Robustness barriers are central in natural and
  man made complex
• systems. They may be physical (skin) or in the
  state space
• (immune system) of a complex, interconnected
  system.

93

• Forest Fires a case where a common disturbance
  type (fires)
• Acts over a broad range of scales (terrestrial
  ecosystems)
• Power law statistics describing the distribution
  of fire sizes.
• Exponents are consistent with the simplest HOT
  model involving
• optimal allocation of resources (suppress
  fires).
• Evolutionary dynamics are much more complex.

94
18 Sep 1998
Forest Fires An Example of Self-Organized
Critical Behavior Bruce D. Malamud, Gleb Morein,
Donald L. Turcotte
4 data sets
95
All four data sets are fit with the PLR model
with a1/2.
4
10
3
10
Rank order
2
10
1
10
0
10
-4
-3
-2
-1
0
1
10
10
10
10
10
10
Size (1000 km2)
96
Forest fires dynamics
Intensity Frequency Extent
97
Los Padres National Forest
Max Moritz
98
Red human ignitions (near roads)
Yellow lightning (at high altitudes in ponderosa
pines)
Brown chaperal Pink Pinon Juniper
Ignition and vegetation patterns in Los Padres
National Forest
99
Santa Monica Mountains
Max Moritz and Marco Morais
100
SAMO Fire History
101
Fires are compact regions of nontrivial area.
Fires 1930-1990
Fires 1991-1995
102
4 Science data sets LPNF HFIREs (SA2)
Cumulative P(size)
PLR
SM
size
Rescaling data for frequency and large size
cutoff gives excellent agreement, except for the
SM data set
103
We are developing a realistic fire spread model
HFIREs
GIS data for Landscape images
104
Models for Fuel Succession
Regrowth modeled by vegetation succession models
105
1996 Calabasas Fire
Historical fire spread
Simulated fire spread
Suppression?
106
HFIREs Simulation Environment

• Topography and vegetation initialized with
  recent observations
• (100 m GIS resolution) for Santa Monica
  Mountains
• Weather based on historical data (SA rate
  treated as a separate
• parameter)
• Fire spread modeled using Rothermel equations
• Fuel regrowth based on succession models

• 8 ignitions per year
• Weather sampled stochastically from distribution
  (4 day SA at
• prescribed rate)
• Fire terminates in a cell when rate of spread
  (RoS) falls below
• a specified value

• Generate 600 year catalogs, omit data for first
  100 years in our
• statistics

107
Preliminary results from the HFIRES simulations
(no extreme weather conditions included)
(we have generated many 600 year catalogs varying
both extreme weather and suppression)
108
Fire scar shapes are compact
Data typical five year period
HFIREs simulations typical five year period
109
4
10
PLR
3
10
HFIRE SA2, RoS .033 m/s, FC46 yr
2
10
1
10
LPNF
0
10
-4
-3
-2
-1
0
1
10
10
10
10
10
10
Excellent agreement between data, HFIREs and the
PLR HOT model
110
4 Science LPNF Hfire (SA2)
PLR
SM

• small incomplete
• large short catalog, or aggressive human
  intervention
• (inhomogeneous)

SM discrepancy?
111
Deviations from typical regional values for
suppression (RoS) and the number of extreme
weather events (SA), lead to deviations from the
a1/2 fit, and unrealistic values of the fire
cycle (FC)
3
10
SA 1, 2, 4, 6
2
10
1
10
SA0, vary stopping rate
0
10
0
1
2
3
4
5
10
10
10
10
10
10
112
SA0, ?.65
Increased rate of SA leads to flatter curves,
shorter fire cycles
SA2, ?.5
SA4, ?.5
Type conversion!
SA6, ?.3
3
10
2
10
1
10
0
10
2
3
4
5
10
10
10
10
113
What is the optimization problem?
(we have not answered this question for fires
today)
Plausibility Argument

• 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

HFIREs Simulations

• 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

114
The shape of trees by Karl Niklas
Simulations of selective pressure shaping
early plants

• 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

Our hypothesis is that robustness in an uncertain
environment is the dominant force shaping
complexity in most biological, ecological,
and technological systems
115
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