Are Forest Fires
HOT?
A mostly critical
look at physics,
phase transitions,
and universality
Jean Carlson, UCSB
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, selforganization, etc.
Physicists emphasize emergent complexity via
self-organization of a homogeneous substrate
near a critical or bifurcation point (SOC/EOC)
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?
Highly
Optimized
Tolerance (HOT)
•
•
•
•
•
Complex systems in biology, ecology, technology,
sociology, economics, …
are driven by design or evolution to highperformance 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!”
“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).
Robustness of
HOT systems
Fragile
Robust
(to known and
designed-for
uncertainties)
Fragile
(to unknown
or rare
perturbations)
Robust
Uncertainties
Complexity
Robustness
Interconnection
Aim: simplest
possible story
The simplest possible spatial model of HOT.
Square site
percolation
or
simplified
“forest fire”
model.
Carlson and Doyle,
PRE, Aug. 1999
empty square lattice
occupied sites
not connected
connected clusters
connected clusters
Draw lattices
without lines.
20x20 lattice
.4
.6
Density = fraction of occupied sites
.2
.8
Assume one
“spark” hits the
lattice at a single
site.
A “spark” that hits
an empty site does
nothing.
A “spark” that hits
a cluster causes
loss of that cluster.
Think of (toy)
forest fires.
Think of (toy)
forest fires.
yield
density
loss
Yield = the density after one spark
yield
density
loss
1
no sparks
0.9
0.8
0.7
Y=
0.6
yield
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
 = density
1
Average over configurations.
yield =
density=.5
4 * .25  2 * .375
0.2917 
6
avg avg
trees spark
1
no sparks
0.9
“critical point”
0.8
Y=
(avg.)
yield
0.7
0.6
0.5
0.4
N=100
0.3
sparks
0.2
0.1
0
0
0.2
0.4
0.6
0.8
 = density
1
1
0.9
Y=
0.8
(avg.)
yield
0.7
“critical point”
limit
N
0.6
0.5
0.4
0.3
0.2
c = .5927
0.1
0
0
0.2
0.4
0.6
0.8
 = density
1
Y
Fires
don’t
matter.
Cold

Y
Everything burns.
Burned

Critical point
Y

This picture is very generic and “universal.”
Y
critical
phase
transition

Statistical physics:
Phase transitions,
criticality, and
power laws
Thermodynamics
and statistical
mechanics
Mean field theory
Renormalization group  Universality classes
Power laws
Fractals
Self-similarity
“hallmarks”or
“signatures”
of criticality
Statistical Mechanics
Microscopic
models
Distributions and correlations
clusters


log(prob)
Ensemble Averages
(all configurations
are equally likely)
log(size)
 = correlation length

Renormalization group
low density
criticality
high density
Criticality
Fractal
and
self-similar
Percolation
  cluster
all sites
occupied
Percolation
P( ) =
probability a site
is on the  cluster
1
P( )
critical
point c
0

no  cluster
1
miss  full
cluster density
Y =
( 1-P() ) 
hit 
cluster
lose 
cluster
+ P() (  -P() )
= P()2
Y
P()
c

yield
Tremendous attention has
been given to this point.
density
2
10
Power laws
cumulative
frequency
Criticality
1
10
0
10
-1
10
0
10
1
10
2
10
3
10
cluster size
4
10
Average
cumulative
distributions
fires
clusters
size
high density
cumulative
frequency
low density
Power laws:
only at the
critical point
cluster size
Percolation has been
studied in many settings.
Connected clusters are
abstractions of cascading
events.
Other lattices
Higher dimensions
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
Self-organized criticality
(SOC)
Create a dynamical
system around the
critical point
yield
density
Self-organized criticality (SOC)
N  N lattice
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.
2
1  n  N 2
Use n  .1N in examples .
lattice
distribution
fire
density
yield
fires
0.8
0.6
0.4
0.2
0
0
200
400
600
800
3
10
-.15
2
10
1
10
0
10 0
10
1
10
2
10
3
10
4
10
1000
18 Sep 1998
Forest Fires: An Example of Self-Organized
Critical Behavior
Bruce D. Malamud, Gleb Morein, Donald L. Turcotte
4 data sets
3
10
2
10
-1/2
1
10
SOC FF
0
10
-2
10
-1
10
0
10
1
10
2
10
Exponents are way off
3
10
4
10
Edge-of-chaos, criticality,
self-organized criticality
(EOC/SOC)
Essential claims:
• Nature is adequately described yield
by generic configurations (with
generic sensitivity).
• Interesting phenomena is at
criticality (or near a bifurcation).
yield
density
• 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.
Self-similarity?
?
Forget random,
generic
configurations.
Would you design a
system this way?
What about high
yield configurations?
Barriers
What about high
yield configurations?
Barriers
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
• Rare, nongeneric, measure zero.
• Structured, stylized configurations.
• Essentially ignored in stat. physics.
• Ubiquitous in
• engineering
• biology
• geophysical phenomena?
What about high
yield configurations?
Highly Optimized Tolerance (HOT)
critical
Cold
Burned
Why power laws?
Almost any
distribution
of sparks

Optimize
Yield
Power law
distribution
of events
both analytic and numerical results.
Special cases
Singleton
(a priori
known spark)
Uniform
spark
Optimize
Yield
Optimize
Yield
No fires
Uniform
grid
Special cases
No fires
In both cases, yields 1 as N .
Uniform
grid
Generally….
1.
2.
3.
4.
Gaussian
Exponential
Power law
….
Optimize
Yield
Power law
distribution
of events
Probability distribution (tail of normal)
2.9529e-016
0.1902
High probability region
5
10
15
20
25
30
5
2.8655e-011
10
15
20
25
30
4.4486e-026
Grid design:
optimize the
position of “cuts.”
cuts = empty sites in
an otherwise fully
occupied lattice.
Compute the global optimum for this constraint.
Optimized grid
Small events likely
large events
are unlikely
density = 0.8496
yield = 0.7752
Optimized grid
density = 0.8496
yield = 0.7752
1
0.9
0.8
High yields.
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
Local incremental
algorithm
“grow” one site at a
time to maximize
incremental (local)
yield
density= 0.8
yield = 0.8
“grow” one site at a
time to maximize
incremental (local)
yield
density= 0.9
yield = 0.9
“grow” one site at a
time to maximize
incremental (local)
yield
Optimal
density= 0.97
yield = 0.96
“grow” one site at a
time to maximize
incremental (local)
yield
Several small events
A very
large
event.
Optimal
density= 0.9678
yield = 0.9625
“grow” one site at a
time to maximize
incremental (local)
yield
At density  explores only
(1   ) N 2
choices out of a possible
Very local and limited optimization,
yet still gives very high yields.
2


N


 (1   ) N 2 


Small events likely
large events
are unlikely
Optimal
density= 0.9678
yield = 0.9625
“grow” one site at a
time to maximize
incremental (local)
yield
At density  explores only
(1   ) N 2
choices out of a possible
Very local and limited optimization,
yet still gives very high yields.
2


N


 (1   ) N 2 


“optimized”
1
0.9
High yields.
0.8
0.7
0.6
At almost all densities.
0.5
0.4
random
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
density
1
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
density
1
0
10
“critical”
Cumulative
distributions
-1
10
“grown”
-2
10
grid
-3
10
-4
10
0
10
1
10
2
10
3
10
0
10
“critical”
-1
10
Cum.
Prob.
“grown”
-2
10
-3
10
All produce
Power laws
grid
-4
10
0
10
1
10
2
3
10
10
size
10
optimal
density= 0.9678
yield = 0.9625
10
10
Local Incremental Algorithm
10
This shows various
stages on the way to
the “optimal.”
Density is shown.
10
10
10
0
-2
-4
.9
-6
.8
-8
-10
.7
-12
10
0
10
1
10
2
10
3
Power laws are inevitable.
Gaussian
log(prob>size)
Improved design,
more resources
log(size)
SOC and HOT have very
different power laws.
d
d=1
HOT
d
d=1
SOC
d 1

10
1

d
• HOT  decreases with dimension.
• SOC increases with dimension.
• HOT yields compact events of nontrivial size.
• SOC has infinitesimal, fractal events.
HOT
SOC
infinitesimal
size
large
A HOT forest fire abstraction…
Fire suppression
mechanisms must
stop a 1-d front.
Burnt regions are 2-d
Optimal strategies
must tradeoff
resources with risk.
Generalized
“coding” problems
Optimizing d-1 dimensional
cuts in d dimensional spaces.
Data compression
Web
Fires
6
5
Cumulative
(Crovella)
4
Frequency
Data
compression
WWW files
Mbytes
(Huffman)
d=1
d=0
3
2
Forest fires
1000 km2
1
(Malamud)
0
d=2
Los Alamos fire
-1
-6
-5
-4
-3
-2
-1
0
1
Decimated data
Size of events
Log (base 10)
(codewords, files, fires)
2
6
Web files
5
Codewords
4
Cumulative
Frequency
-1
3
Fires
2
-1/2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
Size of events
Log (base 10)
2
Data + PLR HOT Model
6
DC
5
WWW
4
3
FF
2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
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
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
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
Robust,
yet fragile?
Extreme robustness and extreme hypersensitivity.
Small
flaws
Robust,
yet fragile?
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
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
If probability of sparks changes.
disaster
Conserved?
Sensitivity to:
sparks
assumed
p(i,j)
flaws
Design for worst case
scenario: no knowledge
of sparks distribution
Can eliminate
sensitivity to:
Uniform
grid
assumed
p(i,j)
Eliminates power laws as well.
Thick open barriers.
There is a yield penalty.
Can reduce
impact of:
flaws
Optimized
Critical percolation
and SOC forest fire
models
HOT forest fire models
• SOC & HOT have completely different characteristics.
• SOC vs HOT story is consistent across different models.
Characteristic Critical
HOT
Densities
Yields
Robustness
Low
Low
Generic
High
High
Robust, yet fragile
Events/structure
Generic, fractal
self-similar
Structured, stylized
self-dissimilar
External behavior Complex
Internally
Simple
Statistics
Nominally simple
Complex
Power laws
Power laws
only at criticality at all densities
Characteristic
Critical
HOT
Densities
Yields
Robustness
Low
Low
Generic
High
High
Robust, yet fragile.
Generic, fractal
self-similar
Structured, stylized
self-dissimilar
Events/structure
Characteristics
External behavior Complex
Internally
Simple
Nominally simple
Complex
Statistics
Power laws
at all densities
Toy models
Power laws
only at criticality
?
• Power systems
• Computers
Examples/
• Internet
• Software
Applications
• Ecosystems
• Extinction
• Turbulence
Characteristic
Critical
HOT
Densities
Yields
Robustness
Low
Low
Generic
High
High
Robust, yet fragile.
Events/structure
Generic, fractal
self-similar
Structured, stylized
self-dissimilar
External behavior Complex
Internally
Simple
Nominally simple
Complex
Statistics
Power laws
at all densities
Power laws
only at criticality
But when we look in detail
at any of these examples...
…they have all the HOT
features...
• Power systems
• Computers
• Internet
• Software
• Ecosystems
• Extinction
• Turbulence
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.
The real work is…
• New Internet protocol design
• Forest fire suppression, ecosystem
management
• Analysis of biological regulatory networks
• Convergent networking protocols
• etc
Forest fires
dynamics
Weather
Spark sources
Intensity
Frequency
Extent
Flora and fauna
Topography
Soil type
Climate/season
Los Padres National Forest
Max Moritz
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
4
10
California brushfires
FF  = 2
3
10
=1
2
10
1
10
0
10
-1
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
Santa Monica Mountains
Max Moritz and Marco Morais
SAMO Fire History
Fires are compact regions
of nontrivial area.
Fires 1930-1990
Fires 1991-1995
We are developing realistic fire spread models
GIS data for
Landscape images
Models for Fuel Succession
1996 Calabasas Fire
Historical fire spread
Simulated fire spread
Suppression?
Preliminary Results from the HFIRES simulations
(no Extreme Weather conditions included)
Fire scar shapes are compact
Data: typical five year period
HFIREs Simulations: typical five
year period
Excellent agreement with data for realistic values of the parameters
Agreement with the PLR HOT theory
based on optimal allocation of resources
α=1/2
α=1
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
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