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, self-organization, 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 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!”

“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

Robust

Fragile

Robust(to known anddesigned-foruncertainties)

Fragile(to unknown

or rareperturbations)

Uncertainties

ComplexityRobustness

Aim: simplest possible story

Interconnection

Square site percolation or simplified “forest fire” model.

The simplest possible spatial model of HOT.

Carlson and Doyle,PRE, Aug. 1999

empty square lattice occupied sites

connectednot connected clusters

connected clusters

Draw lattices without lines.

20x20 lattice

Density = fraction of occupied sites

.2

.4 .6

.8

A “spark” that hits an empty site does

nothing.

Assume one “spark” hits the

lattice at a single site.

A “spark” that hits a cluster causes

loss of that cluster.

Think of (toy) forest fires.

yield density loss

Think of (toy) forest fires.

Yield = the density after one spark

yield density loss

0 0.2 0.4 0.6 0.8 10

0.1

0.2

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0.5

0.6

0.7

0.8

0.9

1

Y =yield

= density

no sparks

density=.5

yield =

6

375.*225.*40.2917

Average over configurations.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

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1

Y =(avg.)yield

= density

“critical point”

N=100

no sparks

sparks

sparktreesavgavg

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

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1

limit N

“critical point”Y =(avg.)yield

= density

c = .5927

Cold

Fires don’t matter.

Y

Y

Burned

Everything burns.

Critical point

Y

critical phase transition

This picture is very generic and “universal.”

Y

Statistical physics:Phase transitions,

criticality, and power laws

Thermodynamics and statistical

mechanics

Mean field theory

Renormalization group Universality classes

Power lawsFractalsSelf-similarity

“hallmarks”or“signatures”of criticality

Statistical Mechanics

Microscopic models

Ensemble Averages (all configurations are equally likely)

Distributions

log(size)

log(prob)

clusters

= correlation length

and correlations

Renormalization group

low density high densitycriticality

Fractaland

self-similar

Criticality

Percolation

cluster

critical

point c

Percolation

P( ) =

probability a site is on the cluster

P( )

0

1

1

no cluster

all sites occupied

Y = ( 1-P() ) + P() ( -P() )

P()

c

miss cluster

fulldensity

hit cluster

lose cluster

= - P()2

Y

yiel

d

density

Tremendous attention has been given to this point.

Tremendous attention has been given to this point.

100

101

102

103

104

10-1

100

101

102

Power laws

Criticality

cluster size

cumulativefrequency

Averagecumulativedistributions

clusters

fires

size

Power laws: only at the critical pointlow density

high density

cluster size

cumulativefrequency

Other lattices

Percolation has been studied in many settings.

Higher dimensionsConnected clusters areabstractions of cascading events.

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)

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.

.examplesin1.Use 2Nn

latticeNN

21 Nn

lattice

fire

distribution

density

yieldfires

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

100

101

102

103

104

100

101

102

103

-.15

Forest Fires: An Example of Self-Organized Critical BehaviorBruce D. Malamud, Gleb Morein, Donald L. Turcotte

18 Sep 1998

4 data sets

10-2

10-1

100

101

102

103

104

100

101

102

103

SOC FF

Exponents are way off

-1/2

Edge-of-chaos, criticality, self-organized criticality

(EOC/SOC)

yieldyield

density

Essential claims:

• Nature is adequately described by generic configurations (with generic sensitivity).

• Interesting phenomena is at criticality (or near a bifurcation).

• 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?

What about high yield

configurations?

?

Forget random, generic

configurations.

Would you design a system this way?

Barriers

Barriers

What about high yield

configurations?

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1

• Rare, nongeneric, measure zero.• Structured, stylized configurations.• Essentially ignored in stat. physics.• Ubiquitous in

• engineering• biology• geophysical phenomena?

What about high yield

configurations?

critical

Cold

Highly Optimized Tolerance (HOT)

Burned

Why power laws?Why power laws?

Almost any distribution

of sparks

OptimizeYield

Power law distribution

of events

both analytic and numerical results.

Special casesSpecial cases

Singleton(a priori

known spark)

Uniformspark

OptimizeYield

Uniformgrid

OptimizeYield

No fires

Special casesSpecial cases

No fires

Uniformgrid

In both cases, yields 1 as N .

Generally….Generally….

1. Gaussian2. Exponential3. Power law4. ….

OptimizeYield

Power law distribution

of events

5 10 15 20 25 30

5

10

15

20

25

30

0.1902 2.9529e-016

2.8655e-011 4.4486e-026

Probability distribution (tail of normal)

High probability region

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

density = 0.8496yield = 0.7752

Small events likely

large events are unlikely

0 0.2 0.4 0.6 0.8 10

0.1

0.2

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0.5

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1

random

grid

High yields.

Optimized grid

density = 0.8496yield = 0.7752

“grow” one site at a time to maximize incremental (local) yield

Local incrementalalgorithm

density= 0.8yield = 0.8

“grow” one site at a time to maximize incremental (local) yield

density= 0.9yield = 0.9

“grow” one site at a time to maximize incremental (local) yield

Optimal density= 0.97yield = 0.96

“grow” one site at a time to maximize incremental (local) yield

Optimaldensity= 0.9678yield = 0.9625

“grow” one site at a time to maximize incremental (local) yield

At density explores only

2)1( Nchoices out of a possible

2

2

)1( N

N

Very local and limited optimization, yet still gives very high yields.

Several small events

A very large event.

Optimaldensity= 0.9678yield = 0.9625

2)1( Nchoices out of a possible

2

2

)1( N

N

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

0 0.2 0.4 0.6 0.8 10

0.1

0.2

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0.5

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1

random

“optimized”

density

High yields.

At almost all densities.

0 0.2 0.4 0.6 0.8 10

0.1

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0.6

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random

density

Very sharp “phase transition.”

optimized

100

101

102

103

10-4

10-3

10-2

10-1

100

grid

“grown”

“critical”Cumulativedistributions

100

101

102

103

10-4

10-3

10-2

10-1

100

grid

“grown”

“critical”

size

Cum.Prob.

All producePower laws

100

101

102

103

10-12

10-10

10-8

10-6

10-4

10-2

100

Local Incremental Algorithm

This shows various stages on the way to the “optimal.”

Density is shown.

.9

.8

.7

optimaldensity= 0.9678yield = 0.9625

Gaussian

log(size)

log(prob>size)

Power laws are inevitable.

Improved design,more resources

HOT

SOC

d=1

dd=1d

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

SOC and HOT have very different power laws.

1

d 1

10

d

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

HOT

SOC

sizeinfinitesimal large

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.

Generalized “coding” problems

Fires

Web

Data compressionOptimizing d-1 dimensional cuts in d dimensional spaces.

-6 -5 -4 -3 -2 -1 0 1 2-1

0

1

2

3

4

5

6

Size of events

Cumulative

Frequency

Decimated dataLog (base 10)

Forest fires1000 km2

(Malamud)

WWW filesMbytes

(Crovella)

Data compression

(Huffman)

(codewords, files, fires)

Los Alamos fire

d=0d=1

d=2

-6 -5 -4 -3 -2 -1 0 1 2-1

0

1

2

3

4

5

6

Size of events

FrequencyFires

Web filesCodewords

Cumulative

Log (base 10)

-1/2

-1

-6 -5 -4 -3 -2 -1 0 1 2-1

0

1

2

3

4

5

6

FF

WWWDC

Data + PLR HOT Model

HOT

SOC

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

SOC and HOT are extremely different.

SOC HOT & Data

Max event size Infinitesimal LargeLarge event shape Fractal Compact

Slope Small LargeDimension d d-1 1/d

SOC and HOT are extremely different.

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?

Robust, yet fragile?

Extreme robustness and extreme hypersensitivity.

Small flaws

0 0.2 0.4 0.6 0.8 10

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If probability of sparks changes.

disaster

Conserved?

Sensitivity to:

sparks

flawsassumed

p(i,j)

Can eliminate sensitivity to: assumed

p(i,j)

Uniformgrid

Eliminates power laws as well.

Design for worst case scenario: no knowledge of sparks distribution

Can reduce impact of: flaws

Thick open barriers.

There is a yield penalty.

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.

Characteristic Critical HOT

Densities Low HighYields Low HighRobustness Generic Robust, yet fragile

Events/structure Generic, fractal Structured, stylizedself-similar self-dissimilar

External behavior Complex Nominally simpleInternally Simple Complex

Statistics Power laws Power lawsonly at criticality at all densities

Characteristic Critical HOT

Densities Low HighYields Low HighRobustness Generic Robust, yet fragile.

Events/structure Generic, fractal Structured, stylizedself-similar self-dissimilar

External behavior Complex Nominally simpleInternally Simple Complex

Statistics Power laws Power lawsonly at criticality at all densities

Characteristics

Toy models

?• Power systems• Computers• Internet• Software • Ecosystems • Extinction• Turbulence

Examples/Applications

• 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 HighYields Low HighRobustness Generic Robust, yet fragile.

Events/structure Generic, fractal Structured, stylizedself-similar self-dissimilar

External behavior Complex Nominally simpleInternally Simple Complex

Statistics Power laws Power lawsonly at criticality at all densities

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

IntensityFrequency

Extent

WeatherSpark sources

Flora and fauna

TopographySoil type

Climate/season

Los Padres National ForestMax Moritz

Yellow: lightning (at high altitudes in ponderosa pines)

Red: human ignitions(near roads)

Ignition and vegetation patterns in Los Padres National Forest

Brown: chaperalPink: Pinon Juniper

10-5

10-4

10-3

10-2

10-1

100

10-1

100

101

102

103

104

FF = 2 = 1

California brushfires

Santa Monica Mountains Max Moritz and Marco Morais

SAMO Fire History

Fires 1991-1995

Fires 1930-1990

Fires are compact regions of nontrivial area.

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)

Data: typical five year period HFIREs Simulations: typical five year period

Fire scar shapes are compact

Excellent agreement with data for realistic values of the parameters

Agreement with the PLR HOT theory based on optimal allocation of resources

α=1

α=1/2

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:

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 selectivepressure shaping earlyplants

Our hypothesis is that robustness in an uncertain environment is thedominant force shaping complexity in most biological, ecological, andtechnological systems