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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

Fragile

Fragile

(to unknown

or rare

perturbations)

Robust

(to known and

designed-for

uncertainties)

Uncertainties

Robust

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

connected clusters

Draw lattices without lines.

Assume one “spark” hits the lattice at a single site.

A “spark” that hits an empty site does nothing.

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

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

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

clusters

log(prob)

log(size)

Statistical Mechanics

Microscopic models

Distributions

and correlations

Ensemble Averages (all configurations are equally likely)

= correlation length

Percolation

cluster

all sites occupied

1

critical point c

0

1

no cluster

Percolation

P( ) = probability a site is on the cluster

P( )

miss cluster

full

density

hit cluster

lose cluster

Y

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

= - P()2

P()

c

Higher dimensions

Other lattices

Percolation has been studied in many settings.

Connected clusters are

abstractions of cascading

events.

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

Self-organized criticality (SOC)

Iterate on:

- Pick n sites at random, and grow new trees on any which are empty.
- Spark 1 site at random. If occupied, burn connected cluster.

18 Sep 1998

Forest Fires: An Example of Self-Organized Critical Behavior

Bruce D. Malamud, Gleb Morein, Donald L. Turcotte

4 data sets

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).

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.

?

Forget random, generic configurations.

What about high yield configurations?

Would you design a system this way?

Rare, nongeneric, measure zero.

- Structured, stylized configurations.
- Essentially ignored in stat. physics.
- Ubiquitous in
- engineering
- biology
- geophysical phenomena?

What about high yield configurations?

both analytic and numerical results.

Why power laws?

Optimize

Yield

Almost any

distribution

of sparks

Power law

distribution

of events

Optimize

Yield

No fires

Optimize

Yield

Uniform

grid

Special cases

Singleton

(a priori

known spark)

Uniform

spark

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

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 = 0.8496

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

Optimal

density= 0.9678

yield = 0.9625

Several small events

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

A very large event.

At density explores only

choices out of a possible

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

Optimal

density= 0.9678

yield = 0.9625

large events are unlikely

Small events likely

“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.

“optimized”

1

0.9

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

High yields.

At almost all densities.

optimal

density= 0.9678

yield = 0.9625

.9

.8

.7

0

10

-2

10

-4

10

Local Incremental Algorithm

-6

10

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

Density is shown.

-8

10

-10

10

-12

10

0

1

2

3

10

10

10

10

d

d

HOT

SOC and HOT have very different power laws.

d=1

SOC

d=1

- HOT decreases with dimension.
- SOC increases with dimension.

HOT yields compact events of nontrivial size.

- SOC has infinitesimal, fractal events.

HOT

SOC

large

infinitesimal

size

Burnt regions are 2-dA HOT forest fire abstraction…

Fire suppression mechanisms must stop a 1-d front.

Optimal strategies must tradeoff resources with risk.

FiresGeneralized “coding” problems

Data compression

Optimizing d-1 dimensional cuts in d dimensional spaces.

Web

Los Alamos fire

6

Data compression

(Huffman)

WWW files

Mbytes

(Crovella)

5

Cumulative

d=1

d=0

4

3

Frequency

Forest fires

1000 km2

(Malamud)

2

1

d=2

0

-1

-6

-5

-4

-3

-2

-1

0

1

2

Decimated data

Log (base 10)

Size of events

(codewords, files, fires)

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

Uniform

grid

Design for worst case scenario: no knowledge

of sparks distribution

assumed

p(i,j)

Can eliminate sensitivity to:

Eliminates power laws as well.

Optimized

HOT forest fire models

Critical percolation and SOC forest fire models

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

Characteristic Critical HOT

Densities Low High

Yields Low High

Robustness Generic Robust, yet fragile

Events/structure Generic, fractalStructured, stylized

self-similar self-dissimilar

External behavior ComplexNominally simple

Internally SimpleComplex

Statistics Power lawsPower laws

only at criticality at all densities

Characteristic Critical HOT

Densities Low High

Yields Low High

Robustness Generic Robust, yet fragile.

Events/structure Generic, fractalStructured, stylized

self-similar self-dissimilar

External behavior ComplexNominally simple

Internally SimpleComplex

Statistics Power lawsPower laws

only at criticality at all densities

- Power systems
- Computers
- Internet
- Software
- Ecosystems
- Extinction
- Turbulence

Characteristics

Examples/

Applications

Toy models

?

Characteristic Critical HOT

Densities Low High

Yields Low High

Robustness Generic Robust, yet fragile.

Events/structure Generic, fractalStructured, stylized

self-similar self-dissimilar

External behavior ComplexNominally simple

Internally SimpleComplex

Statistics Power lawsPower 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...

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

Weather

Spark sources

Flora and fauna

Topography

Soil type

Climate/season

Forest fires dynamicsIntensity

Frequency

Extent

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

Santa Monica Mountains

Max Moritz and Marco Morais

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

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

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