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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.

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slide1
Are Forest Fires

HOT?

A mostly critical

look at physics,

phase transitions,

and universality

Jean Carlson, UCSB

background
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
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
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, 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
Robustness of HOT systems

Fragile

Fragile

(to unknown

or rare

perturbations)

Robust

(to known and

designed-for

uncertainties)

Uncertainties

Robust

slide7
Robustness

Complexity

Interconnection

Aim: simplest possible story

slide8
The simplest possible spatial model of HOT.

Square site percolation or simplified “forest fire” model.

Carlson and Doyle,

PRE, Aug. 1999

slide9
empty square lattice

occupied sites

slide10
not

connected

connected clusters

slide11
connected clusters

Draw lattices without lines.

slide13
.4

.6

Density = fraction of occupied sites

.8

.2

slide14
Assume one “spark” hits the lattice at a single site.

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

slide17
yield

density

loss

Think of (toy) forest fires.

slide18
yield

density

loss

Yield = the density after one spark

slide19
no sparks

1

0.9

0.8

0.7

Y=

yield

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.2

0.4

0.6

0.8

1

 = density

slide20
yield =

Average over configurations.

density=.5

slide21
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

slide22
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

slide23
Y

Fires don’t matter.

Cold

slide27
Statistical physics:

Phase transitions, criticality, and power laws

slide28
Thermodynamics and statistical mechanics

Mean field theory

Renormalization group  Universality classes

Power laws

Fractals

Self-similarity

“hallmarks”or

“signatures”

of criticality

slide29
clusters

log(prob)

log(size)

Statistical Mechanics

Microscopic models

Distributions

and correlations

Ensemble Averages (all configurations are equally likely)

 = correlation length

renormalization group
criticality

high density

low density

Renormalization group
slide31
Fractal

and

self-similar

Criticality

slide34
Percolation

  cluster

slide35
all sites occupied

1

critical point c

0

1

no  cluster

Percolation

P( ) = probability a site is on the  cluster

P( )

slide36
miss  cluster

full

density

hit  cluster

lose  cluster

Y

Y =( 1-P() ) + P() (  -P() )

=  - P()2

P()

c

slide38
Power laws

Criticality

cumulative

frequency

cluster size

slide39
Average

cumulative

distributions

fires

clusters

size

slide40
high density

low density

cumulative

frequency

Power laws: only at the critical point

cluster size

slide41
Higher dimensions

Other lattices

Percolation has been studied in many settings.

Connected clusters are

abstractions of cascading

events.

slide42
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

slide43
Self-organized criticality (SOC)

yield

Create a dynamical system around the critical point

density

slide44
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.
slide45
lattice

distribution

fire

density

yield

fires

slide46
3

10

2

10

1

10

0

10

0

1

2

3

4

10

10

10

10

10

-.15

slide47
18 Sep 1998

Forest Fires: An Example of Self-Organized Critical Behavior

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

4 data sets

slide48
3

10

SOC FF

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

slide49
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).
slide50
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.
slide52
?

Forget random, generic configurations.

What about high yield configurations?

Would you design a system this way?

slide53
Barriers

What about high yield configurations?

Barriers

slide54
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

slide55
Rare, nongeneric, measure zero.
  • Structured, stylized configurations.
  • Essentially ignored in stat. physics.
  • Ubiquitous in
    • engineering
    • biology
    • geophysical phenomena?

What about high yield configurations?

slide57
both analytic and numerical results.

Why power laws?

Optimize

Yield

Almost any

distribution

of sparks

Power law

distribution

of events

slide58
Optimize

Yield

No fires

Optimize

Yield

Uniform

grid

Special cases

Singleton

(a priori

known spark)

Uniform

spark

slide59
Special cases

No fires

In both cases, yields 1 as N .

Uniform

grid

slide60
Generally….

Optimize

Yield

  • Gaussian
  • Exponential
  • Power law
  • ….

Power law

distribution

of events

slide61
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

slide62
Grid design: optimize the position of “cuts.”

cuts = empty sites in an otherwise fully occupied lattice.

Compute the global optimum for this constraint.

slide63
large events are unlikely

Optimized grid

Small events likely

density = 0.8496

yield = 0.7752

slide64
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

slide65
Local incremental

algorithm

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

slide66
density= 0.8

yield = 0.8

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

slide67
density= 0.9

yield = 0.9

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

slide68
Optimal

density= 0.97

yield = 0.96

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

slide69
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.

slide70
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.

slide71
“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.

slide72
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

slide73
“critical”

0

10

Cumulative

distributions

-1

10

“grown”

-2

10

grid

-3

10

-4

10

0

1

2

3

10

10

10

10

slide74
“critical”

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

slide75
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

slide76
Power laws are inevitable.

Improved design,

more resources

Gaussian

log(prob>size)

log(size)

slide77
d

d

HOT

SOC and HOT have very different power laws.

d=1

SOC

d=1

  • HOT  decreases with dimension.
  • SOC increases with dimension.
slide78
HOT yields compact events of nontrivial size.
  • SOC has infinitesimal, fractal events.

HOT

SOC

large

infinitesimal

size

a hot forest fire abstraction
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.

generalized coding problems
FiresGeneralized “coding” problems

Data compression

Optimizing d-1 dimensional cuts in d dimensional spaces.

Web

slide81
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)

slide82
-1

-1/2

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)

slide83
Data + PLR HOT Model

6

DC

5

WWW

4

3

FF

2

1

0

-1

-6

-5

-4

-3

-2

-1

0

1

2

slide86
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
slide89
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

slide90
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

slide92
Conserved?

sparks

assumed

p(i,j)

flaws

Sensitivity to:

slide93
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.

slide94
Thick open barriers.

There is a yield penalty.

flaws

Can reduce impact of:

slide95
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.
slide96
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

slide97
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

?

slide98
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
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
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

Flora and fauna

Topography

Soil type

Climate/season

Forest fires dynamics

Intensity

Frequency

Extent

slide103
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

slide104
4

10

3

10

2

 = 1

10

1

10

0

10

-1

10

-5

-4

-3

-2

-1

0

10

10

10

10

10

10

California brushfires

FF  = 2

santa monica mountains
Santa Monica Mountains

Max Moritz and Marco Morais

slide107
Fires are compact regions of nontrivial area.

Fires 1930-1990

Fires 1991-1995

slide110
1996 Calabasas Fire

Historical fire spread

Simulated fire spread

Suppression?

slide111
Preliminary Results from the HFIRES simulations

(no Extreme Weather conditions included)

slide112
Fire scar shapes are compact

Data: typical five year period

HFIREs Simulations: typical five

year period

slide114
Agreement with the PLR HOT theory

based on optimal allocation of resources

α=1/2

α=1

slide115
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
slide116
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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