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John DoyleControl and Dynamical
Systems Caltech
Research interests
• Complex networks applications– Ubiquitous, pervasive, embedded control,
computing, and communication networks– Biological regulatory networks
• New mathematics and algorithms– robustness analysis – systematic design– multiscale physics
Collaboratorsand contributors
(partial list)
Biology: Csete,Yi, Borisuk, Bolouri, Kitano, Kurata, Khammash, El-Samad, …
Alliance for Cellular Signaling: Gilman, Simon, Sternberg, Arkin,…HOT: Carlson, Zhou,…Theory: Lall, Parrilo, Paganini, Barahona, D’Andrea, …Web/Internet: Low, Effros, Zhu,Yu, Chandy, Willinger, …Turbulence: Bamieh, Dahleh, Gharib, Marsden, Bobba,…Physics: Mabuchi, Doherty, Marsden, Asimakapoulos,…Engineering CAD: Ortiz, Murray, Schroder, Burdick, Barr, …Disturbance ecology: Moritz, Carlson, Robert, …Power systems: Verghese, Lesieutre,…Finance: Primbs, Yamada, Giannelli,……and casts of thousands…
Background reading online• On website accessible from SFI talk abstract• Papers with minimal math
– HOT and power laws– Chemotaxis, Heat shock in E. Coli– Web & Internet traffic, protocols, future issues
• Thesis: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization
• Recommended books– A course in Robust Control Theory, Dullerud and
Paganini, Springer– Essentials of Robust Control, Zhou, Prentice-Hall– Cells, Embryos, and Evolution, Gerhart and Kirschner
+ Regulatory InteractionsMass Transfer in Metabolism*
Biochemical Network: E. Coli Metabolism
* from: EcoCYC by Peter Karp
From Adam Arkin
SuppliesMaterials &
Energy
SuppliesMaterials &
Energy
SuppliesRobustness
SuppliesRobustness
Complexity RobustnessComplexity Robustness
ComplexityRobustness
Transcription/translation
MicrotubulesNeurogenesisAngiogenesis
Immune/pathogenChemotaxis
….
Regulatory feedback control
An apparent paradox
Component behavior seems to be gratuitously uncertain, yet the systems have robust performance.
Mutation
Selection
Darwinian evolution uses selection on random mutations
to create complexity.
Transcription/translation
MicrotubulesNeurogenesisAngiogenesis
Immune/pathogenChemotaxis
….
Regulatory feedback control
• Such feedback strategies appear throughout biology (and advanced technology).
• Gerhart and Kirschner (correctly) emphasis that this “exploratory” behavior is ubiquitous in biology…
• …but claim it is rare in our machines.
• This is true of primitive, but not advanced, technologies.
• Robust control theory provides a clear explanation.
Component behavior seems to be gratuitously uncertain, yet the systems have robust performance.
Overview
• Without extensive engineering theory and math, even reverse engineering complex engineering systems would be hopeless. (Let alone actual design.)
• Why should biology be much easier? • With respect to robustness and complexity, there is too
much theory, not too little.
Overview
• Two great abstractions of the 20th Century:– Separate systems engineering into control, communications,
and computing• Theory
• Applications
– Separate systems from physical substrate
• Facilitated massive, wildly successful, and explosive growth in both mathematical theory and technology…
• …but creating a new Tower of Babel where even the experts do not read papers or understand systems outside their subspecialty.
“Any sufficiently advanced technology is indistinguishable from magic.”
Arthur C. Clarke
“Those who say do not know, those who know do not say.”
Zen saying
“Any sufficiently advanced technology is indistinguishable from magic.”
Arthur C. Clarke
Today’s goal• Introduce basic ideas about robustness and complexity• Minimal math• Hopefully familiar (but unconventional) example
systems• Caveat: the “real thing” is much more complicated• Perhaps any such “story” is necessarily misleading• Hopefully less misleading than existing popular
accounts of complexity and robustness
Complexity and robustness
• Complexity phenotype : robust, yet fragile• Complexity genotype: internally complicated• New theoretical framework: HOT (Highly optimized
tolerance, with Jean Carlson, Physics, UCSB)• Applies to biological and technological systems
– Pre-technology: simple tools– Primitive technologies use simple strategies to build fragile
machines from precision parts.– Advanced technologies use complicated architectures to create
robust systems from sloppy components…– … but are also vulnerable to cascading failures…
Robust, yet fragile phenotype
• Robust to large variations in environment and component parts (reliable, insensitive, resilient, evolvable, simple, scaleable, verifiable, ...)
• Fragile, often catastrophically so, to cascading failures events (sensitive, brittle,...)
• Cascading failures can be initiated by small perturbations (Cryptic mutations,viruses and other infectious agents, exotic species, …)
• There is a tradeoff between – ideal or nominal performance (no uncertainty) – robust performance (with uncertainty)
• Greater “pheno-complexity”= more extreme robust, yet fragile
Robust, yet fragile phenotype
• Cascading failures can be initiated by small perturbations (Cryptic mutations,viruses and other infectious agents, exotic species, …)
• In many complex systems, the size of cascading failure events are often unrelated to the size of the initiating perturbations
• Fragility is interesting when it does not arise because of large perturbations, but catastrophic responses to small variations
Complicated genotype
• Robustness is achieved by building barriers to cascading failures
• This often requires complicated internal structure, hierarchies, self-dissimilarity, layers of feedback, signaling, regulation, computation, protocols, ...
• Greater “geno-complexity” = more parts, more structure• Molecular biology is about biological simplicity, what
are the parts and how do they interact.• If the complexity phenotypes and genotypes are linked,
then robustness is the key to biological complexity.• “Nominal function” may tell little.
Transcription/translation
MicrotubulesNeurogenesisAngiogenesis
Immune/pathogenChemotaxis
….
Regulatory feedback control
An apparent paradox
Component behavior seems to be gratuitously uncertain, yet the systems have robust performance.
Mutation
Selection
Darwinian evolution uses selection on random mutations
to create complexity.
Tempenviron
Tempcell
Folded Proteins
Unfolded Proteins Aggregates
Loss of ProteinFunction
Networkfailure
Death
Cell
Tempenviron
Tempcell
Folded Proteins
Unfolded Proteins Aggregates
Loss of ProteinFunction
Networkfailure
Death
Cell
How does the cell build “barriers” (in state space) to stop
this cascading failure event?
Tempenviron
Folded Proteins
Tempcell
Insulate &Regulate
Temp
Tempenviron
Folded Proteins
Tempcell
Thermo-tax
Tempenviron
Tempcell
Folded Proteins
Unfolded Proteins Aggregates
More robust ( Temp stable)
proteins
Tempenviron
Tempcell
Folded Proteins
Unfolded Proteins Aggregates
• Key proteins can have multiple (allelic or paralogous) variants• Allelic variants allow populations to adapt• Regulated multiple gene loci allow individuals to adapt
-1/T
21o
Log of E. ColiGrowthRate
37o
46o
Heat Shock Response
RTAEev
42o
-1/T
21o
Log of E. ColiGrowthRate
37o
42o
46o
Robustness/performance tradeoff?
Tempenviron
Tempcell
Folded Proteins
Unfolded Proteins
Refold denatured proteins
Heat shock response involves complex feedback
and feedforward control.
Alternative strategies
• Robust proteins– Temperature stability
– Allelic variants
– Paralogous isozymes
• Regulate temperature• Thermotax• Heat shock response
– Up regulate chaperones and proteases
– Refold or degraded denatured proteins
Why does biology (and advanced technology)
overwhelmingly opt for the complex control
systems instead of just robust components?
E. Coli Heat Shock (with Kurata, El-Samad, Khammash, Yi)
unfoldPDnaK :
dependent T
DnaK:32
320 32
free
0FtsH
FtsHDnaK ::32
protease:32
0DnaK freeDnaK
1k 2k
distk
3k
03.0
1
s
D n a k t r a n s l a t i o n & t r a n s c r i p t i o n
d y n a m i c s
1r
2r
rateon translati
dependent T 32
raten degradatio 32
protease
rpoH gene
Transcription
32 mRNA
hsp1 hsp2
Transcription & Translation
FtsHLonDnaKGroLGroS
Chaperones
Proteases
-
- Translation
32
Heat
Heat stabilizes32
Heat
Outer Feedback Loop
Local Loop
Feedforward
Heater
Thermostat
Added mass
Moves the center of mass forward.
Tail
Moves the center of pressure aft.
Thus stabilizing forward flight.
At the expense of extra weight and drag.
For minimum weight & drag, (and other performance issues)
eliminate fuselage and tail.
Why do we love building robust systems from highly uncertain
and unstable components?
P- +
(disturbance)d
r( )y P r d
Assumptions on components:• Everything just numbers • Uncertainty in P• Higher gain = more uncertain
( )y P P r d
1 21 2
1 2
P PP P
P P
G-
K
+
dr
P- +
(disturbance)d
r
11y GSr Sd S r Sd
K
1
1S
GK
Negative feedback
( )y G r GK y d
( )y P r d
11y GSr Sd S r Sd
K
1
1S
GK
G-
K
+
dr y
11 1
11
G GKK
S y rK
Results for y (1/K )r:• high gain• low uncertainty• d attenuated
S = sensitivity function
Design recipe:• 1 >> K >> 1/G • G >> 1/K >> 1• G maximally uncertain!• K small, low uncertainty
Results for y (1/K )r:• high gain• low uncertainty• d attenuated
Extensions to:• Dynamics• Multivariable• Nonlinear• Structured uncertainty
All cost more computationally.
G-
K
+
dr y
Design recipe:• 1 >> K >> 1/G • G >> 1/K >> 1• G maximally uncertain!• K small, low uncertainty
G-
K
r y
Transcription/translationMicrotubule formation
NeurogenesisAngiogenesis
Antibody productionChemotaxis
….
Regulatory feedback control
Uncertain high gain
Summary
• Primitive technologies build fragile systems from precision components.
• Advanced technologies build robust systems from sloppy components.
• There are many other examples of regulator strategies deliberately employing uncertain and stochastic components…
• …to create robust systems.• High gain negative feedback is the most powerful
mechanism, and also the most dangerous.• In addition to the added complexity, what can go
wrong?
G-
K
y
1
1y d
F
1d if F 1
F
( )y F y d
+
(disturbance)d
F
y+
d
F GK
1
1y d
F
F
y+
d
If y, d and F are just numbers:
S = sensitivity function
S measures disturbance rejection.
It’s convenient to study ln(S).
1
1
yS
d F
P
N
o
eg
si
ativ
tive
e ( 0) ln
( 0) ln( ) 0 Disturbance ampli
( ) 0 Disturbance attenuated
fiedF
F S
S
F
F
P
N
o
eg
si
ativ
tive
e ( 0) ln
( 0) ln( ) 0 Disturbance ampli
( ) 0 Disturbance attenuated
fiedF
F S
S
F
F
ln(S)
F
F < 0ln(S) < 0
attenuation
F > 0ln(S) > 0
amplification
ln( |S| )
1
1
yS
d F
P
N
o
eg
si
ativ
tive
e ( 0) ln
( 0) ln( ) 0 Disturbance ampli
( ) 0 Disturbance attenuated
fiedF
F S
S
F
F
P
N
o
eg
si
ativ
tive
e ( 0) ln
( 0) ln( ) 0 Disturbance ampli
( ) 0 Disturbance attenuated
fiedF
F S
S
F
F
ln(S)
F ln(S)
extreme robustnessextreme robustness
F 1 ln(S)
extreme sensitivityextreme sensitivity
F
1
1
yS
d F
If these model physical processes, then d and y are signals and F is an operator. We can still define
S( = |Y( /D( |where E and D are the Fourier transforms of y and d. ( If F is linear, then S is independent of D.)
Under assumptions that are consistent with F and d modeling physical systems (in particular, causality), it is possible to prove that:
0)(log dS
(Bode, ~1940)
Fy+d
1
1S
F
log|S |he amplification (F>0) must atleast balance the attenuation (F<0).
( 0) ln( ) 0 attenuate
( 0) ln( ) 0 amplify
F
F S
S
( 0) ln( ) 0 attenuate
( 0) ln( ) 0 amplify
F
F S
S
log|S |
ln|S|
F
Negative feedback
Positive feedback
log|S |
ln|S|
F
Negative feedbackRobust
Positive feedback
…yetfragile
Robustness of HOT systems
Robust
Fragile
Robust(to known anddesigned-foruncertainties)
Fragile(to unknown
or rareperturbations)
Uncertainties
Feedback and robustness
• Negative feedback is both the most powerful and most dangerous mechanism for robustness.
• It is everywhere in engineering, but appears hidden as long as it works.
• Biology seems to use it even more aggressively, but also uses other familiar engineering strategies:– Positive feedback to create switches (digital systems)– Protocol stacks– Feedforward control– Randomized strategies– Coding
ComplexityRobustness
Current research
• So far, this is all undergraduate level material• Current research involves lots of math not
traditionally thought of as “applied”• New theoretical connections between robustness,
evolvability, and verifiability• Beginnings of a more integrated theory of control,
communications and computing• Both biology and the future of ubiquitous,
embedded networking will drive the development of new mathematics.
Robustness of HOT systems
Robust
Fragile
Robust(to known anddesigned-foruncertainties)
Fragile(to unknown
or rareperturbations)
Uncertainties
Robustness of HOT systems
Robust
Fragile
Chess Meteors
Humans
Archaea
Robustness of HOT systems
Robust
Fragile
Chess Meteors
Humans
Archaea
Humans + machines?
Machines
Robust
Fragile
Uncertainty
Diseases of complexity
CancerEpidemics
Viral infectionsAuto-immune disease
Robust
Fragile
Sources of uncertainty
• In a system– Environmental perturbations– Component variations
• In a model– Parameter variations– Unmodeled dynamics– Assumptions– Noise
( )F
Robust
Fragile
Sources of uncertainty
( ) ?F
( ) ?F
Typically NP hard.
• If true, there is always a short proof.• Which may be hard to find.
, ( ) ?F
Typically coNP hard.
• More important problem.• Short proofs may not exist.
Fundamental asymmetries* • Between P and NP• Between NP and coNP
Fundamental asymmetries* • Between P and NP• Between NP and coNP
* Unless they’re the same…
• Standard techniques include relaxations, Grobner bases, resultants, numerical homotopy, etc…
• Powerful new method based on real algebraic geometry and semidefinite programming (Parrilo, Shor, …)
• Nested series of polynomial time relaxations search for polynomial sized certificates
• Exhausts coNP (but no uniform bound)• Relaxations have both computational and physical
interpretations• Beats gold standard algorithms (eg MAX CUT)
handcrafted for special cases• Completely changes the P/NP/coNP picture
How do we prove that , ( ) ?F
Bacterial chemotaxis
Random walk
Ligand Motion Motor
Bacterial chemotaxis (Yi, Huang, Simon, Doyle)
pCheY
Ligand
SignalTransduction
gradient
Biased random walk
Motion Motor
pCheYSignal
Transduction
MotorLigand Motion
High gain (cooperativity)
“ultrasensitivity”
References:Cluzel, Surette, Leibler
pCheYSignal
Transduction
+CH3R
ATP ADPP
~
flagellarmotor
Z
Y
PY
~
PiB
B~P
Pi
CW-CH3
ATP
WA
MCPs
WA
+ATT
-ATT
MCPsSLOW
FAST
ligand binding motor
Motor
References:Cluzel, Surette, Leibler + Alon, Barkai, Bray, Simon, Spiro, Stock, Berg, …
+CH3R
ATP ADPP
~
flagellarmotor
Z
Y
PY
~
PiB
B~P
Pi
CW-CH3
ATP
WA
MCPs
WA
+ATT
-ATT
MCPsSLOW
FAST
ligand binding
motor
ATP ADPP
~
flagellarmotor
Z
Y
PY
~
Pi
CW
ATP
WA
MCPs
WA
+ATT
-ATT
MCPs
FAST
ligand binding
motor
Fast (ligand and phosphorylation)
0 1 2 3 4 5 6
0
1
0 1 2 3 4 5 6
Time (seconds)
No methylation
Barkai, et al
Short time Yp response
Che Yp
Ligand
Extend run(more ligand)
+CH3R
ATP ADPP
~B
B~P
Pi
-CH3
ATP
WA
MCPs
WA
MCPsSLOW
Slow (de-) methylation dynamics
+CH3R
ATP ADPP
~
flagellarmotor
Z
Y
PY
~
PiB
B~P
Pi
CW-CH3
ATP
WA
MCPs
WA
+ATT
-ATT
MCPsSLOW
FAST
ligand binding
motor
0 1000 2000 3000 4000 5000 6000 7000
01
3
5
0 1000 2000 3000 4000 5000 6000 7000Time (seconds)
No methylation
B-L
Long time Yp response
No methylation
Extend run(more ligand)
Tumble(less ligand)
Ligand
Biologists call this “perfect adaptation”
• Methylation produces “perfect adaptation” by integral feedback.• Integral feedback is ubiquitous in both engineering systems and
biological systems.• Integral feedback is necessary for robust perfect adaptation.
Tumbling bias
pCheY
SignalTransduction
Motor
Perfect adaptation is necessary …
pCheYligand
pCheY
Tumbling bias
ligand
Perfect adaptation is necessary …
…to keep CheYp in the responsive range of the motor.
Fine tuned or robust ?
• Maybe just not the right question.
• Fine tuned for robustness…
• …with resource costs and new fragilities as the price.
+ Regulatory InteractionsMass Transfer in Metabolism*
Biochemical Network: E. Coli Metabolism
* from: EcoCYC by Peter Karp
From Adam Arkin
SuppliesMaterials &
Energy
SuppliesMaterials &
Energy
SuppliesRobustness
SuppliesRobustness
Complexity RobustnessComplexity Robustness
What about ?
• Information & entropy
• Fractals & self-similarity
• Chaos
• Criticality and power laws
• Undecidability
• Fuzzy logic, neural nets, genetic algorithms
• Emergence
• Self-organization
• Complex adaptive systems
• New science of complexity
• Not really about complexity
• These concepts themselves are “robust, yet fragile”
• Powerful in their niche
• Brittle (break easily) when moved or extended
• Some are relevant to biology and engineering systems
• Comfortably reductionist
• Remarkably useful in getting published
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?
-6 -5 -4 -3 -2 -1 0 1 2-1
0
1
2
3
4
5
6
Size of events
Frequency
Decimated dataLog (base 10)
Forest fires1000 km2
(Malamud)
WWW filesMbytes
(Crovella)
Data compression
(Huffman)
Los Alamos fire
Cumulative
Size of events x vs. frequency
log(size)
)1()( xxpdx
dPlog(probability)
log(Prob > size)
xPlog(rank)
-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
The HOT view of power laws
• Engineers design (and evolution selects) for systems with certain typical properties:
• Optimized for average (mean) behavior
• Optimizing the mean often (but not always) yields high variance and heavy tails
• Power laws arise from heavy tails when there is enough aggregate data
• One symptom of “robust, yet fragile”
Source coding for data compression
Based on frequencies of source word occurrences,
Select code words.
To minimize message length.
Shannon coding
• Ignore value of information, consider only “surprise”• Compress average codeword length (over stochastic
ensembles of source words rather than actual files)• Constraint on codewords of unique decodability• Equivalent to building barriers in a zero dimensional tree• Optimal distribution (exponential) and optimal cost are:
DataCompression
length log( )
exp( )i i
i i
l p
p cl
Avg. length =
log( )
i i
i i
p l
p p
0 1 2-1
0
1
2
3
4
5
6
DC
Data
Avg. length =
log( )
i i
i i
p l
p p
How well does the model predict the data?
length log(
exp( )
)i i
i i
l p
p cl
0 1 2-1
0
1
2
3
4
5
6
DC
Data + Modellength log(
exp( )
)i i
i i
l p
p cl
Avg. length =
log( )
i i
i i
p l
p p
How well does the model predict the data?
Not surprising, because the file was compressed using
Shannon theory.
Small discrepancy due to integer lengths.
Web layout as generalized “source coding”
• Keep parts of Shannon abstraction:– Minimize downloaded file size– Averaged over an ensemble of user access
• But add in feedback and topology, which completely breaks standard Shannon theory
• Logical and aesthetic structure determines topology
• Navigation involves dynamic user feedback • Breaks standard theory, but extensions are
possible• Equivalent to building 0-dimensional
barriers in a 1- dimensional tree of content
document
split into N files to minimize download time
A toy website model(= 1-d grid HOT design)
# links = # files
split into N files to minimize download time
Forest fires dynamics
IntensityFrequency
Extent
WeatherSpark sources
Flora and fauna
TopographySoil type
Climate/season
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 compression
• Optimizing d-1 dimensional cuts in d dimensional spaces…
• To minimize average size of files or fires, subject to resource constraint.
• Models of greatly varying detail all give a consistent story.
• Power laws have 1/d.• Completely unlike criticality.
d = 0 data compressiond = 1 web layoutd = 2 forest fires
1
(1 )0d
i ip l c d
1
d
1
( ) dP l l
exp( )
0i ip cl
d
Theory
-6 -5 -4 -3 -2 -1 0 1 2-1
0
1
2
3
4
5
6
FF
WWWDC
Data
-6 -5 -4 -3 -2 -1 0 1 2-1
0
1
2
3
4
5
6
FF
WWWDC
Data + Model/Theory
Forest fires?
Burnt regions are 2-d
Fire suppression mechanisms must stop a 1-d front.
Forest fires?
Geography could make d <2.
California geography:further irresponsible speculation
• Rugged terrain, mountains, deserts• Fractal dimension d 1?• Dry Santa Ana winds drive large ( 1-d) fires
-6 -5 -4 -3 -2 -1 0 1 2-1
0
1
2
3
4
5
6
FF(national)
d = 2
Data + HOT Model/Theory
d = 1
California brushfires
-6 -5 -4 -3 -2 -1 0 1 2-1
0
1
2
3
4
5
6
Data + HOT+SOC
d = 1
SOC FFd = 2
.15
Critical/SOC exponents are way off
SOC < .15
Data: > .5
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
HOT FFd = 2
Additional 3 data sets
Fires 1991-1995
Fires 1930-1990
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
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
yetfragile
Robust
Gaussian,Exponential
Log(event sizes)
Log(freq.) cumulative
Gaussian
log(size)
log(prob>size)
Power laws are inevitable.
Improved design,more resources
Power laws summary
• Power laws are ubiquitous• 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 complex system as HOT is just the
beginning of study.• The real work is in new Internet protocol design,
forest fire suppression strategies, etc…
Universal network behavior?
demand
throughputCongestion
induced “phase
transition.”
Similar for:• Power grid?• Freeway traffic?• Gene regulation?• Ecosystems?• Finance?
Web/Internet?demand
thro
ughp
utCongestion induced “phase transition.”
Power laws
log(file size)
log(
P>
)
2
3 H
random networks
log(thru-put)
log(demand)
Networks Making a “random network:”• Remove protocols
– No IP routing
– No TCP congestion control
• Broadcast everything
Many orders of magnitude slower
BroadcastNetwork
Networks
random networks
real networks
HOTlog(thru-put)
log(demand)
BroadcastNetwork
HOT
Turbulence
flow
pressure drop
random pipes
streamlined pipes
HOT turbulence?Robust, yet
fragile?
• Through streamlined design• High throughput• Robust to bifurcation transition (Reynolds number)• Yet fragile to small perturbations• Which are irrelevant for more “generic” flows
HOT
flow
pressure drop
random pipes
streamlined pipes
Shear flow turbulence summary
• Shear flows are ubiquitous and important
• HOT may be a unifying perspective
• Chaos is interesting, but may not be very important for many important flows
• Viewing a turbulent or transitioning flow as HOT is just the beginning of study
random
designed
HOTYield,flow, …
Densities, pressure,…
The yield/density curve predicted using random ensembles is way off.
Similar for:• Power grid• Freeway traffic• Gene regulation• Ecosystems• Finance?
pipes
channelswings
Turbulence in shear flows
Turbulence is thegraveyard of theories.
Hans Liepmann Caltech
Kumar Bobba, Bassam Bamieh
Chaos and turbulence
• The orthodox view:
• Adjusting 1 parameter (Reynolds number) leads to a bifurcation cascade to chaos
• Turbulence transition is a bifurcation
• Turbulent flows are chaotic, intrinsically nonlinear
• There are certainly many situations where this view is useful.
velocitylow high
equilibriumequilibrium periodicperiodic chaoticchaotic
pressure drop
averageflow
speed
“random” pipe
pressure (drop)
flow(averagespeed)
laminar
turbulent
bifurcation
Random pipes are like bluff bodies.
pressure
flowTypical flow
pipes
channels
wingsStreamline
log(pressure)
log(flow)laminar
turbulent
“theory”
experiment
Random pipe
streamlined pipe
log(Re)
log(flow)
Random pipe
streamlined pipe
210 310 410 510
log(Re)
Random pipe
streamlined pipe
210 310 410 510
It can be promoted (or delayed!)with tiny perturbations.
This transition is extremely delicate(and controversial).
Transition to turbulence is promoted (occurs at lower speeds) by
Surface roughnessInlet distortionsVibrationsThermodynamic fluctuations?Non-Newtonian effects?
None of which makes much difference for “random” pipes.
Random pipe
210 310 410 510
Shark skin delays transition to turbulence
log(pressure)
log(flow)
water
80 ppm Guar
It can be reduced with small amounts of polymers.
HOT turbulence?Robust, yet
fragile?
• Through streamlined design• High throughput• Robust to bifurcation transition (Reynolds number)• Yet fragile to small perturbations• Which are irrelevant for more “generic” flows
HOT
flow
pressure drop
random pipes
streamlined pipes
streamwise
Couette flow
upflow
high-speedregion
downflow
low speedstreaks
From Kline
Streamwiseconstantperturbation
Spanwiseperiodic
Streamwiseconstantperturbation
Spanwiseperiodic
w
vu
flow
velocity
z
yx
flow
position
z
y
x
flowposition
flow
w
v
u
velocity
0u
2 2 2
2 2 2
/1
/
/
x y z
x y z
x y z
u u u u x
v v v v y pt R x y z
w w w w z
1uu u p u
t R
( , , )u u v w0u v w
x y z
z
yx
flow
w
vu
flow
velocityposition
0x
0u
2 2 2
2 2 2
/1
/
/
x y z
x y z
x y z
u u u u x
v v v v y pt R x y z
w w w w z
1uu u p u
t R
( , , )u u v w0u v w
x y z
z
yx
flow
w
vu
flow
velocityposition
0x
2d NS
0u
2 2 2
2 2 2
/1
/
/
x y z
x y z
x y z
u u u u x
v v v v y pt R x y z
w w w w z
1uu u p u
t R
( , , )u u v w
,
( , , )
v wz y
y x t
2
1
1
u u uu
t z y y z R
t z y y z R
0u v w
x y z
2
1
1
u u uu
t z y y z R
t z y y z R
,
( , , )
v wz y
y x t
2d-3c model
z
yx
flow
position
0x
2 dimensionsw
vu
flow
velocity
3 components
2
1
1
u u uu
t z y y z R
t z y y z R
,
( , , )
v wz y
y x t
2d-3c model
0x
These equations are globally stable!Laminar flow is global attractor.
t
energy
2RR
Total energy3R
(Bamieh and Dahleh)
0 200 400 600 800 100010
-10
10-5
100
105
t
ener
gyenergyN=10R=1000t=1000alpha=2
Total energy
vortices
What you’ll see next.
( , , )z y t
( , , )u z y t
( , , )z y t
( , , )u z y t
Log-log plot of time response.
Random initial conditions on
( , , 0)z y t concentrated at lower boundary.
( , , )z y t
( , , )u z y t
( , , )z y t
( , , )u z y t
Exponential decay.
Long range correlation.
Streamwise streaks.
HOT turbulence?Robust, yet
fragile?
• Through streamlined design• High throughput• Robust to bifurcation transition (Reynolds number)• Yet fragile to small perturbations• Which are irrelevant for more “generic” flows
HOT
flow
pressure drop
random pipes
streamlined pipes
Complexity, chaos and criticality
• The orthodox view:– Power laws suggest criticality
– Turbulence is chaos
• HOT view:– Robust design often leads to power laws
– Just one symptom of “robust, yet fragile”
– Shear flow turbulence is noise amplification
• Other orthodoxies:– Dissipation, time irreversibility, ergodicity and mixing
– Quantum to classical transitions
– Quantum measurement and decoherence
Epilogue
• HOT may make little difference for explaining much of traditional physics lab experiments,
• So if you’re happy with orthodox treatments of power laws, turbulence, dissipation, quantum measurement, etc then you can ignore HOT.
• Otherwise, the differences between the orthodox and HOT views are large and profound, particularly for…
• Forward or reverse (eg biology) engineering complex, highly designed or evolved systems,
• But perhaps also, surprisingly, for some foundational problems in physics