Uncertainty management in complex systems: Mathematics foundations

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Uncertainty management in complex
systemsMathematics foundations

• John Doyle
• Control and Dynamical Systems

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3
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2
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Frequency of outages gt N
1
10
US Power outages 1984-1997
0
10
4
5
6
7
10
10
10
10
N of customers affected by outage
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Frequency of outages gt N
N of customers affected by outage
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Size of events vs. frequency
log(Prob gt size)
log(size)
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Robust, yet fragile
Robust
Good (small events)
Log(freq.) cumulative
Bad (large events)
yet fragile
Log(event sizes)
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Log(freq.) cumulative
Fat tails
Log(event sizes)
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web traffic
Is streamed out on the net.
Web client
Web servers
Creating internet traffic
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Network protocols.
Files
HTTP
TCP
IP
packets
packets
packets
packets
packets
packets
Routers
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web traffic
Lets look at some web traffic
Is streamed out on the net.
Web client
Web servers
Creating internet traffic
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Data compression (Huffman)
WWW files Mbytes (Crovella)
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Cumulative
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Frequency
Forest fires 1000 km2 (Malamud)
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1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
Size of events
Decimated data
(codewords, files, fires)
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Examples of fat tail distributions

• Power outages, forest fires, web files
• UNIX files, CPU utilization
• Meteor impacts, earthquakes
• Deaths and dollars lost due to man-made disasters
• Deaths and dollars lost due to natural disasters
• Word rank in English (Zipfs law)
• Income and wealth of individuals and companies
• Variations in stock prices and federal budgets
• Masses or sizes of objects in this room
• Ecosystem and specie extinction events?
• Large scale phenomena far from Gaussian or
  exponential

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Consequences of fat-tail web traffic

• Most web file transfers are small, but
• Most packets are in very large files!
• With current protocols (TCP Reno with drop
  tails), during congestion
• Small file packets are queued behind large
• Unnecessary delays
• Exactly the opposite of what you want
• Promising alternatives
• Generalized coding theory

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A toy website model( 1-d grid HOT design)
document
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Data compression
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Forest fires?
Fire suppression mechanisms must stop a 1-d front.
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d-dimensional
li volume enclosed ri barrier density
pi Probability of event
Resource/loss relationship
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PLR optimization
Minimize expected loss
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PLR optimization
? 0 data compression ? 1 web layout ?
2 forest fires
? dimension
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Data
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DC
5
WWW
4
3
FF
2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
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Data Model
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DC
5
WWW
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3
FF
2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
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Forest fires?
Fire suppression mechanisms must stop a 1-d front.
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Forest fires?
Geography could make ? lt2.
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California geographyfurther irresponsible
speculation

• Rugged terrain, mountains, deserts
• Fractal dimension ? ? 1?

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Data Model
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5
California brushfires
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3
FF (national)
2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
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Trends

• Information technology allows us to create
  systems with bewildering complexity.
• Networking which is
• Ubiquitous, pervasive
• Convergent, heterogeneous
• Hierarchical, multiscale
• Biology is shifting from an exclusive focus on
  the molecular basis of life to systems questions.
  
• Modeling, analysis, and simulation-based design
  of complex systems.
• Simulation as basis for policy decisions.

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Trends

• Anything we can imagine, we can build.
• Robustness and reliability become the dominant
  design challenges.
• Cascading failures of highly interconnected
  complex systems (infrastructure).
• Theoretical foundation is fragmented into fairly
  isolated technical disciplines computational
  complexity, information theory, control theory,
  dynamical systems.
• New science of complexity lacks rigor and
  relevance.
• But the need for a new science remains.

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Control Theory
Information Theory
Computational
Theory of Complex systems?
Complexity
Statistical Physics
Dynamical Systems
1 dimension ?
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Control Theory
Information Theory
Congestion Control
Source Coding
Web/Internet Traffic
Dynamics
Power Laws
Statistical Physics
Dynamical Systems
1 dimension ?
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Universal network behavior?
Congestion induced phase transition.
throughput

• Similar for
• Power grid?
• Freeway traffic?
• Gene regulation?
• Ecosystems?
• Finance?

demand
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Networks

• Making a random network
• Remove protocols
• No IP routing
• No TCP congestion control
• Broadcast everything
• ? Many orders of magnitude slower

log(thru-put)
log(demand)
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Networks
HOT
log(thru-put)
log(demand)
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Turbulence
Log(flow)
HOT
log(pressure drop)
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The yield/density curve predicted using random
ensembles is way off.

• Similar for
• Power grid
• Freeway traffic
• Gene regulation
• Ecosystems
• Finance?

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Application domains

• Web/Internet and convergent, ubiquitous
  networking
• Power and transportation systems
• Simulation-based design of complex systems
• Biological regulatory networks and evolution
• Turbulence in shear flows
• Ecosystems and global change
• Financial and economic systems
• Natural and man-made disasters
• Quantum networks and computation

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Extensions and related research

• General theory (Carlson, Chandy, many
  collaborators)
• More realistic models of websites and forests
• Generalized rate distortion theory (Michelle
  Effros)
• Physical fundamentals of information and
  computation (Hideo Mabuchi)
• New TCP/IP protocols (Steven Low)
• New browser/server designs
• Robustness properties of biological networks
• Unified underlying mathematical framework

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Unified underlying mathematical framework

• HOT (Highly optimized tolerance)
• Robust, yet fragile
• Power laws and phase transitions (Stat. Phys.)
• Designing past bifurcations (Dyn. Syst.)
• Disturbance rejection but noise amplification
  (Control)
• HOT systems become high-gain, low-rank noise
  amplifiers
• Simple models in the right coordinates
• New more rigorous approach to multiscale
  phenomena in turbulence, quantum measurement,
  statistical mechanics, biology,

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Network coding/control (Effros, Low)

• Generalized source coding layout plus
  compression
• Generalized rate distortion theory
• Generalized channel coding/control
• Joint source/channel issues IP level channel
  losses of packets is due primarily to congestion
  from the sources
• Control and coding are intertwined
• Preliminary results and promising new directions
• From control of IP to control using IP?!?!
• Similar for biological regulatory networks? (gene
  regulation, signal transduction, neural coding)

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Physical fundamentals of information and
computation (Mabuchi)

• Control, feedback, interconnection, robustness,
  measurement, etc., of quantum systems
• Tools from robust and optimal control,
  non-self-adjoint operator theory Hankel
  operators and model reduction, robustness
  analysis, error bounds, feedback design, implicit
  (behavioral) interconnection, stochastic control,
  dynamic programming,
• Preliminary results and promising new approach
• A rigorous (and practically useful) treatment of
  time irreversibility dissipation, entropy,
  quantum measurement, etc.?

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Control Theory
Information Theory
Computational
Theory of Complex systems?
Complexity
Statistical Physics
Dynamical Systems
1 dimension ?
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Robust, yet fragile
Robust
Good (small events)
Log(freq.) cumulative
Bad (large events)
yet fragile
Log(event sizes)
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Analysis
Log(freq.) cumulative
Log(event sizes)
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Analysis
co-NP
Log(freq.) cumulative
NP
Log(event sizes)
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Control Theory
Information Theory
Theory of Complex systems?
Statistical Physics
Dynamical Systems
1 dimension ?
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Asymmetry between NP vs. co-NP

• Given a propositional formula P(x).
• Have to convince you that it is satisfiable.
• Simple. Just produce a valid assignment. (NP)
• But how do I convince you that it is not? (co-NP)
• Nothing better than try all the solutions

Complementary problems, but very different.
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Traveling Salesman
Nonnegativity
Co-NPC
NPC
P
NP
Co-NP
Primes
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Optimization problems
Upper bounds are in co-NP.
Lower bounds are in NP. Given by feasible points.
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Robustness problems
Bad events are in NP.
Nominal System
Robustness measures are in co-NP.
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• How to compute upper bounds? (co-NP)
• Dual of convex relaxations.
• In particular, semidefinite programming.
• Example standard m upper bound.

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Improving bounds

• Branch Bound.

• NP side
• Local search.
• Monte Carlo.

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More powerful bounds for the co-NP side? (Pablo
Parrilo)
Semialgebraic geometry convex optimization

• Polynomial time computation.
• Never worse than the standard.
• Exhausts co-NP.

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Some examples

• Nonlinear dynamical systems
• Lyapunov function computation
• Robust bifurcation analysis
• MAX CUT
• Structured singular value bounds