COMPLEXITY IN COMPUTER NETWORKS

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COMPLEXITY IN COMPUTER NETWORKS
Sergi Valverde
Joint Work with Ricard V. Solé
Complex Systems Research Group (UPC)
e-mail sergi_at_complex.upc.es
www http//complex.upc.es/sergi
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Evidences of Complex Behavior
Self-Similarity, Scaling Laws, 1/f Noise
I. Csabai (1994)
Poisson Assumption is No Longer Valid
H. Takayasu et al (1997)
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Critical Point Phenomena
Complexity in Computer Networks
Ferromagnetism
Magnetization Temperature
Local Interactions
Order
Thermal Noise
Disorder
Phase Transition
T Tc
An iron atom is a very complicated thing (lots of
e-)
A piece of iron involves a huge number of atoms
etc...
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Critical Point Phenomena
Complexity in Computer Networks
Ising Model The importance of Interactions
Simplicity
Consider the simplest model that reproduces the
essential features
Universality
The model is a general framework for explaining
other physical phenomena (f.ex fluids, social
behavior, galaxies)
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Ising Model
Complexity in Computer Networks
Complexity and Criticality
T lt Tc
T Tc
T gt Tc
Fractals, 1/f Noise, Scaling
Why Macroscopic Complexity? Because the
Microscopic Local Interactions
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Ising Model
Complexity in Computer Networks
At the Critical Point...
Magnetization Shows Complex Fluctuations
Island Sizes Distribution is a Scaling Law
Model Matches Real System at Criticality
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Road Traffic
Complexity in Computer Networks
Nagel-Schreckenberg Model (1992)
Acceleration if (v lt gap) v (vmax, v1)
Avoid Collisions if (v gt gap) v gap
Randomization if (random() lt 0.5) v max(v-1,
0)
Movement x x v
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Road Traffic
Complexity in Computer Networks
Maximum Efficiency at the Critical Point
Free Phase
Congested Phase
Maximum flow at critical point
Traffic Management wants to keep a freeway in
the regime of maximum flow
Flow us Vehicle Density
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Road Traffic
Complexity in Computer Networks
Complexity Criticality (Again)
System at Critical Density
Traffic Jams Emerge as Fractal, Branching Waves
They might originate simply from one car slowing
down
These Structures Cannot Be Understood in Terms of
Properties of the Vehicles
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Road Traffic
Complexity in Computer Networks
Efficiency and Unpredictability Connected by
Phase Transition
Management Measures May Even Have Consecuences
Opposite to Their Intention !
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Phase transitions in Computer Networks
Ohira-Sawatari Deterministic Model (1998)
Parameters L, l
Two Kinds of Nodes
Routers
Hosts (Random Sources)
Unbounded Queues
Routing
L 5
Take Shortest Paths
Avoid Overloaded Links
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Routing Example
Complexity in Computer Networks
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Ohira-Sawatari Model
Complexity in Computer Networks
Phase Transition
Increasing Size
Deterministic (A) us Non-deterministic (B)
Mean Latency us Packet Injection Rate
Non-Linear Response, Sharp Transition
Non-Deterministic Model (B) Shifting of the
Transition!
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Information Transfer Phase Transitions
Model 1 (Solé-Valverde, 1999)
New Parameter r (Density of Hosts)
Set Periodic Boundary Conditions
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Model 1
Complexity in Computer Networks
Maximum Efficiency at the Critical Point
MEASURE OF EFFICIENCY
Exhibits a Similar Transition than the Previous
Studies (Ohira-Sawatari)
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Model 1
Complexity in Computer Networks
Mean Field Approximation
Packets Are Removed If There Isnt Free Space for
Movement
N gt L2
N L2
Mean Field Simulation Approach the Same Values
for High Densities of Hosts
N lt L2
Phase Diagram
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Model 1
Complexity in Computer Networks
Self-Affine Queue Activity at Criticality
Power Spectrum of the Signal Yields 1/f Noise
H. Takayasu et al (1997)
Sudden Peaks of Congestion (Storms)
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Model 1
Complexity in Computer Networks
Complexity Criticality
Distribution of Queue Lengths, Scaling Law near
Critical Point
Distribution of Latencies, Lognormal near
Critical Point
Log-normality Proposed in Literature
(Social Dilemas and Internet Congestion,
Huberman, Science 1997)
Scaling means Long-Range Correlations
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Model 1
Complexity in Computer Networks
Efficiency and Unpredictability Connected by
Phase Transition (as in the Road Traffic Model)
Variance Indicates the Critical Point with a
Sharp Maximum
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Self Organized Criticality
Complexity in Computer Networks
Sandpile Model (Per Bak et al, 1987)
A Dynamical Theory for Complexity
Simplicity Grains interact and may cause each
other to topple
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Sandpile Model
Complexity in Computer Networks
The system reaches criticality by itself
Many natural systems show critical behavior
Avalanches are emergent, complex phenomena
But its not the unique way to reach complexity!
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Self Organized Criticality in Computer Networks
Model 2 (Solé-Valverde, 2000)
Hosts Control Their Own Packet Injection Rate
l no longer is an external parameter
Additive Increase, Multiplicative Decrease
( Analysis of the Increase and Decrease
Algorithms for Congestion Avoidance in CN, Raj
Jain, 1989)
Depends on the local congestion around host
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Model 2
Complexity in Computer Networks
The System Reaches Criticality
Individual Behavior Displays Complex Fluctuations
We note that N L2 as predicted by Mean Field
Theory of Model 1
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Model 2
Complexity in Computer Networks
The System Self-Organises towards Criticality
Log-Normality Scaling Law
Distribution of Queue Lengths
Distribution of Latencies
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Future Work
Complexity in Computer Networks
Topology Attacks
How Influences the Traffic Dynamics?
Comparison between Model and Reality
Perform Experiments in Multiprocessors
Gather Observations from Internet
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Complexity in Computer Networks
THANK YOU
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