Predrag R. Jelenkovic´ - Electrical Engineeringdpwe/linked/eerschday-2008-09/jelenkovic.pdf ·...

Complex Networks

Predrag R. Jelenkovic

Dept. of Electrical EngineeringColumbia University , NY 10027, USA

[email protected]

September 2008

Jelenkovic (Columbia University) Large Scale Systems September 2008 1 / 24

Outline

1 Complex NetworksExamples of Complex NetworksCharacteristics of Complex NetworksTimingDesign Principles

2 Power Laws are Ubiquitous in Complex SystemsPower Law (Heavy-Tailed) Distributions100+ Years of Repeated Observations of Power LawsFinding Universal Laws

Outline

Examples of Complex Networks

Common examplesBio-molecular networks: gene regulatory networksThe brain circuitry: the most complex computer of allMassively parallel computers: single computer will havemillion/billion+ CPUsComputer networks: wireless networks (Ad Hoc, Sensor), theInternet, World Wide WebLarge integrated circuits of (nano-devices)Socio-economic systems: the economy and stock marketsSocial networks

Outline

Characteristics of Complex Networks

Typical characteristicsHave millions, billions or more interacting componentsWiring of these networks usually evolves randomly(evolution/mutaton, WWW growth)Interactions between components are randomSpatial and temporal parameters are highly variable (number ofmolecules, user behavior)New statistical laws are needed for these large complex systems

Outline

Timing

Why emphasizing these systems now?The engineered systems are approaching the same level ofcomplexity as the natural ones (e.g., biological)The remaining problems in natural systems often involve systemlevel issues (e.g., systems biology)The accumulated scientific knowledge and technology(computational power) seems ready to tackle these problems

Timing

Outline

Design Principles

Modularity and Reliability/RobustnessBio-molecular networks are highly modular and hierarchical, similarly asthe men-made systemsIndividual components are unreliable but the overall performance ishighly reliable/robustBio-networks are highly adaptable, self-correctingWe should be able to learn some design principles from nature: Possiblynew architectural designs are needed that achieve reliability out ofextremely unreliable components (e.g., nano-devices)

Scalability

The systems of today, e.g. wireless networks, should operate in thefuture on much larger scaleSmall inefficiencies may accumulate and deteriorate the performance ofthe system as a wholeOptimal designs for small systems may not be optimal for the large onessince their complexities may become prohibitively large

Design Principles

Scalability

Design Principles

Scalability

Design Principles

Scalability

Design Principles

Scalability

Design Principles

Scalability

Design Principles

Scalability

Design Principles

Multiple Spatial and Temporal ScalesDifferent dynamics on different scalesNew analytical/mathematical tools may be needed

Statistical InvariantsPower law (heavy-tailed) distributions are universally observed

Design Principles

Outline

Power Law Distributions

A random variable X has a power law tail if

for α > 0

P[X > x ] ∼ Hxα

log P[X > x ] ≈ −αlog x

Figure: Size vs rank of the 135 largestU.S. cities in 1991, Gabaix (1999).

Power law distribution is a straight line on the log-log plot

Power Law Distributions

A random variable X has a power law tail if

for α > 0

P[X > x ] ∼ Hxα

log P[X > x ] ≈ −αlog x

Figure: Size vs rank of the 135 largestU.S. cities in 1991, Gabaix (1999).

Power law distribution is a straight line on the log-log plot

Sample path characteristic of power laws

Large values occur frequently

Sample path ofpower lawversusexponentialdistribution.

Figure: Distribution of video scenes,Jelenkovic et al. (1997).

Outline

100+ Years of Repeated Observations of Power Laws

Distribution of incomeStarting with Pareto (1897)

World income distribution

Figure: Zipf’s plot of the 30th-85th percentiles of the world income distribution(GDP per capita) in 1960 and 1997.

Neuron spiking time-series

Figure: Power law inter-spike distribution in depressed rats, Bershadskii(2001).

File sizes

Figure: Distribution of computer files in COMET lab at Columbia University(α = 1.44), Jelenkovic and Momcilovic (2001)

Visitors and pages per Web site

Figure: Distributions of the number of pages and visitors per Web site.

Scale free networks

Power law random graphsPower law connectivity of complex networks came as a surprise toresearchers accustomed to the traditional random networks.

Traditional random graph - Erdos-Renyi modelVS scale free graph - Barabási model

Figure: Concentrated degreedistribution (Poisson)

Figure: Power law degreedistribution

Graphs of Internet topology and interacting proteins in yeast

Figure: Image credit: Internet Mapping Project of Lumeta Corporation;Scientific American

Science citation index

Outline

Finding Universal Statistical Laws

Are there universal statistical laws governing these systems?Can we sue these laws to better understand or modify thebio-molecular circuits?Can we designing new control mechanisms based on thesecharacteristics for men-made systems, e.g.: Using power lawtopology to design better communication protocols andinformation storage/content distribution mechanisms, etc.

Thank youQuestions?

Predrag R. Jelenkovic´ - Electrical Engineeringdpwe/linked/eerschday-2008-09/jelenkovic.pdf ·...

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