Complex Systems Complex Networks

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Complex SystemsComplex Networks

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Complex Systems

• What is the definition of complex systems?
• Is there any difference between the complex
  systems and complicated systems?

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Ants
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Ants

• Task allocation? a process of continual
  adjustment
• The number of workers engaged in a specific task?
  appropriate to the current condition
• The queen does not decide which worker does what!
• Small piles of the mixed seed are placed outside
  the nest mound
• Away from the foraging trails
• In front of scouting patrollers
• ? active recruitment of foragers takes place
• Toothpicks are placed near the nest entrance
• ? the number of nest maintenance workers
  increases
• The task allocation achieved without any central
  control.
• The individual ant only perceives local
  information from the ants nearby through chemical
  and tactile communication.
• This cooperative behavior of an ant colony
  results from local interaction between its
  members not from central controller ?emergent
  behavior

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Emergent behavior

• Defined as a large scale effects of locally
  interacting agents that are often surprising and
  hard to predict even in the case of simple
  interactions.
• Cannot predict the emergent behavior just by
  analyzing the interaction between each element.
• A system such as an ant colony, which consists of
  large populations of connected agent (that is,
  collections of interacting elements), is said to
  be complex if there exists an emergent global
  dynamics resulting from the actions of its parts
  rather than being imposing a central controller.

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Complex Systems

• From Sync by Steven Strogatz "Every decade or
  so, a grandiose theory comes along, bearing
  similar aspirations and often brandishing an
  ominous-sounding C-name. In the 1960 it was
  cybernetics. In the '70s it was catastrophe
  theory. Then came chaos theory in the '80s and
  complexity theory in the '90s."
• Various informal descriptions of complex systems
  have been put forward, and these may give some
  insight into their properties. A special edition
  of Science about complex systems Science Vol.
  284. No. 5411 (1999). highlighted several of
  these
• A complex system is a highly structured system,
  which shows structure with variations (Goldenfeld
  and Kadanoff)
• A complex system is one whose evolution is very
  sensitive to initial conditions or to small
  perturbations, one in which the number of
  independent interacting components is large, or
  one in which there are multiple pathways by which
  the system can evolve (Whitesides and Ismagilov)
• A complex system is one that by design or
  function or both is difficult to understand and
  verify (Weng, Bhalla and Iyengar)
• A complex system is one in which there are
  multiple interactions between many different
  components (D. Rind)
• Complex systems are systems in process that
  constantly evolve and unfold over time (W. Brian
  Arthur).

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SummaryProperties of Complex Systems

• large number of elements in the system
• Nonlinear interaction between each element
• Emergent behavior Universal behavior or order
• Collective behavior
• The behavior which cannot be predicted from the
  interaction between each element
• Feedback interaction
• Adaptation
• Open System
• Edge of Chaos Self-Organizing system
• More is Different.

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Related Topics
Bio-systems
Fractal
Computer Science
Complex systems
Statistical Mechanics

Networks
chaos
Social Systems
Non-linear dynamics
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Examples
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Examples
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Examples
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Example
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Examples

• Synchronization of average velocities in
  neighboring lanes in congested traffic
• On a crowded sidewalk, pedestrians walking in
  opposite directions tends to form lanes along
  which walkers move in the same direction
• Social networks citation networks, WWW
• Stock market

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How to study the complex systems

• Assume a simplified system
• Identical elements
• Representative agent model in microeconomics
• Each agent has the same properties to maximize
  its efficiency
• Probabilistic interactions or movements
• Brownian motion by random walk

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Complexity

• Complexitythe amount of information to
  characterize a system
• Complexity measure
• A measure to quantify the complexity
• Ex. Computational complexity, information
  complexity
• Kolmogorov measurecomplexity measure of a string
  Ziv-Lepel algorithm
• The length of the program to generate the string
• Grassberger measure forecast complexity
• Excess entropy

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Complexity

• The complelxity can be changed by scale.
• Small scale? increase the complexity

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Emergent Macroscopic order from Complex Dynamics

• Power Law Scale invariant universality
• Earthquake Gutenberg-Richters law
• Energy of the earthquake1/frequency

The power-law implies that there is some
universal law which governs the earthquake,
regardless of the energy scale.
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Paretos Principle

• The universal law in social and economic system.
• 80-20 rule
• 80 of income in Italy is received by 20 of
  Italian population.
• 80 of consequences stem from 20 of the causes.
• The same rule has been observed over many
  different countries which have different
  histories and social environments.

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Economics

• Mandelbrot
• Study the price changes of cotton
• Power-law
• The price change cannot be described by random
  walk or normal distribution
• The similar behavior is found in many stock
  exchanges, foreign exchanges and on-line markets

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Zipfs Law

• Originated from Linguistics
• The frequency of any word is inversely
  proportional to its ranks in the frequency table.
• The same law can be found in the residence
  distribution of the democratic countries. (Not in
  the socialist states)
• This demonstrate that the free movement can
  create some universal law

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Other Examples
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And others

• Distribution of the file size transmitted through
  the Internet
• Size of the instructions in computer
• Size of the sand on the beach
• Number of species in each genus
• Degree distribution of the Internet, www, etc.