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Complex Systems Complex 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 forager Nest maintenance worker patroller

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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."Steven Strogatz 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:Science Science Vol. 284. No. 5411 (1999) 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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Summary Properties 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 Complex systems chaos Non-linear dynamics Social Systems Statistical Mechanics Fractal Computer Science Networks Bio-systems

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Examples Fractals in Nature

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Examples Nonlinear Systems Strange attractors Strange attractors

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A Few Good Man Robert Wagner Austin Powers: The spy who shagged me Wild Things Lets make it legal Barry Norton What Price Glory Monsieur Verdoux

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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 earthquake~1/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.

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Swarms MONT 104Q – Mathematical Journeys, November 2015.

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