Statistics of Extreme Fluctuations in Task Completion Landscapes

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Statistics of Extreme Fluctuations in Task
Completion Landscapes

• Hasan Guclu (LANL)
• with
• G. Korniss (Rensselaer)

Isaac Newton Institute, Cambridge, UK June
26-30, 2006
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Motivation and introduction

• Synchronization is a fundamental problem in
  coupled multi-component systems.
• Small-World networks help autonomous
  synchronization. But what about extreme
  fluctuations? Extreme fluctuations are to be
  avoided for scalability and stability.
• We discuss to what extent SW couplings lead to
  suppression of the extreme fluctuations.
• One typical example of task-completion systems is
  Parallel Discrete-Event Simulation (PDES).
• Stochastic time increments in task completion
  system correspond to noise in the associated
  surface growth problem. We used both exponential
  (short-tailed) and power-law noise
  (heavy-tailed).

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Distribution of maxima for i.i.d. random variables
Fisher-Tippett (Gumbel)
Fréchet Distribution
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Generalized extreme-value distribution (GEVDM)
Castillo, Galambos (1988,1989)
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Models
Original (1D Ring)
Small-world network
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Dynamics in the network and observables
Coarse-grained equation of motion
Original (KPZ/EW)
SW Network
Hastings, PRL 91, 098701 (2003) Kozma, Hastings,
Korniss, PRL 92, 108701 (2003)
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1D ring distribution of maxima
Raychaudhuri, PRL, 01
Majumdar and Comtet (2004)
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Exponential noise individual height distributions
Fisher-Tippett Type I (Gumbel)
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Exponential noise maximum height distributions
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Power-law noise in SW network (p0.1 )
Fréchet Distribution
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Power-law noise in SW network
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Extreme fluctuations in scale-free network (exp
noise)
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Extreme fluctuations in scale-free network
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Extreme fluctuations in scale-free network
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Summary

• Small-World links introduces a finite effective
  correlation length, so the system can be divided
  into small quasi-independent blocks.
• When the interaction topology in a network is
  changed from regular lattice into small-world or
  scale-free, the extreme fluctuations diverge
  weakly (logarithmically) with the system size
  when the noise in the system is short-tailed and
  diverge in the power-law fashion when the noise
  is heavy-tailed noise.
• The extreme statistics is governed by
  Fisher-Tippet Type I (Gumbel) distribution when
  noise in the system is exponential or Gaussian
  and Fréchet distribution in the case of power-law
  noise.
• Refs H. Guclu, G. Korniss, PRE 69, 065104
  (2004) H. Guclu and G. Korniss, FNL 5, L43
  (2005).

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An incomplete collaboration network of the
workshop