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Statistics of Extreme Fluctuations in Task Completion Landscapes

Statistics of Extreme Fluctuations in Task Completion Landscapes. Hasan Guclu (LANL) with G. Korniss (Rensselaer). Isaac Newton Institute, Cambridge, UK; June 26-30, 2006. Motivation and introduction. Synchronization is a fundamental problem in coupled multi-component systems.

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Statistics of Extreme Fluctuations in Task Completion Landscapes

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  1. Statistics of Extreme Fluctuations in Task Completion Landscapes Hasan Guclu (LANL) with G. Korniss (Rensselaer) Isaac Newton Institute, Cambridge, UK; June 26-30, 2006

  2. 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).

  3. Distribution of maxima for i.i.d. random variables Fisher-Tippett (Gumbel) Fréchet Distribution

  4. Generalized extreme-value distribution (GEVDM) Castillo, Galambos (1988,1989)

  5. Models Original (1D Ring) Small-world network

  6. 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)

  7. 1D ring: distribution of maxima Raychaudhuri, PRL, ’01 Majumdar and Comtet (2004)

  8. Exponential noise: individual height distributions Fisher-Tippett Type I (Gumbel)

  9. Exponential noise: maximum height distributions

  10. Power-law noise in SW network (p=0.1 ) Fréchet Distribution

  11. Power-law noise in SW network

  12. Extreme fluctuations in scale-free network (exp noise)

  13. Extreme fluctuations in scale-free network

  14. Extreme fluctuations in scale-free network

  15. 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, PRE69, 065104 (2004); H. Guclu and G. Korniss, FNL5, L43 (2005).

  16. An incomplete collaboration network of the workshop

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