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. 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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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

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).


Distribution of maxima for i i d random variables

Distribution of maxima for i.i.d. random variables

Fisher-Tippett (Gumbel)

Fréchet Distribution


Generalized extreme value distribution gevd m

Generalized extreme-value distribution (GEVDM)

Castillo, Galambos (1988,1989)


Models

Models

Original (1D Ring)

Small-world network


Dynamics in the network and observables

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)


1d ring distribution of maxima

1D ring: distribution of maxima

Raychaudhuri, PRL, ’01

Majumdar and Comtet (2004)


Statistics of extreme fluctuations in task completion landscapes

Exponential noise: individual height distributions

Fisher-Tippett Type I (Gumbel)


Statistics of extreme fluctuations in task completion landscapes

Exponential noise: maximum height distributions


Power law noise in sw network p 0 1

Power-law noise in SW network (p=0.1 )

Fréchet Distribution


Statistics of extreme fluctuations in task completion landscapes

Power-law noise in SW network


Extreme fluctuations in scale free network exp noise

Extreme fluctuations in scale-free network (exp noise)


Extreme fluctuations in scale free network

Extreme fluctuations in scale-free network


Extreme fluctuations in scale free network1

Extreme fluctuations in scale-free network


Summary

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).


An incomplete collaboration network of the workshop

An incomplete collaboration network of the workshop


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