1. Statistics of Extreme Fluctuations in Task Completion Landscapes
Hasan Guclu (LANL)
with
G. Korniss (Rensselaer)
We worked on a very simple model of task completion system with many components. By changing the topology of the system we can go from strong correlation to weak correlation and look at behavior of the extreme-value properties. This models also serves as a test system to study synchronization in coupled multi-component systems.
This is joint work with Gyorgy Korniss from Rensselaer Polytechnic Institute.
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)
This is my most crowded slide, I promise.
Fréchet Distribution

4. Generalized extreme-value distribution (GEVDM)
Domain of Attraction
Distribution
Maximal
Minimal
Exponential
Gumbel
Weibull
Normal
Uniform
Lognormal
Gamma
GumbelM
Gumbelm
WeibullM
Weibullm
FréchetM
Fréchet
Fréchetm
The extreme-value limit distributions are also called Fisher-Tippet Type I (Gumbel), Type II (Frechet), and Type III (Weibull). As Zoltan Racz said Gumbel popularized the Type I distribution and so most of the times it is called Gumbel distribution. Gumbel’s book in extreme statistics was also the only book for about twenty years until Galambos’ book.
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
We have a simple incrementing minima dynamics. At every time step the minima in the system are incremented by a random amount drawn from a distribution.
Mention Dasgupta’s talk.
Hastings, PRL 91, (2003); Kozma, Hastings, Korniss, PRL 92, (2003)

7. 1D ring: distribution of maxima
You remember this Airy distribution function from Alain Comtet’s talk, maximal height distribution of fluctuating interfaces.
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)
Scale-free network was proposed as a model for many real-life networks in which we have power-law degree distributions without a specific scale. There are a couple ways to generate it artificially. One method also known as preferential attachment or rich get richer mechanism was invented by Barabasi and Albert. In this method the network is grown one node at a time and each node has m stubs. The newly added nodes connect themselves to m other nodes in the network with probability proportional to the degree of the already existing nodes. Thus higher-degree nodes get more and more links and become hubs, very high-degree nodes.
The scale-free networks are “ultra-small” networks because the average shortest path changes with the system size very very slowly meaning double logarithmically when the power-law exponent is between two and three. For most real-life networks such as Internet the exponent is between two and three. For this artificially generated network the exponent is three although because of the finite-size effects the exponent is less than three.

13. Extreme fluctuations in scale-free network
Scale-free tree (m=1) s marginally stable. Although the width goes to infinity as the system size goes to infinity but very slowly.

14. Extreme fluctuations in scale-free network
Scale-free network is not so bad in terms of agreement with the extreme-value distribution. The problem is mostly for the small values of the argument.

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, PRE 69, (2004); H. Guclu and G. Korniss, FNL 5, L43 (2005).

16. An incomplete collaboration network of the workshop