Advanced

Generalizations of the Discrete Bak-Sneppen Model

Jönsson, John LU (2020) In Bachelor's Theses in Mathematical Sciences MASK11 20201
Mathematical Statistics
Abstract
Consider n vertices arranged in a circle where each vertex is given a fitness from {0,1}. At each discrete time step, one of the vertices with fitness equal to zero (unless there are none of those, then pick a vertex with fitness equal to one) is selected with equal probability. Then this vertex and the two neighbouring vertices are each given a new fitness from a Bernoulli(p) distribution independently of each other, for some p in [0,1]. This model is known as the discrete Bak-Sneppen model. What happens to the fraction of ones (vertices with fitness one) as n and the time t goes to infinity? How does this quantity depend on p? Is there a pc in (0,1) such that this quantity is equal to one for p > pc and less than one for p < pc? In this... (More)
Consider n vertices arranged in a circle where each vertex is given a fitness from {0,1}. At each discrete time step, one of the vertices with fitness equal to zero (unless there are none of those, then pick a vertex with fitness equal to one) is selected with equal probability. Then this vertex and the two neighbouring vertices are each given a new fitness from a Bernoulli(p) distribution independently of each other, for some p in [0,1]. This model is known as the discrete Bak-Sneppen model. What happens to the fraction of ones (vertices with fitness one) as n and the time t goes to infinity? How does this quantity depend on p? Is there a pc in (0,1) such that this quantity is equal to one for p > pc and less than one for p < pc? In this paper we prove upper bounds for pc for generalized versions of this model. We alsoprovide a number of experimental results, as well as a quick summary of what hasbeen done in the past. (Less)
Popular Abstract
The Bak-Sneppen model is a simple model of co-evolution. It can be viewed as a simplified version of how different species interact with each other in nature. It takes into account the randomness of evolution as well as the idea of the survival of the fittest. Even though this model is very simple compared to the real world, it is not yet fully understood. Maybe in order to understand it better we should first try to understand an even simpler model, namely the discrete Bak-Sneppen model. The main topic of this paper will be to generalize the discrete Bak-Sneppen modeland to prove relevant properties to it. We will also provide experimental results of these properties as well as for the properties that are left unproven.
Please use this url to cite or link to this publication:
author
Jönsson, John LU
supervisor
organization
course
MASK11 20201
year
type
M2 - Bachelor Degree
subject
keywords
discrete, Bak-Sneppen, model, generalized, probability, Markov
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFMS-4045-2020
ISSN
1654-6229
other publication id
2020:K10
language
English
id
9012003
date added to LUP
2020-06-12 11:43:13
date last changed
2020-06-15 15:37:58
@misc{9012003,
  abstract     = {Consider n vertices arranged in a circle where each vertex is given a fitness from {0,1}. At each discrete time step, one of the vertices with fitness equal to zero (unless there are none of those, then pick a vertex with fitness equal to one) is selected with equal probability. Then this vertex and the two neighbouring vertices are each given a new fitness from a Bernoulli(p) distribution independently of each other, for some p in [0,1]. This model is known as the discrete Bak-Sneppen model. What happens to the fraction of ones (vertices with fitness one) as n and the time t goes to infinity? How does this quantity depend on p? Is there a pc in (0,1) such that this quantity is equal to one for p > pc and less than one for p < pc? In this paper we prove upper bounds for pc for generalized versions of this model. We alsoprovide a number of experimental results, as well as a quick summary of what hasbeen done in the past.},
  author       = {Jönsson, John},
  issn         = {1654-6229},
  keyword      = {discrete,Bak-Sneppen,model,generalized,probability,Markov},
  language     = {eng},
  note         = {Student Paper},
  series       = {Bachelor's Theses in Mathematical Sciences},
  title        = {Generalizations of the Discrete Bak-Sneppen Model},
  year         = {2020},
}