Generalizations of the Discrete Bak-Sneppen Model
(2020) In Bachelor's Theses in Mathematical Sciences MASK11 20201Mathematical 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:
http://lup.lub.lu.se/student-papers/record/9012003
- author
- Jönsson, John LU
- supervisor
- organization
- course
- MASK11 20201
- year
- 2020
- 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}}, language = {{eng}}, note = {{Student Paper}}, series = {{Bachelor's Theses in Mathematical Sciences}}, title = {{Generalizations of the Discrete Bak-Sneppen Model}}, year = {{2020}}, }