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Observations on versions of the Discrete Bak-Sneppen Model

Winqvist, Frank LU (2025) In Bachelor's Thesis in Mathematical Sciences MASK11 20251
Mathematical Statistics
Abstract
The discrete Bak-Sneppen model is a Markov chain describing the co-evolution of n species arranged in a one-dimensional circular lattice with periodic boundary conditions. Each species i at time t is assigned a fitness value xi(t) ∈ {0,1} and the system evolves through local interactions: at each time step, a species with fitness 0 is selected, and its fitness—along with that of its two neighbors—is updated using independent Bernoulli random variables with parameter p, which we will denote by Be(p). It has been proven by Volkov in [7] that there exists some p⋄, such that, whenever p > p⋄, the fraction of ones in the system (denoted νn(p)) converges to 1 as n → ∞.
In this work, we provide the necessary background to understand how this... (More)
The discrete Bak-Sneppen model is a Markov chain describing the co-evolution of n species arranged in a one-dimensional circular lattice with periodic boundary conditions. Each species i at time t is assigned a fitness value xi(t) ∈ {0,1} and the system evolves through local interactions: at each time step, a species with fitness 0 is selected, and its fitness—along with that of its two neighbors—is updated using independent Bernoulli random variables with parameter p, which we will denote by Be(p). It has been proven by Volkov in [7] that there exists some p⋄, such that, whenever p > p⋄, the fraction of ones in the system (denoted νn(p)) converges to 1 as n → ∞.
In this work, we provide the necessary background to understand how this model arose. We also explore generalizations of the update rules, including cases where the neighbors are replaced using the same Be(p) variable, as well as cases where the left and right neighbors are replaced independently by Be(pL) and Be(pR), respectively, where pL and pR are probabilities in [0,1].
Our results combine analytical proofs with numerical simulations, offering insight into the critical behavior and phase transitions of the model. (Less)
Popular Abstract
How can we model the evolution of species? While evolution is often thought of as a slow, incremental process, Per Bak and Kim Sneppen (1993) proposed a model that captures the idea of sudden, dramatic shifts. This phenomenon can be seen in events like the Cambrian explosion, when life on Earth rapidly diversified in a relatively short time. However, theoriginal Bak-Sneppen modelisdifficult to analyze rigorously. To make progress,researchers introduced a discrete version of the model that simplifies the dynamics while preserving its core ideas. In this thesis, we investigate two special modifications of the discrete Bak-Sneppen model, with the goal of gaining deeper insights into its structure and behavior.
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author
Winqvist, Frank LU
supervisor
organization
course
MASK11 20251
year
type
M2 - Bachelor Degree
subject
keywords
Discrete Bak-Sneppen model, self-organized criticality
publication/series
Bachelor's Thesis in Mathematical Sciences
report number
LUNFMS-4079-2025
ISSN
1654-6229
other publication id
2025:K6
language
English
id
9191249
date added to LUP
2025-06-11 15:37:25
date last changed
2025-06-11 15:37:25
@misc{9191249,
  abstract     = {{The discrete Bak-Sneppen model is a Markov chain describing the co-evolution of n species arranged in a one-dimensional circular lattice with periodic boundary conditions. Each species i at time t is assigned a fitness value xi(t) ∈ {0,1} and the system evolves through local interactions: at each time step, a species with fitness 0 is selected, and its fitness—along with that of its two neighbors—is updated using independent Bernoulli random variables with parameter p, which we will denote by Be(p). It has been proven by Volkov in [7] that there exists some p⋄, such that, whenever p > p⋄, the fraction of ones in the system (denoted νn(p)) converges to 1 as n → ∞.
In this work, we provide the necessary background to understand how this model arose. We also explore generalizations of the update rules, including cases where the neighbors are replaced using the same Be(p) variable, as well as cases where the left and right neighbors are replaced independently by Be(pL) and Be(pR), respectively, where pL and pR are probabilities in [0,1].
Our results combine analytical proofs with numerical simulations, offering insight into the critical behavior and phase transitions of the model.}},
  author       = {{Winqvist, Frank}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Thesis in Mathematical Sciences}},
  title        = {{Observations on versions of the Discrete Bak-Sneppen Model}},
  year         = {{2025}},
}