Observations on versions of the Discrete Bak-Sneppen Model - Winqvist,Frank (Bachelor's Thesis in Mathematical Sciences, 2025)
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.
Keywords: Discrete Bak-Sneppen model; self-organized criticality