On Rank Driven Dynamical Systems (original) (raw)

Criticality in a dynamics ruled evolutionary model

Communications in Nonlinear Science and Numerical Simulation, 2018

We propose a model of dynamics-ruled evolution inspired by Bak-Sneppen model. We argue that the only way for the ecological system to find the least or most fit species is to infer it from dynamics. Thus, instead of punishing the species which is 'least fit', we punish one which ' appears to be least fit'. We find that the model still evolves to a critical state. The detailed dynamics does not seem to affect the presence of a critical state.

Self-organized criticality in evolutionary systems with local interaction

2001

This paper studies a stylized model of local interaction where agents choose from an ever increasing set of vertically ranked actions, e.g. technologies. The driving forces of the model are infrequent upward shifts (``updates''), followed by a rapid process of local imitation (``diffusion''). Our main focus is on the regularities displayed by the long-run distribution of diffusion waves and their implication on the performance of the system. By integrating analytical techniques and numerical simulations, we come to the following two main conclusions. (1) If dis-coordination costs are sufficiently high, the system behaves critically, in the sense customarily used in physics. (2) The performance of the system is optimal at the frontier of the critical region. Heuristically, this may be interpreted as an indication that (performance-sensitive) evolutionary forces induce the system to be placed ``at the edge of order and chaos''

Self-organized criticality in a model of biological evolution with long-range interactions

Physica A: Statistical Mechanics and its Applications, 2000

In this work we study the effects of introducing long range interactions in the Bak-Sneppen (BS) model of biological evolution. We analyze a recently proposed version of the BS model where the interactions decay as r −α ; in this way the first nearest neighbors model is recovered in the limit α → ∞, and the random neighbors version for α = 0. We study the space and time correlations and analize how the critical behavior of the system changes with the parameter α. We also study the sensibility to initial conditions of the model using the spreading of damage technique. We found that the system displays several distinct critical regimes as α is varied from α = 0 to α → ∞ In recent years an increasing numbers of systems that present Self Organized Criticality [1,2] have been widely investigated. The general approach of statistical physics, where simple models try to catch the essencial ingredientes responsable for a given complex behavior has turned out to be very powerful for the study of this kind of problems. In particular Bak and Sneppen have introduced a simple model which has shown to be able to reproduce evolutionary features such as punctuacted equilibrium . Altough this model does not intend to give an accurate description of darwinian evolution, it catches into a single and very simple scheme (it is based on very simple dynamical rules) several features that are expected to be present in evolutionary processes, that is, punctuated equilibrium [3], Self Organized Criticality (SOC) [3] and weak sensitivity to initial conditions (WSIC) [5,6] (i.e., chaotic behaviour where the trajectories depart with a power law of the time instead of exponentially). In this sense, one important question arises about the robustness of such properties against modifications (i.e., complexifying) of the simple dynamical rules on which the model is based. The original model, hereafter referred as the first nearest neighbors (FNN) version [3], includes only nearest neighbors interactions in a one dimensional chain. This model presents SOC and weak sensitivity to initial conditions . On the other hand, another version of the model with interactions between sites randomly chosen in the lattice (and therefore it can be regarded as a mean field version of the FNN), hereafter referred as the random negihbors (RN) version [7], does not present SOC . Moreover, it is not expected (and we shall show in this work that it is indeed the case) to present WSIC.

A stochastic model of evolution

2009

We propose a stochastic model for evolution. Births and deaths of species occur with constant probabilities. Each new species is associated with a fitness sampled from the uniform distribution on [0, 1]. Every time there is a death event then the type that is killed is the one with the smallest fitness. We show that there is a sharp phase transition when the birth probability is larger than the death probability. The set of species with fitness higher than a certain critical value approach an uniform distribution. On the other hand all the species with fitness less than the critical disappear after a finite (random) time. 1. Introduction. Consider a discrete time model that starts from the empty set. At each time n ≥ 1 with probability p there is a birth of a new species and with probability q = 1 − p there is a death of a species (if the system is not empty). Hence, the total number of species at time n is a random walk on the positive integers which jumps to the right with probability p and to the left with probability q. When the random walk is at 0 then it jumps to 1 with probability p or stays at 0 with probability 1 − p. Each new species is associated with a random number. This random number is sampled from the uniform distribution on [0, 1]. We think of the random number associated with a given species as being the fitness of the species. These random numbers are independent of each other and of everything else. Every time there is a death event then the type that is killed is the one with the smallest fitness. This is similar to a model introduced by Liggett and Schinazi (2009) for a different question. Take p in (1/2, 1) and let f c = 1 − p p. Note that f c is in (0, 1). Let L n and R n be the set of species alive at time n whose fitness is lower and higher than f c , respectively.

Natural Models for Evolution on Networks

Computing Research Repository, 2011

Evolutionary dynamics have been traditionally studied in the context of homogeneous populations, mainly described my the Moran process. Recently, this approach has been generalized in \cite{LHN} by arranging individuals on the nodes of a network. Undirected networks seem to have a smoother behavior than directed ones, and thus it is more challenging to find suppressors/amplifiers of selection. In this paper

Self-Organized Network Evolution Coupled to Extremal Dynamics

Topologica, 2008

The interplay between topology and dynamics in complex networks is a fundamental but widely unexplored problem. Here, we study this phenomenon on a prototype model in which the network is shaped by a dynamical variable. We couple the dynamics of the Bak-Sneppen evolution model with the rules of the so-called fitness network model for establishing the topology of a network; each vertex is assigned a 'fitness', and the vertex with minimum fitness and its neighbours are updated in each iteration. At the same time, the links between the updated vertices and all other vertices are drawn anew with a fitness-dependent connection probability. We show analytically and numerically that the system self-organizes to a non-trivial state that differs from what is obtained when the two processes are decoupled. A power-law decay of dynamical and topological quantities above a threshold emerges spontaneously, as well as a feedback between different dynamical regimes and the underlying correlation and percolation properties of the network.

Self-organized evolution in a socioeconomic environment

Physical Review E, 2000

We propose a general scenario to analyze technological changes in socio-economic environments. We illustrate the ideas with a model that incorporating the main trends is simple enough to extract analytical results and, at the same time, sufficiently complex to display a rich dynamic behavior. Our study shows that there exists a macroscopic observable that is maximized in a regime where the system is critical, in the sense that the distribution of events follow power laws. Computer simulations show that, in addition, the ...

Evolutionary Dynamics on Degree-Heterogeneous Graphs

Physical Review Letters, 2006

The evolution of two species with different fitness is investigated on degree-heterogeneous graphs. The population evolves either by one individual dying and being replaced by the offspring of a random neighbor (voter model dynamics) or by an individual giving birth to an offspring that takes over a random neighbor node (invasion process dynamics). The fixation probability for one species to take over a population of N individuals depends crucially on the dynamics and on the local environment. Starting with a single fitter mutant at a node of degree k, the fixation probability is proportional to k for voter model dynamics and to 1=k for invasion process dynamics.

A new formal approach to evolutionary processes in socioeconomic systems

Journal of Evolutionary Economics, 2013

Generalized Darwinian evolutionary theory has emerged as central to the description of economic process (e.g., Aldrich et al., J Evol Econ 18:577-596, 2008). Just as Darwinian principles provide necessary, but not sufficient, conditions for understanding the dynamics of social entities, so too the asymptotic limit theorems of information theory instantiate another set of necessary conditions that constrain socioeconomic evolution. These restrictions can be formulated as a statistics-like analytic toolbox for the study of empirical data that is consistent with generalized Darwinism, but escapes the intellectual straightjacket of replicator dynamics. The formalism is a coevolutionary theory in which punctuated convergence to temporary quasiequilibira is inherently nonequilibrium, involving highly dynamic 'languages' rather than system stable points.

Asymptotic behaviour of a discrete dynamical system generated by a simple evolutionary process

International Journal of Applied Mathematics and Computer Science, 2004

A simple model of phenotypic evolution is introduced and analysed in a space of population states. The expected values of the population states generate a discrete dynamical system. The asymptotic behaviour of the system is studied with the use of classical tools of dynamical systems. The number, location and stability of fixed points of the system depend on parameters of a fitness function and the parameters of the evolutionary process itself. The influence of evolutionary process parameters on the stability of the fixed points is discussed. For large values of the standard deviation of mutation, fixed points become unstable and periodical orbits arise. An analysis of the periodical orbits is presented.