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Bak, P. & Snepen, K. Punctuated equilibrium and criticality in a simple model of evolution. Phys. Rev. Lett. 59, 381-384
Abstract
A simple and robust model of biological evolution of an ecology of interacting species is introduced. The model self-organizes into a critical steady state with intermittent coevolutionary avalanches of all sizes; i.e., it exhibits ``punctuated equilibrium'' behavior. This collaborative evolution is much faster than non-cooperative scenarios since no large and coordinated, and hence prohibitively unlikely, mutations are involved.
... By the early 1980s, the RG approach was largely complete and became the common language of physicists when discussing phase transitions and critical phenomena. In subsequent years, the realm of applicability of the RG rapidly grew; it now includes critical dynamics [29,30] and Chapter 5 in [16], strongly non-equilibrium phase transitions, diffusionlimited chemical reactions [1-4,31], driven diffusive systems [32,33], percolation [5], roughening of fluctuating surfaces, growth processes and propagation of fronts [6-10,34-36], turbulence and turbulent transport [37,38], systems with self-organized criticality [11,[39][40][41][42][43][44][45][46] and random walks and anomalous diffusion in random media [12,[47][48][49][50][51][52] (not an exhaustive list). ...
... The concept of self-organized criticality (SOC) was famously introduced to explain the abundance of phenomena with the IR range power-law scaling of spatial, temporal or spatial-temporal correlations in Nature [11,[39][40][41][42][43][44][45][46]. Unlike equilibrium systems that undergo phase transitions when a tuning parameter made to arrive at its critical value, non-equilibrium systems with SOC are believed to evolve to their critical state without any external tuning taking place. ...
... However, it is the case where only the non-linearity is relevant, i.e., model (43) should coincide with the pure Hwa-Kardar model (36). This means that d and d ⊥ are two independent dimensions, see (39), and additional canonical symmetry arises. ...
This paper is concerned with intriguing possibilities for non-conventional critical behavior that arise when a nearly critical strongly non-equilibrium system is subjected to chaotic or turbulent motion of the environment. We briefly explain the connection between the critical behavior theory and the quantum field theory that allows the application of the powerful methods of the latter to the study of stochastic systems. Then, we use the results of our recent research to illustrate several interesting effects of turbulent environment on the non-equilibrium critical behavior. Specifically, we couple the Kazantsev–Kraichnan “rapid-change” velocity ensemble that describes the environment to the three different stochastic models: the Kardar–Parisi–Zhang equation with time-independent random noise for randomly growing surface, the Hwa–Kardar model of a “running sandpile” and the generalized Pavlik model of non-linear diffusion with infinite number of coupling constants. Using field-theoretic renormalization group analysis, we show that the effect can be quite significant leading to the emergence of induced non-linearity or making the original anisotropic scaling appear only through certain “dimensional transmutation”.
... Many authors have applied the Pareto model in their research, e.g., Burroughs and Tebbens [4] described earthquake and wildfire observations, and Schroeder et al. [5] analyzed plate fault data. We can also refer to previous studies [6][7][8] in this area. Some authors have proposed modified or generalized versions of Pareto to respond to more flexible models to describe data from different scientific fields. ...
... where U is a random instance from a standard uniform distribution, is solved in terms of X ; see Eq. (6). ...
Models of various physical, biological, and artificial phenomena follow a power law over multiple magnitudes. This article presents a modified Pareto model with an upside-down shape and an adjustable right tail. The moments, quantiles, failure rate, mean residual life, and quantile residual life functions are examined. In addition, some stochastic ordering characteristics of the proposed model are investigated. The estimation of the parameters using the maximum likelihood estimator, the mean square error, and the Anderson–Darling estimator is explored, and a simulation study is conducted to analyze their behavior. Finally, we compare the proposed model with alternative methods for describing a dataset on the strength of carbon fiber and a dataset on customer waiting times in a bank.
... New discoveries would then generate an expanding space of opportunities that are only available to us in the moment we "unlock" what is adjacent to them. Kauffman's AP has seen many interesting applications ranging from biology 24,28 and economics 9,29 to models of discovery and innovation processes. Among these, of particular interest is the recently proposed Urn Model with Triggering (UMT) 1,6,30 . ...
The Heaps' law, which characterizes the growth of novelties, has triggered new mathematical descriptions, based on urn models or on random walks, of the way we explore the world. However, an often-overlooked aspect is that novelties can also arise as new combinations of existing elements. Here we propose to study novelties as $n \ge 1$ consecutive elements appearing for the first time in a sequence, and we introduce the $n^{\text{th}}$-order Heaps' exponents to measure the pace of discovery of novelties of any order. Through extensive analyses of real-world sequences, we find that processes displaying the same pace of discovery of single items can instead differ at higher orders. We then propose to model the exploration dynamics as an edge-reinforced random walk with triggering on a network of relations between items which elvolves over time. The model reproduces the observed properties of higher-order novelties, and reveals how the space of possibilities expands over time along with the exploration process.
... Previously (Harper 2017a) this model was introduced and shown to produce the behavior of stasis and punctuation as represented by the distribution of maximum urban area population magnitudes over time. For a more detailed description of this model the empirical data for maximum urban area magnitude, C max , were natural log-transformed to reveal a stepwise pattern suggestive of punctuated equilibrium as first suggested by Eldredge and Gould (1972) and later incorporated into a more general description of system behavior by the late Per Bak, in his book entitled, How Nature Works: The Science of Self-Organized Criticality (1996) and in a number of research papers, for example Bak and Sneppen (1993). However, with the publication of the paper 'Integration of Disparate Processes' there arose a question as to how the variables, N, K, and T are related to one another and what interchangeable units could be used to show these relationships. ...
The present Yearbook is subtitled Entropy and Destabilization. It is the tenth in the series. The study of the forms and causes of destabilization is extremely important, because no regime, no society, no system is immune to destabilization. Destabilization, or at least the threat of it, is an inevitable stage in the historical development of any society. The question is to what extent a society is capable of resisting it, how institutionalized and adaptive it is. The articles of this issue are devoted to the various manifestations of destabilization, its different forms, patterns and causes in the past and present. The issue consists of four sections: (I) Historical Aspects; (II) Social and Cultural Aspects; (III) Factors of Destabilization; (IV) Reviews and Notes. We hope that this issue will be interesting and useful both for historians and mathematicians, as well as for all those dealing with various social and natural sciences.
... Although SOC is treated as a rather isolated concept after its first discovery in statistical physics Bak, Tang, and Wiesenfeld (1987), subsequent analyses demonstrate SOC as relevant with ordinary continuous phase transitions into infinitely many absorbing states Dickman, Muñoz, Vespignani, and Zapperi (2000); Dickman, Vespignani, and Zapperi (1998); Narayan and Middleton (1994); Sornette, Johansen, and Dornic (1995). Specifically, SOC models can be subdivided into two families, which we refer to as external dynamics family (e.g., Bak-Sneppen model Bak and Sneppen (1993)) and conserved field family (e.g., sandpile models such as Manna model Manna (1991) and Bak-Tang-Wiesenfeld model Bak et al. (1987)). The second family, being the main theoretical source of studying SOC in neural dynamics, corresponds to absorbing-state transitions since it can represent any system with conserved local dynamics and continuous transitions to absorbing states Dickman et al. (2000); Lübeck (2004). ...
Criticality is hypothesized as a physical mechanism underlying efficient transitions between cortical states and remarkable information processing capacities in the brain. While considerable evidence generally supports this hypothesis, non-negligible controversies persist regarding the ubiquity of criticality in neural dynamics and its role in information processing. Validity issues frequently arise during identifying potential brain criticality from empirical data. Moreover, the functional benefits implied by brain criticality are frequently misconceived or unduly generalized. These problems stem from the non-triviality and immaturity of the physical theories that analytically derive brain criticality and the statistic techniques that estimate brain criticality from empirical data. To help solve these problems, we present a systematic review and reformulate the foundations of studying brain criticality, i.e., ordinary criticality (OC), quasi-criticality (qC), self-organized criticality (SOC), and self-organized quasi-criticality (SOqC), using the terminology of neuroscience. We offer accessible explanations of the physical theories and statistic techniques of brain criticality, providing step-by-step derivations to characterize neural dynamics as a physical system with avalanches. We summarize error-prone details and existing limitations in brain criticality analysis and suggest possible solutions. Moreover, we present a forward-looking perspective on how optimizing the foundations of studying brain criticality can deepen our understanding of various neuroscience questions.
Evolvability is recognized as a core feature of living systems. Evolution is seen as phylogenetic adaptation, and the salient features of evolutionary systems, which display true complexity, are described. Darwin’s theory is discussed, and concepts like exaptation, punctuated equilibrium, and altruism (group and kin selection). Models of evolution are described and discussed. Finally, evolutionary computing, such as genetic algorithms, is described.
Massively multiplayer online games (MMOGs) played on the Web provide a new form of social, computer-mediated interactions that allow the connection of millions of players worldwide. The rules governing team-based MMOGs are typically complex and non-deterministic giving rise to an intricate dynamical behavior. However, due to the novelty and complexity of MMOGs their behavior is understudied. In this paper, we investigate the MMOG World of Tanks (WOT) Blitz by using a combined approach based on data science and complex adaptive systems. We analyze data on the population level to get insight into organizational principles of the game and its game mechanics. For this reason, we study the scaling behavior and the predictability of system variables. As a result, we find a power-law behavior on the population level revealing long-range interactions between system variables. Furthermore, we identify and quantify the predictability of summary statistics of the game and its decomposition into explanatory variables. This reveals a heterogeneous progression through the tiers and identifies only a single system variable as key driver for the win rate.
We study a self-organized critical system coupled to an isotropic random fluid environment. The former is described by a strongly anisotropic continuous (coarse-grained) model introduced by Hwa and Kardar [Phys. Rev. Lett. 62 1813 (1989); Phys. Rev. A 45 7002 (1992)]; the latter is described by the stirred Navier–Stokes equation due to Forster, Nelson and Stephen [Phys. Rev. A 16 732 (1977)]. The full problem of two coupled stochastic equations is represented as a field theoretic model, which is shown to be multiplicatively renormalizable. The corresponding renormalization group equations possess a semi-infinite curve of fixed points in the four-dimensional space of the model parameters. The whole curve is infrared attractive for realistic values of parameters; its endpoint corresponds to the purely isotropic regime where the original Hwa-Kardar nonlinearity becomes irrelevant. There, one is left with a simple advection of a passive scalar field by the external environment. The main critical dimensions are calculated to the leading one-loop order (first terms in the ε=4-d expansion); some of them are appear to be exact in all orders. They remain the same along that curve, which makes it reasonable to interpret it as a single universality class. However, the correction exponents do vary along the curve. It is therefore not clear whether the curve survives in all orders of the renormalization group expansion or shrinks to a single point when the higher-order corrections are taken into account.
Chapter 3 is devoted to the study of the existence of solutions to nonlinear diffusion equations with a time dependent nonlinearity whose potential is not coercive, and it is treated in the duality (L∞)′− L∞.
We show that certain extended dissipative dynamical systems naturally evolve into a critical state, with no characteristic time or length scales. The temporal ''fingerprint'' of the self-organized critical state is the presence of flicker noise or 1/f noise; its spatial signature is the emergence of scale-invariant (fractal) structure.
We show that dynamical systems with spatial degrees of freedom naturally evolve into a self-organized critical point. Flicker noise, or 1/f noise, can be identified with the dynamics of the critical state. This picture also yields insight into the origin of fractal objects.
A variant of Kauffman's {ital NKC}-model for the coevolution of haploid organisms is shown to have two phases: a {ital frozen} phase in which all species eventually reach local fitness maxima and stop evolving, and a {ital chaotic} phase in which a fraction of all species is at local maxima, while another fraction evolves towards maxima. In doing so, they set other species back in evolution, thereby maintaining a steady fraction of evolving species. The evolutionary activity of the steady state is a natural order parameter for the ecosystem. Closed expressions are given for this order parameter and for the system's relaxation time. The latter quantity diverges at the phase boundary, showing the system is critical there. All results were obtained analytically for the maximally rugged case of {ital K}+1={ital N}, and to leading order in {ital N}, the number of genes in a species.
We believe that punctuational change dominates the history of life: evolution is concentrated in very rapid events of speciation (geologically instantaneous, even if tolerably continuous in ecological time). Most species, during their geological history, either do not change in any appreciable way, or else they fluctuate mildly in morphology, with no apparent direction. Phyletic gradualism is very rare and too slow, in any case, to produce the major events of evolution. Evolutionary trends are not the product of slow, directional transformation within lineages; they represent the differential success of certain species within a clade—speciation may be random with respect to the direction of a trend (Wright's rule).
As an a priori bias, phyletic gradualism has precluded any fair assessment of evolutionary tempos and modes. It could not be refuted by empirical catalogues constructed in its light because it excluded contrary information as the artificial result of an imperfect fossil record. With the model of punctuated equilibria, an unbiased distribution of evolutionary tempos can be established by treating stasis as data and by recording the pattern of change for all species in an assemblage. This distribution of tempos can lead to strong inferences about modes. If, as we predict, the punctuational tempo is prevalent, then speciation—not phyletic evolution—must be the dominant mode of evolution.
We argue that virtually none of the examples brought forward to refute our model can stand as support for phyletic gradualism; many are so weak and ambiguous that they only reflect the persistent bias for gradualism still deeply embedded in paleontological thought. Of the few stronger cases, we concentrate on Gingerich's data for
Hyopsodus
and argue that it provides an excellent example of species selection under our model. We then review the data of several studies that have supported our model since we published it five years ago. The record of human evolution seems to provide a particularly good example: no gradualism has been detected within any hominid taxon, and many are long-ranging; the trend to larger brains arises from differential success of essentially static taxa. The data of molecular genetics support our assumption that large genetic changes often accompany the process of speciation.
Phyletic gradualism was an a priori assertion from the start—it was never “seen” in the rocks; it expressed the cultural and political biases of 19th century liberalism. Huxley advised Darwin to eschew it as an “unnecessary difficulty.” We think that it has now become an empirical fallacy. A punctuational view of change may have wide validity at all levels of evolutionary processes. At the very least, it deserves consideration as an alternate way of interpreting the history of life.
THE 'Game of Life'1,2 is a cellular automaton, that is, a lattice system in which the state of each lattice point is determined by local rules. It simulates, by means of a simple algorithm, the dynamical evolution of a society of living organisms. Despite its simplicity, the complex dynamics of the game are poorly understood. Previous interest in 'Life' has focused on the generation of complexity in local configurations; indeed, the system has been suggested to mimic aspects of the emergence of complexity in nature1,2. Here we adopt a different approach, by using concepts of statistical mechanics to study the system's long-time and large-scale behaviour. We show that local configurations in the "Game of Life" self-organize into a critical state. Such self-organized criticality provides a general mechanism for the emergence of scale-free structures3-5, with possible applications to earth-quakes6,7, cosmology8, turbulence9, biology and economics10. By contrast to these previous studies, where a local quantity was conserved, 'Life' has no local conservation laws and therefore represents a new type of universality class for self-organized criticality. This refutes speculations that self-organized criticality is a consequence of local conservation11, and supports its relevance to the natural phenomena above, as these do not involve any locally conserved quantities. The scaling is universal in the sense that the exponents that characterize correlation functions do not depend on details of the local rules.
We introduce a broadened framework to study aspects of coevolution based on the NK class of statistical models of rugged fitness landscapes. In these models the fitness contribution of each of N genes in a genotype depends epistatically on K other genes. Increasing epistatic interactions increases the rugged multipeaked character of the fitness landscape. Coevolution is thought of, at the lowest level, as a coupling of landscapes such that adaptive moves by one player deform the landscapes of its immediate partners. In these models we are able to tune the ruggedness of landscapes, how richly intercoupled any two landscapes are, and how many other players interact with each player. All these properties profoundly alter the character of the coevolutionary dynamics. In particular, these parameters govern how readily coevolving ecosystems achieve Nash equilibria, how stable to perturbations such equilibria are, and the sustained mean fitness of coevolving partners. In turn, this raises the possibility that an evolutionary metadynamics due to natural selection may sculpt landscapes and their couplings to achieve coevolutionary systems able to coadapt well. The results suggest that sustained fitness is optimized when landscape ruggedness relative to couplings between landscapes is tuned such that Nash equilibria just tenuously form across the ecosystem. In this poised state, coevolutionary avalanches appear to propagate on all length scales in a power law distribution. Such avalanches may be related to the distribution of small and large extinction events in the record.
A new class of interface growth models is proposed, where global equilibration of the driving force is achieved between each local deposition. Two such models are studied numerically, and it is seen that roughness can occur with higher exponents than in situations where global equilibration of the driving force is not established. In particular, we have found a new universality class of growth models which in one dimension gives self-affine interfaces with roughness exponent chi = 0.63 +/- 0.02.
We study roughening interfaces that become self-organized critical by a rule similar to that of invasion percolation. We demonstrate that there is a fundamental difference between transient and critical dynamical exponents. The exponents break the Galilean invariance and temporal multiscaling is observed. We show that the activity along the interface exhibits nontrivial power law correlations in both space and time even though only quenched Gaussian noise is applied. The results are compared with simulations where spatial power law correlated noise is used as input.
DOI:https://doi.org/10.1103/PhysRevLett.70.3833