coevolutionary dynamics on scale free networks we investigate bak sneppen coevolution models on scale free networks with various degree exponents ensuremath gamma including random networks for ensuremath gamma g3 the critical fitness value f c approaches a nonzero finite value in the limit n ensuremath rightarrow ensuremath infty whereas f c approaches zero as 2l ensuremath gamma ensuremath leqslant 3 these results are explained by showing analytically f c n ensuremath simeq a k 1 2 n on the networks with size n the avalanche size distribution p s shows the normal power law behavior for ensuremath gamma g3 in contrast p s for 2l ensuremath gamma ensuremath leqslant 3 has two power law regimes one is a short regime for small s with a large exponent ensuremath tau 1 and the other is a long regime for large s with a small exponent ensuremath tau 2 phantom rule 0 3em 0ex ensuremath tau 1 g ensuremath tau 2 the origin of the two power regimes is explained by the dynamics on an artificially made star linked network

quadratic serendipity finite elements over convex polyhedra summary the conventional approach to construct quadratic elements for an n sided polygon will yield n n 1 2 shape functions which increases the computational effort it is well known that the serendipity elements based on isoparametric formulation suffers from mesh distortion floater and lai proposed a systematic way to construct higher order approximations over arbitrary polygons using the generalized barycentric coordinates and triangular coordinates this approach ensures 2n shape functions with nodes only on the boundary of the polygon in this paper we extend the polygonal splines approach to three dimensions and construct serendipity shape functions over hexahedra and convex polyhedra this is done by expressing the shape functions using the barycentric coordinates and the local tetrahedral coordinates the quadratic shape functions possess kronecker delta property and satisfy constant linear and quadratic precision the accuracy and the convergence properties of the quadratic serendipity shape elements are demonstrated with a series of standard patch tests the numerical results show that the quadratic serendipity elements pass the patch test yield optimal convergence rate and can tolerate extreme mesh distortion this article is protected by copyright all rights reserved

quasi concave density estimation maximum likelihood estimation of a log concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum shannon entropy problem closely related maximum renyi entropy estimators that impose weaker concavity restrictions on the fitted density are also considered notably a minimum hellinger discrepancy estimator that constrains the reciprocal of the square root of the density to be concave a limiting form of these estimators constrains solutions to the class of quasi concave densities

breakdown properties of location estimates based on halfspace depth and projected outlyingness we describe multivariate generalizations of the median trimmed mean and w estimates the estimates are based on a geometric construction related to projection pursuit they are both affine equivariant coordinate free and have high breakdown point the generalization of the median has a breakdown point of at least 1 d 1 in dimension d and the breakdown point can be as high as 1 3 under symmetry in contrast various estimators based on rejecting apparent outliers and taking the mean of the remaining observations have breakdown points not larger than 1 d 1 in dimension d

general notions of statistical depth function statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data here we introduce several general structures for depth functions classify many existing examples as special cases and establish results on the possession or lack thereof of four key properties desirable for depth functions in general roughly speaking these properties may be described as affine invariance maximality at center monotonicity relative to deepest point and vanishing at infinity this provides a more systematic basis for selection of a depth function in particular from these and other considerations it is found that the halfspace depth behaves very well overall in comparison with various competitors

a technique to develop mesh distortion immune finite elements three sufficient conditions are proposed for an iso parametric finite element formulation to be immune to mesh distortion effects even for extremely distorted element geometries for which the jacobian geometric mapping goes negative illustrative element formulations are presented to test the sufficiency of the proposed conditions numerical results confirm that the illustrative elements are indeed capable of giving mesh distortion immune solution for extreme mesh distortions this has been possible partly because the stiffness integral is free of the jacobian term in the new formulation

rotationally invariant distortion resistant finite elements the predictive capability of conventional iso parametric finite elements deteriorates with mesh distortion in the case of geometrically non linear analysis changes in geometry causing severe distortion can result in negative jacobian mapping between the local and global systems resulting in numerical breakdown this paper presents a finite element formulation that is resistant to irregular mesh geometries and large element distortions whilst remaining invariant to rigid body motion the predictive capabilities of the family of finite elements are demonstrated using a series of geometrically non linear analyses including an elastic cantilever beam and an elasto plastic double notched specimen

multivariate analysis by data depth descriptive statistics graphics and inference with discussion and a rejoinder by liu and singh a data depth can be used to measure the depth or outlyingness of a given multivariate sample with respect to its underlying distribution this leads to a natural center outward ordering of the sample points based on this ordering quantitative and graphical methods are introduced for analyzing multivariate distributional characteristics such as location scale bias skewness and kurtosis as well as for comparing inference methods all graphs are one dimensional curves in the plane and can be easily visualized and interpreted a sunburst plot is presented as a bivariate generalization of the box plot dd depth versus depth plots are proposed and examined as graphical inference tools some new diagnostic tools for checking multivariate normality are introduced one of them monitors the exact rate of growth of the maximum deviation from the mean while the others examine the ratio of the overall dispersion to the dispersion of a certain central region the affine invariance property of a data depth also leads to appropriate invariance properties for the proposed statistics and methods

an unsymmetric 4 node 8 dof plane membrane element perfectly breaking through macneal s theorem summary among numerous finite element techniques few models can perfectly without any numerical problems break through macneal s theorem any 4 node 8 dof membrane element will either lock in in plane bending or fail to pass a c0 patch test when the element s shape is an isosceles trapezoid in this paper a 4 node plane quadrilateral membrane element is developed following the unsymmetric formulation concept which means two different sets of interpolation functions for displacement fields are simultaneously used the first set employs the shape functions of the traditional 4 node bilinear isoparametric element while the second set adopts a novel composite coordinate interpolation scheme with analytical trail function method in which the cartesian coordinates x y and the second form of quadrilateral area coordinates qacm ii s t are applied together the resulting element us atfq4 exhibits amazing performance in rigorous numerical tests it is insensitive to various serious mesh distortions free of trapezoidal locking and can satisfy both the classical first order patch test and the second order patch test for pure bending furthermore because of usage of the second form of quadrilateral area coordinates qacm ii the new element provides the invariance for the coordinate rotation it seems that the behaviors of the present model are beyond the well known contradiction defined by macneal s theorem copyright 2015 john wiley sons ltd

classifier selection using the predicate depth typically one approaches a supervised machine learning problem by writing down an objective function and finding a hypothesis that minimizes it this is equivalent to finding the maximum a posteriori map hypothesis for a boltzmann distribution however map is not a robust statistic we present an alternative approach by defining a median of the distribution which we show is both more robust and has good generalization guarantees we present algorithms to approximate this median one contribution of this work is an efficient method for approximating the tukey median the tukey median which is often used for data visualization and outlier detection is a special case of the family of medians we define however computing it exactly is exponentially slow in the dimension our algorithm approximates such medians in polynomial time while making weaker assumptions than those required by previous work

weighted scale free networks in euclidean space using local selection rule received 23 july 2005 revised manuscript received 21 july 2006 published 22 september 2006a spatial scale free network is introduced and studied whose motivation originated in the growing internetas well as airport networks we argue that in these real world networks a new node necessarily selects one ofits neighboring local nodes for connection and is not controlled by preferential attachment as in the barabasi albert ba model this observation is mimicked in our model where the nodes pop up at randomly locatedpositions in the euclidean space and are connected to one end of the nearest link in spite of this crucialdifference it is observed that the leading behavior of our network is like that of the ba model de ning the linkweight as an algebraic power of its euclidean length the weight distribution and the nonlinear dependence ofthe nodal strength on the degree are analytically calculated it is claimed that a power law decay of the linkweights with time ensures such nonlinear behavior switching off the euclidean space from the same modelyields a much simpler de nition of the ba model where numerical effort grows linearly with

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

a 20 node hexahedron element with enhanced distortion tolerance this paper presents a distortion resistant 20 node hexahedron element that employs two different sets of shape functions for the trial and test functions the formulation seeks to satisfy the continuity and completeness requirements by exploiting the intrinsic properties of these two sets of shape functions several test problems are used to assess the performance of the element under various mesh distortions the ability of the proposed as well as the classical 20 node element to maintain solution accuracy under severe mesh distortions has been studied the proposed element exhibits a very high tolerance to mesh distortions in particular for problems involving linear and quadratic displacement fields the element is capable of reproducing exact solution under all admissible geometrical distortions of the mesh for test problems involving higher order displacement fields the performance of the present element is in general better than that of the classical element copyright 2004 john wiley sons ltd

multivariate quantiles and multiple output regression quantiles from l1 optimization to halfspace depth a new multivariate concept of quantile based on a directional version of koenker and bassett s traditional regression quantiles is introduced for multivariate location and multiple output regression problems in their empirical version those quantiles can be computed efficiently via linear programming techniques consistency bahadur representation and asymptotic normality results are established most importantly the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of tukey this relation does not only allow for efficient depth contour computations by means of parametric linear programming but also for transferring from the quantile to the depth universe such asymptotic results as bahadur representations finally linear programming duality opens the way to promising developments in depth related multivariate rank based inference

from depth to local depth a focus on centrality aiming at analyzing multimodal or nonconvexly supported distributions through data depth we introduce a local extension of depth our construction is obtained by conditioning the distribution to appropriate depth based neighborhoods and has the advantages among others of maintaining affine invariance and applying to all depths in a generic way most importantly unlike their competitors which for extreme localization rather measure probability mass the resulting local depths focus on centrality and remain of a genuine depth nature at any locality level we derive their main properties establish consistency of their sample versions and study their behavior under extreme localization we present two applications of the proposed local depth for classification and for symmetry testing and we extend our construction to the regression depth context throughout we illustrate the results on several datasets both artificial and real univariate and multivariate supplementary materials for this artic

permutation tests for equality of distributions in high dimensional settings motivated by applications in high dimensional settings we suggest a test of the hypothesis h 0 that two sampled distributions are identical it is assumed that two independent datasets are drawn from the respective populations which may be very general in particular the distributions may be multivariate or infinite dimensional in the latter case representing for example the distributions of random functions from one euclidean space to another our test uses a measure of distance between data this measure should be symmetric but need not satisfy the triangle inequality so it is not essential that it be a metric the test is based on ranking the pooled dataset with respect to the distance and relative to any fixed data value and repeating this operation for each fixed datum a permutation argument enables a critical point to be chosen such that the test has concisely known significance level conditional on the set of all pairwise distances

robust estimate of excitations in mechanical systems using m estimators theoretical background and numerical applications abstract model based methods are often used along with least squares to estimate or to identify in equivalent but in more engineering terms dynamic forces parameters and malfunctions in mechanical systems starting from experimental vibrations the effectiveness of these methods broadly proven and documented by several cases of study can be reduced if the model of the system is not accurate or if the experimental data are corrupted by noise especially if the mean value of the noise is not null or if biases are present a possible solution is the use of robust estimation techniques instead of traditional least squares in the ambit of model based identification the author proposes the application of the m estimators and discusses the problems related to their application to excitation identification in mechanical systems in this paper the necessary theory is presented in detail introducing several concepts of statistics in order to properly introduce the concept of robust estimation and the required algorithms based on iterative re weighted least squares are described then the different types of m estimators proposed in literature are introduced their performances with regard to mechanical applications are evaluated by means of a theoretical analysis and a couple of simple numerical examples a single input single output and a multiple inputs multiple outputs systems moreover the problem of the scale parameter which is not discussed in literature for complex numbers as the vibrations are conveniently represented is analyzed and a solution is proposed using a concept related to the data depth

extremal dynamics on complex networks analytic solutions the bak sneppen model displaying punctuated equilibria in biological evolution is studied on random complex networks by using the rate equation and the random walk approaches we obtain the analytic solution of the fitness threshold x c to be 1 k f 1 where k f k 2 k k in the quenched annealed updating case where k n is the n text th moment of the degree distribution thus the threshold is zero finite for the degree exponent ensuremath gamma l3 ensuremath gamma g3 for the quenched case in the thermodynamic limit the theoretical value x c fits well to the numerical simulation data in the annealed case only avalanche size defined as the duration of successive mutations below the threshold exhibits a critical behavior as its distribution follows a power law p a s ensuremath sim s ensuremath 3 2

robust estimation via robust gradient estimation we provide a new computationally efficient class of estimators for risk minimization we show that these estimators are robust for general statistical models under varied robustness settings including in the classical huber e contamination model and in heavy tailed settings our workhorse is a novel robust variant of gradient descent and we provide conditions under which our gradient descent variant provides accurate estimators in a general convex risk minimization problem we provide specific consequences of our theory for linear regression and logistic regression and for canonical parameter estimation in an exponential family these results provide some of the first computationally tractable and provably robust estimators for these canonical statistical models finally we study the empirical performance of our proposed methods on synthetic and real data sets and we find that our methods convincingly outperform a variety of baselines

a quadratic plane triangular element immune to quadratic mesh distortions under quadratic displacement fields abstract a quadratic plane triangular element is developed extending the concepts of us quad8 element the element belongs to the broad class of petrov galerkin formulation which is characterized by the choice of test and trial functions from different spaces the performance of the proposed element has been studied for typical static and free vibration problems the element exhibits a high degree of tolerance to mesh distortions as compared to the classical isoparametric quadratic triangular element for undistorted meshes however the performance is similar to that of the classical element

tyler shape depth summaryin many problems from multivariate analysis the parameter of interest is a shape matrix a normalized version of the corresponding scatter or dispersion matrix in this article we propose a notion of depth for shape matrices that involves data points only through their directions from the centre of the distribution we refer to this concept as tyler shape depth since the resulting estimator of shape namely the deepest shape matrix is the median based counterpart of the m estimator of shape due to tyler 1987 besides estimation shape depth like its tyler antecedent also allows hypothesis testing on shape its main benefit however lies in the ranking of the shape matrices it provides the practical relevance of which is illustrated by applications to principal component analysis and shape based outlier detection we study the invariance quasi concavity and continuity properties of tyler shape depth the topological and boundedness properties of the corresponding depth regions and the existence of a deepest shape matrix and we prove fisher consistency in the elliptical case finally we derive a glivenko cantelli type result and establish almost sure consistency of the deepest shape matrix estimator

multidimensional trimming based on projection depth as estimators of location parameters univariate trimmed means are well known for their robustness and efficiency they can serve as robust alternatives to the sample mean while possessing high efficiencies at normal as well as heavy tailed models this paper introduces multidimensional trimmed means based on projection depth induced regions robustness of these depth trimmed means is investigated in terms of the influence function and finite sample breakdown point the influence function captures the local robustness whereas the breakdown point measures the global robustness of estimators it is found that the projection depth trimmed means are highly robust locally as well as globally asymptotics of the depth trimmed means are investigated via those of the directional radius of the depth induced regions the strong consistency asymptotic representation and limiting distribution of the depth trimmed means are obtained relative to the mean and other leading competitors the depth trimmed means are highly efficient at normal or symmetric models and overwhelmingly more efficient when these models are contaminated simulation studies confirm the validity of the asymptotic efficiency results at finite samples

convergence of depth contours for multivariate datasets contours of depth often provide a good geometrical understanding of the structure of a multivariate dataset they are also useful in robust statistics in connection with generalized medians and data ordering if the data constitute a random sample from a spherical or elliptic distribution the depth contours are generally required to converge to spherical or elliptical shapes we consider contour constructions based on a notion of data depth and prove a uniform contour convergence theorem under verifiable conditions on the depth measure applications to several existing depth measures discussed in the literature are also considered

asymmetric dynamics and critical behavior in the bak sneppen model abstract we investigate using mean field theory and simulation the effect of asymmetry on the critical behavior and probability density of bak sneppen models two kinds of anisotropy are investigated i different numbers of sites to the left and right of the central minimum site are updated and ii sites to the left and right of the central site are renewed in different ways of particular interest is the crossover from symmetric to asymmetric scaling for weakly asymmetric dynamics and the collapse of data with different numbers of updated sites but the same degree of asymmetry all non symmetric rules studied fall independent of the degree of asymmetry in the same universality class conversely symmetric variants reproduce the exponents of the original model our results confirm the existence of two symmetry based universality classes for extremal dynamics

distortion immune nine node displacement based quadrilateral thick plate finite elements that satisfy constant bending patch test we employ the linked interpolation concept to develop a higher order 9 node quadrilateral plate finite element with curved sides which passes the constant bending patch test for arbitrary node positions the linked interpolation for the plate displacements is expanded with three bubble parameters to get polynomial completeness necessary to satisfy the patch test in contrast to some other techniques the elements developed in this way retain a symmetric stiffness matrix at a marginal computational expense at the element level two new elements are generated using this concept and tested on several examples with curved sides or some kind of geometric distortion

8 and 12 node plane hybrid stress function elements immune to severely distorted mesh containing elements with concave shapes abstract by simply improving the first version of hybrid stress element method proposed by pian several 8 and 12 node plane quadrilateral elements which are immune to severely distorted mesh containing elements with concave shapes are successfully developed firstly instead of the stresses the stress function is regarded as the functional variable and introduced into the complementary energy functional then the fundamental analytical solutions in global cartesian coordinates of are taken as the trial functions for 2d finite element models and meanwhile the corresponding unknown stress function constants are introduced thus the resulting stress fields must be more reasonable because both the equilibrium and the compatibility relations can be satisfied thirdly by using the principle of minimum complementary energy these unknown stress function constants can be expressed in terms of the displacements along element boundaries which can be interpolated directly by the element nodal displacements finally the complementary energy functional can be rewritten in terms of element nodal displacement vector and thus the element stiffness matrix of such hybrid stress function hs f element is obtained this technique establishes a universal frame for developing reasonable hybrid stress elements based on the principle of minimum complementary energy and the first hybrid stress element proposed by pian is just a special case within this frame following above procedure two 8 node and two 12 node quadrilateral plane elements are constructed by employing different fundamental analytical solutions of airy stress function numerical results show that the 8 node and 12 node models can produce the exact solutions for pure bending and linear bending problems respectively even the element shape degenerates into triangle and concave quadrangle furthermore these elements do not possess any spurious zero energy mode and rotational frame dependence

branching process in a stochastic extremal model we considered a stochastic version of the bak sneppen model sbsm of ecological evolution where the number m of sites mutated in a mutation event is restricted to only two here the mutation zone consists of only one site and this site is randomly selected from the neighboring sites at every mutation event in an annealed fashion the critical behavior of the sbsm is found to be the same as the bs model in dimensions d 1 and 2 however on the scale free graphs the critical fitness value is nonzero even in the thermodynamic limit but the critical behavior is mean field like finally m has been made even smaller than two by probabilistically updating the mutation zone which also shows the original bs model behavior we conjecture that a sbsm on any arbitrary graph with any small branching factor greater than unity will lead to a self organized critical state

on depth and deep points a calculus for a general definition of depth in data analysis a differential like calculus is constructed in which the location case the framework of tukey s median plays a fundamental role similar to that of linear functions in the mathematical analysis as an application a lower bound for maximal regression depth is proved in the general multidimensional case as conjectured by rousseeuw and hubert and others this lower bound is demonstrated to have an impact on the breakdown point of the maximum depth estimator

aspects of robust linear regression section 1 of the paper contains a general discussion of robustness in section 2 the influence function of the hampel rousseeuw least median of squares estimator is derived linearly invariant weak metrics are constructed in section 3 it is shown in section 4 that s estimators satisfy an exact holder condition of order 1 2 at models with normal errors in section 5 the breakdown points of the hampel krasker dispersion and regression functionals are shown to be 0 the exact breakdown point of the krasker welsch dispersion functional is obtained as well as bounds for the corresponding regression functional section 6 contains the construction of a linearly equivariant high breakdown and locally lipschitz dispersion functional for any design distribution in section 7 it is shown that there is no inherent contradiction between efficiency and a high breakdown point section 8 contains a linearly equivariant high breakdown regression functional which is lipschitz continuous at models with normal errors

a partition of unity based fe meshfree quad4 element for geometric non linear analysis the recently published fe meshfree quad4 element is extended to geometrical non linear analysis the shape functions for this element are obtained by combining meshfree and finite element shape functions the concept of partition of unity pu is employed for the purpose the new shape functions inherit their higher order completeness properties from the meshfree shape functions and the mesh distortion tolerant compatibility properties from the finite element fe shape functions updated lagrangian formulation is adopted for the non linear solution several numerical example problems are solved and the performance of the element is compared with that of the well known q4 qm6 and q8 elements the results show that for regular meshes the performance of the element is comparable to that of qm6 and q8 elements and superior to that of q4 element for distorted meshes the present element has better mesh distortion tolerance than q4 qm6 and q8 elements copyright 2009 john wiley sons ltd

on fractile transformation of covariates in regression the need for comparing two regression functions arises frequently in statistical applications comparison of the usual regression functions is not very meaningful in situations where the distributions and the ranges of the covariates are different for the populations for instance in econometric studies the prices of commodities and people s incomes observed at different time points may not be on comparable scales due to inflation and other economic factors in this article we describe a method of standardizing the covariates and estimating the transformed regression function which then become comparable we develop smooth estimates of the fractile regression function and study its statistical properties analytically as well as numerically we also provide a few real examples that illustrate the difficulty in comparing the usual regression functions and motivate the need for the fractile transformation our analysis of the real examples leads to new and useful statistical conclusions that are missed b

penalty based aggregation of multidimensional data abstract research in aggregation theory is nowadays still mostly focused on algorithms to summarize tuples consisting of observations in some real interval or of diverse general ordered structures of course in practice of information processing many other data types between these two extreme cases are worth inspecting this contribution deals with the aggregation of lists of data points in r d for arbitrary d 1 even though particular functions aiming to summarize multidimensional data have been discussed by researchers in data analysis computational statistics and geometry there is clearly a need to provide a comprehensive and unified model in which their properties like equivariances to geometric transformations internality and monotonicity may be studied at an appropriate level of generality the proposed penalty based approach serves as a common framework for all idempotent information aggregation methods including componentwise functions pairwise distance minimizers and data depth based medians it also allows for deriving many new practically useful tools

asymptotic distributions of the maximal depth estimators for regression and multivariate location we derive the asymptotic distribution of the maximal depth regression estimator recently proposed in rousseeuw and hubert the estimator is obtained by maximizing a projection based depth and the limiting distribution is characterized through a max min operation of a continuous process the same techniques can be used to obtain the limiting distribution of some other depth estimators including tukey s deepest point based on half space depth results for the special case of two dimensional problems have been available but the earlier arguments have relied on some special geometric properties in the low dimensional space this paper completes the extension to higher dimensions for both regression and multivariate location models

a quality index based on data depth and multivariate rank tests let f and g be the distribution functions of two given populations on rp p 1 we introduce and study a parameter q q f g which measures the overall outlyingness of population g relative to population f the parameter q can be defined using any concept of data depth its value ranges from 0 to 1 and is 5 when f and g are identical we show that within the dass of elliptical distributions when g departs from f in location or g has a larger spread or both the value of q dwindles down from 5 hence q can be used to detect the loss of accuracy or precision of a manufacturing process and thus it should serve as an important measure in quality assurance this in fact is the reason why we refer to q as a quality index in this article in addition to studying the properties of q we provide an exact rank test for testing q 5 vs q 5 this can be viewed as a multivariate analog of wilcoxon s rank sum test the tests proposed here have power against location change and scale increase simultaneo

regression depth commentaries rejoinder in this article we introduce a notion of depth in the regression setting it provides the rank of any line plane rather than ranks of observations or residuals in simple regression we can compute the depth of any line by a fast algorithm for any bivariate dataset z n of size n there exists a line with depth at least n 3 the largest depth in z n can be used as a measure of linearity versus convexity in both simple and multiple regression we introduce the deepest regression method which generalizes the univariate median and is equivariant for monotone transformations of the response throughout the errors may be skewed and heteroscedastic we also consider depth based regression quantiles they estimate the quantiles of y given x as do the koenker bassett regression quantiles but with the advantage of being robust to leverage outliers we explore the analogies between depth in regression and in location where tukey s halfspace depth is a special case of our general definition also liu s simplicial depth can be extended to the regression framework

bak sneppen model near zero dimension we consider the bak sneppen model near zero dimension where the avalanche exponent tau is close to 1 and the exponents m and sigma are close to 0 we demonstrate that tau 1 m sigma exp m 1 gamma small middle dot small middle dot small middle dot in this limit where gamma is euler s constant the avalanche hierarchy equation is rewritten in a form that makes it possible to find the relation between the critical exponents sigma and m with high accuracy we obtain precise values of the critical exponents for the one and two dimensional bak sneppen model and for the one dimensional anisotropic bak sneppen model

mean field theory for a simple model of evolution a simple dynamical model for darwinian evolution on its slowest time scale is analyzed its mean field theory is formulated and solved a random neighbor version of the model is simulated as is a one dimensional version in one dimension the dynamics can be described in terms of a repetitious random walker and anomalous diffusion with exponent 0 4 in all cases the model self organizes to a robust critical attractor