statistical analysis of airport network of china through the study of airport network of china anc composed of 128 airports nodes and 1165 flights edges we show the topological structure of anc conveys two characteristics of small worlds a short average path length 2 067 and a high degree of clustering 0 733 the cumulative degree distributions of both directed and undirected anc obey two regime power laws with different exponents i e the so called double pareto law in degrees and out degrees of each airport have positive correlations whereas the undirected degrees of adjacent airports have significant linear anticorrelations it is demonstrated both weekly and daily cumulative distributions of flight weights frequencies of anc have power law tails besides the weight of any given flight is proportional to the degrees of both airports at the two ends of that flight it is also shown the diameter of each subcluster consisting of an airport and all those airports to which it is linked is inversely proportional to its density of connectivity efficiency of anc and of its subclusters is measured through a simple definition in terms of that the efficiency of anc s subclusters increases as the density of connectivity does anc is found to have an efficiency of 0 484

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

change of a continuous character caused by gene flow an analytical approach abstract recent results sznajd weron and p ekalski physica a 259 1998 457 have shown that the competition of two forces gene flow and natural selection gfs is responsible for three basic types of population structures using monte carlo simulations a critical value above which there is no gene flow was found in this paper we apply fourier analysis and the properties of convolution of probability density functions in order to find this critical value analytically the obtained critical value for an arbitrarily dimensional system is the same as found in simulations for 1 2 and 3 dimensional systems we also show that in the investigated system there exists a possibility of a dynamical phase transition in one dimension this leads us to the conclusion that different population structures observed in nature correspond to different ground states of the gfs model sznajd weron p ekalski physica a 252 1998 336

evolutive information contained in frequency spectra abstract information concerning important details of an ecosystem s evolution are shown to be contained in an appropriately constructed power spectrum in particular it provides an answer to the question of just why power law behavior vanishes in the absence of darwinian competition

simple classical groups of lie type are determined by their character degrees let g be a finite group denote by irr g the set of all irreducible complex characters of g let cd g be the set of all irreducible complex character degrees of g forgetting multiplicities that is cd g 1 irr g and let x1 g be the set of all irreducible complex character degrees of g counting multiplicities let h be a finite nonabelian simple classical group in this paper we will show that if g is a finite group and x1 g x1 h then g is isomorphic to h in particular this implies that the nonabelian simple classical groups of lie type are uniquely determined by the structure of their complex group algebras

self organized critical model of biological evolution abstract a punctuated equilibrium model of biological evolution with relative fitness between different species being the fundamental driving force of evolution is introduced mutation is modeled as a fitness updating cellular automation process where the change in fitness after mutation follows a gaussian distribution with mean x 0 and standard deviation scaling behaviors are observed in our numerical simulation indicating that the model is self organized critical besides the numerical experiment suggests that models with different x and belongs to the same universality class

rank driven markov processes we study a class of markovian systems of n elements taking values in 0 1 that evolve in discrete time t via randomized replacement rules based on the ranks of the elements these rank driven processes are inspired by variants of the bak sneppen model of evolution in which the system represents an evolutionary fitness landscape and which is famous as a simple model displaying self organized criticality our main results are concerned with long time large n asymptotics for the general model in which at each time step k randomly chosen elements are discarded and replaced by independent u 0 1 variables where the ranks of the elements to be replaced are chosen independently at each time step according to a distribution n on 1 2 n k our main results are that under appropriate conditions on n the system exhibits threshold behavior at s 0 1 where s is a function of n and the marginal distribution of a randomly selected element converges to u s 1 as t and n of this class of models results in the literature have previously been given for special cases only namely the mean field or random neighbor bak sneppen model our proofs avoid the heuristic arguments of some of the previous work and use foster lyapunov ideas our results extend existing results and establish their natural more general context we derive some more specialized results for the particular case where k 2 one of our technical tools is a result on convergence of stationary distributions for families of uniformly ergodic markov chains on increasing state spaces which may be of independent interest

explained randomness in proportional hazards models a coefficient of explained randomness analogous to explained variation but for non linear models was presented by kent the construct hinges upon the notion of kullback leibler information gain kent and o quigley developed these ideas obtaining simple multiple and partial coefficients for the situation of proportional hazards regression their approach was based upon the idea of transforming a general proportional hazards model to a specific one of weibull form xu and o quigley developed a more direct approach more in harmony with the semi parametric nature of the proportional hazards model thereby simplifying inference and allowing for instance the use of time dependent covariates a potential drawback to the coefficient of xu and o quigley is its interpretation as explained randomness in the covariate given time an investigator might feel that the interpretation of the kent and o quigley coefficient as a proportion of explained randomness of time given the covariate is preferable one purpose of this note is to indicate that under an independent censoring assumption the two population coefficients coincide thus the simpler inferential setting for xu and o quigley can also be applied to the coefficient of kent and o quigley our second purpose is to point out that a sample based coefficient in common use in the sas statistical package can be interpreted as an estimate of explained randomness when there is no censoring when there is censoring the sas coefficient would not seem satisfactory in that its population counterpart depends on an independent censoring mechanism however there is a quick fix and we argue in favour of its use

on the phase space approach to complexity we elaborate in some detail on a new phase space approach to complexity due to y c zhang we show in particular that the connection between maximal complexity and power law noise or correlations can be derived from a simple variational principle for a 1d signal we find 1 f noise in accordance with zhang

the screening of species in a darwinistic tree like model of evolution abstract numerical and theoretical investigations of a tree like model of self organized critical evolution are presented the screening of species from the evolution process is numerically studied critical exponents for i the avalanche distribution ii the correlation function iii the order parameter and iv the fraction of unscreened species are given a mean field approximation is developed this theory is quite accurate for describing the critical state but is not able to describe the transient regimes i e the self organized slowing down of both order parameter and fraction of unscreened species towards the critical state

explained variation in survival analysis several measures of explained variation have been suggested for the cox proportional hazards regression model we have categorized these measures into three classes which correspond to three different definitions of multiple r2 of the general linear model in an empirical study we compared the performance of these measures and classified them by their adherence to a set of criteria which we think should be met by a measure of explained variation for survival data we suggest that currently there is no uniformly superior measure particularly as the concepts of either uncensored or censored populations may lead to different choices for uncensored populations a measure by kent and o quigley and the squared rank correlation between survival time and the predictor from a cox regression model appear recommendable choices for the latter censored survival times are terminated using a very recent data augmentation algorithm for multiple imputation under proportional hazards with censored populations schemper s measure v2 could be considered we give an introductory example discuss aspects of application and stress the desirability of routinely evaluating explained variation in studies of survival

non triviality of a discrete bak sneppen evolution model consider the following evolution model proposed in ref 1 by bak and sneppen put n vertices on a circle spaced evenly each vertex represents a certain species we associate with each vertex a random variable representing the state or fitness of the species with values in 0 1 the dynamics proceeds as follows every discrete time step we choose the vertex with minimal fitness and assign to this vertex and to its two neighbours three new independent fitnesses with a uniform distribution on 0 1 a conjecture of physicists based on simulations is that in the stationary regime the one dimensional marginal distributions of the fitnesses converges when n to a uniform distribution on f 1 for some threshold f 1 in this paper we consider a discrete version of this model proposed in ref 2 in this discrete version the fitness of a vertex can be either 0 or 1 the system evolves according to the following rules each discrete time step we choose an arbitrary vertex with fitness 0 if all the vertices have fitness 1 then we choose an arbitrary vertex with fitness 1 then we update the fitnesses of this vertex and of its two neighbours by three new independent fitnesses taking value 0 with probability 0 q 1 and 1 with probability p 1 q we show that if q is close enough to one then the mean average fitness in the stationary regime is bounded away from 1 uniformly in the number of vertices this is a small step in the direction of the conjecture mentioned above and also settles a conjecture mentioned in ref 2 our proof is based on a reduction to a continuous time particle system

asymptotic fitness distribution in the bak sneppen model of biological evolution with four species we suggest a new method to compute the asymptotic fitness distribution in the bak sneppen model of biological evolution as applications we derive the full asymptotic distribution in the four species model and give an explicit linear recurrence relation for a set of coefficients determining the asymptotic distribution in the five species model

avalanche dynamics surface roughening and self organized criticality experiments on a three dimensional pile of rice we present a two dimensional system that exhibits features of self organized criticality the avalanches that occur on the surface of a pile of rice are found to exhibit finite size scaling in their probability distribution the critical exponents are ensuremath tau 1 21 2 for the avalanche size distribution and d 1 99 2 for the cutoff size furthermore the geometry of the avalanches is studied leading to a fractal dimension of the active sites of d b 1 58 2 using a set of scaling relations we can calculate the roughness exponent ensuremath alpha d ensuremath d b 0 41 3 and the dynamic exponent z d 2 ensuremath ensuremath tau 1 56 8 this result is compared with that obtained from a power spectrum analysis of the surface roughness which yields ensuremath alpha 0 42 3 and z 1 5 1 in excellent agreement with those obtained from the scaling relations

quantifying degrees of necessity and of sufficiency in cause effect relationships with dichotomous and survival outcomes we suggest measures to quantify the degrees of necessity and of sufficiency of prognostic factors for dichotomous and for survival outcomes a cause represented by certain values of prognostic factors is considered necessary for an event if without the cause the event cannot develop it is considered sufficient for an event if the event is unavoidable in the presence of the cause necessity and sufficiency can be seen as the two faces of causation and this symmetry and equal relevance are reflected by the suggested measures the measures provide an approximate in some cases an exact multiplicative decomposition of explained variation as defined by schemper and henderson for censored survival and for dichotomous outcomes the measures ranging from zero to one are simple intuitive functions of unconditional and conditional probabilities of an event such as disease or death these probabilities often will be derived from logistic or cox regression models the measures however do not require any particular model the measures of the degree of necessity implicitly generalize the established attributable fraction or risk for dichotomous prognostic factors and dichotomous outcomes to continuous prognostic factors and to survival outcomes in a setting with multiple prognostic factors they provide marginal and partial results akin to marginal and partial odds and hazard ratios from multiple logistic and cox regression properties of the measures are explored by an extensive simulation study their application is demonstrated by three typical real data examples

the origin of power law distributions in self organized criticality the origin of power law distributions in self organized criticality is investigated by treating the variation of the number of active sites in the system as a stochastic process an avalanche is then regarded as a first return random walk process in a one dimensional lattice power law distributions of the lifetime and spatial size are found when the random walk is unbiased with equal probability to move in opposite directions this shows that power law distributions in self organized criticality may be caused by the balance of competitive interactions at the mean time the mean spatial size for avalanches with the same lifetime is found to increase in a power law with the lifetime

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

predictive accuracy and explained variation in cox regression summary we suggest a new measure of the proportion of the variation of possibly censored survival times explained by a given proportional hazards model the proposed measure termed v shares several favorable properties with an earlier v1 but also improves the handling of censoring the statistic contrasts distance measures between individual 1 0 survival processes and fitted survival curves with and without covariate information these distance measures dx and d respectively are themselves informative as summaries of absolute rather than relative predictive accuracy we recommend graphical comparisons of survival curves for prognostic index groups to improve the understanding of obtained values for v dx and d their use and interpretation is exemplified for a yorkshire lung cancer study on survival from this and an overview for several well known clinical data sets we show that the likely amount of relative or absolute predictive accuracy is often low even if there are highly significant and relatively strong prognostic factors

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

probabilistic approach to the bak sneppen model we study here the bak sneppen model a prototype model for the study of self organized criticality in this model several species interact and undergo extinction with a power law distribution of activity bursts species are defined through their fitness whose distribution in the system is uniform above a certain threshold run time statistics is introduced for the analysis of the dynamics in order to explain the peculiar properties of the model this approach based on conditional probability theory takes into account the correlations due to memory effects in this way we may compute analytically the value of the fitness threshold with the desired precision this represents a substantial improvement with respect to the traditional mean field approach

self organized criticality on small world networks abstract we study the btw height model of self organized criticality on a square lattice with some long range connections giving to the lattice the character of small world network we find that as function of the fraction p of long ranged bonds the power law of the avalanche size and lifetime distribution changes following a crossover scaling law with crossover exponents 2 3 and 1 for size and lifetime respectively

self organized critical pinball machine abstract the nature of self organized criticality soc is pin pointed with a simple mechanical model a pinball machine its phase space is fully parameterized by two integer variables one describing the state of an on going game the other describing the state of the machine this is the simplest possible soc system having only two degrees of freedom and no spatial correlations yet is not solvable by analytical means

on a local version of the bak sneppen model a major difficulty in studying the bak sneppen model is in effectively comparing it with well understood models this stems from the use of two geometries complete graph geometry to locate the global fitness minimizer and graph geometry to replace the species in the neighborhood of the minimizer over the years a number of models inspired by bak sneppen were studied usually by introducing different or new features e g discretizing fitness randomized neighbors or population size we present a variant that only uses features present in bak sneppen and whose difference from the bak sneppen is that only the graph geometry is used for the evolution this allows to obtain the stationary distribution through random walk dynamics while preserving the geometric nature of the model we use this to show that for constant degree graphs the stationary fitness distribution converges to an iid law as the number of vertices tends to infinity we also discuss exponential ergodicity through coupling and avalanches for the model

area distribution for directed random walks we study the probability distribution for the area under a directed random walk in the plane the walk can serve as a simple model for avalanches based on the idea that the front of an avalanche can be described by a random walk and the size is given by the area enclosed this model captures some of the qualitative features of earthquakes avalanches and other self organized critical phenomena in one dimension by finding nonlinear functional relations for the generating functions we calculate directly the exponent in the size distribution law and find it to be 4 3

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

dynamic critical approach to self organized criticality a dynamic scaling ansatz for the approach to the self organized critical soc regime is proposed and tested by means of extensive simulations applied to the bak char21 sneppen model bs which exhibits robust soc behavior considering the short time scaling behavior of the density of sites ensuremath rho t below the critical value it is shown that i starting the dynamics with configurations such that ensuremath rho t 0 ensuremath rightarrow 0 one observes an initial increase of the density with exponent ensuremath theta 0 12 2 ii using initial configurations with ensuremath rho t 0 ensuremath rightarrow 1 the density decays with exponent ensuremath delta 0 47 2 it is also shown that the temporal autocorrelation decays with exponent c a 0 35 2 using these dynamically determined critical exponents and suitable scaling relationships all known exponents of the bs model can be obtained e g the dynamical exponent z 2 10 5 the mass dimension exponent d 2 42 5 and the exponent of all returns of the activity ensuremath tau text all 0 39 2 in excellent agreement with values already accepted and obtained within the soc regime

concordance probability and discriminatory power in proportional hazards regression summary the concordance probability is used to evaluate the discriminatory power and the predictive accuracy of nonlinear statistical models we derive an analytical expression for the concordance probability in the cox proportional hazards model the proposed estimator is a function of the regression parameters and the covariate distribution only and does not use the observed event and censoring times for this reason it is asymptotically unbiased unlike harrell s c index based on informative pairs the asymptotic distribution of the concordance probability estimate is derived using u statistic theory and the methodology is applied to a predictive model in lung cancer

limit behavior of the bak sneppen evolution model one of the key problems related to the bak sneppen evolution model on the circle is computing the limit distribution of the fitness at a fixed observation vertex in the stationary regime as the size of the system tends to infinity some simulations have suggested that this limit distribution is uniform on f 1 for some f 2 3 in this article we prove that the mean of the fitness in the stationary regime is bounded away from 1 uniformly in the size of the system thereby establishing the nontriviality of the limit behavior the bak sneppen dynamics can easily be defined on any finite connected graph we also present a generalization of the phase transition result in the context of an increasing sequence of such graphs this generalization covers the multidimentional bak sneppen model as well as the bak sneppen model on a tree our proofs are based on a self similar graphical representation of the avalanches

on rank driven dynamical systems we investigate a class of models related to the bak sneppen bs model initially proposed to study evolution the bs model is extremely simple and yet captures some forms of complex behavior such as self organized criticality that is often observed in physical and biological systems in this model random fitnesses in 0 1 are associated to agents located at the vertices of a graph g their fitnesses are ranked from worst 0 to best 1 at every time step the agent with the worst fitness and some others with a priori given rank probabilities are replaced by new agents with random fitnesses we consider two cases the exogenous case where the new fitnesses are taken from an a priori fixed distribution and the endogenous case where the new fitnesses are taken from the current distribution as it evolves we approximate the dynamics by making a simplifying independence assumption we use order statistics and dynamical systems to define a rank driven dynamical system that approximates the evolution of the distribution of the fitnesses in these rank driven models as well as in the bs model for this simplified model we can find the limiting marginal distribution as a function of the initial conditions agreement with experimental results of the bs model is excellent

individual based predator prey model for biological coevolution fluctuations stability and community structure we study an individual based predator prey model of biological coevolution using linear stability analysis and large scale kinetic monte carlo simulations the model exhibits approximate 1 f noise in diversity and population size fluctuations and it generates a sequence of quasisteady communities in the form of simple food webs these communities are quite resilient toward the loss of one or a few species which is reflected in different power law exponents for the durations of communities and the lifetimes of species the exponent for the former is near ensuremath 1 while the latter is close to ensuremath 2 statistical characteristics of the evolving communities including degree predator and prey distributions and proportions of basal intermediate and top species compare reasonably with data for real food webs

length scales and power laws in the two dimensional forest fire model abstract we re examine a two dimensional forest fire model via monte carlo simulations and show the existence of two length scales with different critical exponents associated with clusters and with the usual two point correlation function of trees we check resp improve previously obtained values for other critical exponents and perform a first investigation of the critical behaviour of the slowest relaxational mode we also investigate the possibility of describing the critical point in terms of a distribution of the global density we find that some qualitative features such as a temporal oscillation and a power law of the cluster size distribution can nicely be obtained from such a model that discards the spatial structure

deterministic bak sneppen model lyapunov spectrum and avalanches as return times abstract a deterministic version of the bak sneppen model is studied the role of the lyapunov spectrum in the onset of scale free behavior is established as well as the measure theoretic nature of the bak sneppen self organized state avalanches are interpreted as return times to a small measure set and the problem of accurate determination of the scaling exponents near the critical barrier is addressed

evolution on trees a brief review about the extinctions of species and the case of genocide in a darwinistic punctuated equilibrium evolution model abstract numerical and theoretical investigations of species evolution in presence of extinction rules are recalled in the framework of punctuated equilibrium evolution models previous results on the bak sneppen model are mentioned together with extensions taking into account a variable number of species conditions for limited growth due to species interactions external fields or intrinsic population density conditions have already been considered an original genocide model is presented as a step towards describing effects due to the extinction of species critical exponents for i the avalanche distribution ii the steady state approach and iii the tree size are given a mean field approximation is shown to be quite accurate for describing the critical state the screening of species from the evolution process is not numerically studied

numerical analysis of the master equation applied to the master equation the usual numerical integration methods such as runge kutta method become inefficient when the rates associated with various transitions differ by several orders of magnitude we introduce an integration scheme that remains stable with much larger time increments than can be used in standard methods when only the stationary distribution is required a direct iteration method is even more rapid this method may be extended to construct the quasistationary distribution of a process with an absorbing state applications to birth and death processes reveal gains in efficiency of two or more orders of magnitude

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

generating synthetic time series from bak sneppen co evolution model mixtures abstract the bak sneppen model of co evolution is used to derive synthetic time series with a priori specified fractal dimension or hurst exponent through a mixing of processes in various lattice dimensions both theoretical and numerical analyses concern the avalanches at the critical threshold and provide a model for time series reconstruction that can be tested as an alternative to the classical fractional brownian motion fbm because of differences in properties new results on critical threshold and avalanche structure are obtained up to euclidean dimension d 6 the method involves a lattice based structure and therefore is suitable for the application of parallel computing

critical properties of the one dimensional forest fire model abstract a one dimensional forest fire model including lightnings is studied numerically and analytically for the tree correlation function a new correlation length with critical exponent v 5 6 is found by simulations a hamiltonian formulation is introduced which enables one to study the stationary state close to the critical point using quantum mechanical perturbation theory with this formulation also the structure of the low lying relaxation spectrum and the critical behaviour of the smallest complex gap are investigated numerically finally it is shown that critical correlation functions can be obtained from a simplified model involving only the total number of trees although such simplified models are unable to reproduce the correct off critical behaviour

symmetric groups are determined by their character degrees abstract let g be a finite group let x 1 g be the first column of the ordinary character table of g in this paper we will show that if x 1 g x 1 s n then g s n as a consequence we show that s n is uniquely determined by the structure of the complex group algebra c s n

entropic sampling and natural selection in biological evolution with a view to connecting random mutation on the molecular level to punctuated equilibrium behaviour on the phenotype level we propose a new model for biological evolution which incorporates random mutation and natural selection in this scheme the system evolves continuously into new configurations yielding non stationary behaviour of the total fitness furthermore both the waiting time distribution of species and the avalanche size distribution display power law behaviours with exponents close to two which are consistent with the fossil data these features are rather robust indicating the key role of entropy

model of pair aggregation on the bethe lattice departamento de ciencias de la tierra universidad de zaragoza 50009 zaragoza spain received 16 july 1996 revised manuscript received 18 november 1996 we extend a recent model of aggregation of pairs of particles analyzing the case in which the supportingframework is a bethe lattice the model exhibits a critical behavior of the percolation theory type s1063 651x 97 03403 x pacs number s 05 50 1qi introduction

three measures of explained variation for correlated survival data under the proportional hazards mixed effects model measures of explained variation are useful in scientific research as they quantify the amount of variation in an outcome variable of interest that is explained by one or more other variables we develop such measures for correlated survival data under the proportional hazards mixed effects model because different approaches have been studied in the literature outside the classical linear regression model we investigate three measures r 2 rres2 and 2 that quantify three different population coefficients we show that although the three population measures are not the same they reflect similar amounts of variation explained by the predictors among the three measures we show that r 2 which is the simplest to compute is also consistent for the first population measure under the usual asymptotic scenario when the number of clusters tends to infinity the other two measures on the other hand all require that in addition the cluster sizes be large we study the properties of the measures both analytically and through simulation studies we illustrate their different usage on a multi center clinical trial and a recurrent events data set copyright 2016 john wiley sons ltd

on singular probability densities generated by extremal dynamics abstract extremal dynamics is the mechanism that drives the bak sneppen model into a self organized critical state marked by a singular stationary probability density p x with the aim of understanding this phenomenon we study the bs model and several variants via mean field theory and simulation in all cases we find that p x is singular at one or more points as a consequence of extremal dynamics furthermore we show that the extremal barrier x i always belongs to the prohibited interval in which p x 0 our simulations indicate that the bak sneppen universality class is robust with regard to changes in the updating rule we find the same value for the exponent for all variants mean field theory which furnishes an exact description for the model on a complete graph reproduces the character of the probability distribution found in simulations for the modified processes mean field theory takes the form of a functional equation for p x

on the thresholds probability densities and critical exponents of bak sneppen like models abstract we report a simple method to accurately determine the threshold and the exponent of the bak sneppen bs model and also investigate the bs universality class for the random neighbor version of the bs model we find the threshold x 0 33332 3 in agreement with the exact result x 1 3 given by mean field theory for the one dimensional original model we find x 0 6672 2 in good agreement with the results reported in the literature for the anisotropic bs model we obtain x 0 7240 1 we study the finite size effect x l x l l observed in a system with l sites and find 1 00 1 for the random neighbor version 1 40 1 for the original model and 1 58 1 for the anisotropic case finally we discuss the effect of defining the extremal site as the one which minimizes a general function f x instead of simply f x x as in the original updating rule we emphasize that models with extremal dynamics have singular stationary probability distributions p x our simulations indicate the existence of two symmetry based universality classes

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

analytic results for scaling function and moments for a different type of avalanche in the bak sneppen evolution model starting from the master equation for the hierarchical structure of avalanches of a different kind within the frame of the bak sneppen evolution model we derive the exact formula of the scaling function describing the probability distribution of avalanches the scaling function displays features required by the scaling ansatz and verified by simulations using the scaling function we investigate the avalanche moment denoted by s k ensuremath delta overline f it is found that for any non negative integer k s k ensuremath delta overline f diverges as ensuremath delta f ensuremath k which gives an infinite group of exact critical exponents simulation outcomes of avalanche moments with k 1 2 3 are found to be consistent with the corresponding analytical results

criticality in simple models of evolution we consider two apparently similar models of biological evolution that have been claimed to exhibit self organized critical behavior a careful reanalysis of these models including several analytic results for one of them suggests that they are qualitatively different we demonstrate the limitations of the mean field description of these systems we argue that a more precise definition of self organized criticality is desirable and establish several criteria in this connection

a note on bias of measures of explained variation for survival data papers evaluating measures of explained variation or similar indices almost invariably use independence from censoring as the most important criterion and they always end up suggesting that some measures meet this criterion and some do not most of the time leading to a conclusion that the first is better than the second as a consequence users are offered measures that cannot be used with time dependent covariates and effects not to mention extensions to repeated events or multi state models we explain in this paper that the aforementioned criterion is of no use in studying such measures because it simply favors those that make an implicit assumption of a model being valid everywhere measures not making such an assumption are disqualified even though they are better in every other respect we show that if these allegedly inferior measures are allowed to make the same assumption they are easily corrected to satisfy the independent from censoring criterion even better it is enough to make such an assumption only for the times greater than the last observed failure time which in contrast with the preferred measures makes it possible to use all the modeling flexibility up to and assume whatever one wants after as a consequence we claim that some of the measures being preferred as better in the existing reviews are in fact inferior

correlations in the bak sneppen model from detrended fluctuation analysis abstract the correlations in the bak sneppen bs model are studied from fluctuations of three variables with detrended fluctuation analysis method similar scaling behaviors of the detrended fluctuations are found for the series of the minimum fitness in one and two dimensional 1d and 2d bs models the root mean square detrended fluctuations increase in a power law with the box size for the series of the location of the minimum fitness and the number of active sites the detrended fluctuations increase first for small box sizes and then saturate for large box sizes the power spectrum for the series of the number of active sites is shown to be a typical 1 f behavior for both the 1d and 2d bs models

huppert s conjecture for alternating groups abstract we prove that the alternating groups of degree at least 5 are uniquely determined up to an abelian direct factor by the set of degrees of their irreducible complex representations this confirms huppert s conjecture for alternating groups

critical thresholds and the limit distribution in the bak sneppen model one of the key problems related to the bak sneppen evolution model is to compute the limit distribution of the fitnesses in the stationary regime as the size of the system tends to infinity simulations in 3 1 4 suggest that the one dimensional limit marginal distribution is uniform on p c 1 for some p c 0 667 in this paper we define three critical thresholds related to avalanche characteristics we prove that if these critical thresholds are the same and equal to some p c we can only prove that two of them are the same then the limit distribution is the product of uniform distributions on p c 1 and moreover p c 0 75 our proofs are based on a self similar graphical representation of the avalanches

different hierarchy of avalanches observed in the bak sneppen evolution model a quantity overline f denoting the average fitness of an ecosystem is introduced in the bak sneppen model through this quantity a different hierarchy of avalanches f 0 avalanche is observed in the evolution of bak sneppen model an exact gap equation governing the self organization of the model is presented in terms of overline f it is found that self organized threshold f c can be exactly obtained two basic exponents of the new avalanche ensuremath tau avalanche distribution and d avalanche dimension are given through simulations of one and two dimensional bak sneppen models it is suggested that overline f may be a good quantity in determining the emergence of criticality

absorbing state phase transitions with extremal dynamics extremal dynamics represents a path to self organized criticality in which the order parameter is tuned to a value of zero the order parameter is associated with a phase transition to an absorbing state given a process that exhibits a phase transition to an absorbing state we define an extremal absorbing process providing the link to the associated extremal nonabsorbing process stationary properties of the latter correspond to those at the absorbing state phase transition in the former studying the absorbing version of an extremal dynamics model allows to determine certain critical exponents that are not otherwise accessible in the case of the bak sneppen bs model the absorbing version is closely related to the f avalanche introduced by paczuski maslov and bak phys rev e 53 414 1996 or in spreading simulations to the bs branching process also studied by these authors the corresponding nonextremal process belongs to the directed percolation universality class we revisit the absorbing bs model obtaining refined estimates for the threshold and critical exponents in one dimension we also study an extremal version of the usual contact process using mean field theory and simulation the extremal condition slows the spread of activity and modifies the critical behavior radically defining an extremal directed percolation universality class of absorbing state phase transitions asymmetric updating is a relevant perturbation for this class even though it is irrelevant for the corresponding nonextremal class

self organized percolation growth in regular and disordered lattices abstract the self organized percolation process sop is a growth model in which a critical percolation state is reached through self organization by controlling the number of sites or bonds in the growth front of the aggregate the system is spontaneously driven to a stationary state that corresponds to approximately the percolation threshold of the lattice topology and percolation process the sop model is applied here to site and bond percolation in several regular lattices in two and three dimensions triangular honeycomb and simple cubic as well as in a disordered network voronoi delaunai based on these results we propose the use of this growth algorithm as a plausible model to describe the dynamics and the anomalous geometrical properties of some natural processes

bak sneppen type models and rank driven processes the bak sneppen model is a well known stochastic model of evolution that exhibits self organized criticality only a few analytical results have been established for it so far we report a surprising connection between bak sneppen type models and more tractable markov processes that evolve without any reference to an underlying topology specifically we show that in the case of a large number of species the long time behavior of the fitness profile in the bak sneppen model can be replicated by a model with a purely rank based update rule whose asymptotics can be studied rigorously

avalanches and correlations in driven interface depinning we study the critical behavior of a driven interface in a medium with random pinning forces by analyzing spatial and temporal correlations in a lattice model recently proposed by sneppen phys rev lett bf 69 3539 1992 the static and dynamic behavior of the model is related to the properties of directed percolation we show that due to the interplay of local and global growth rules the usual method of dynamical scaling has to be modified we separate the local from the global part of the dynamics by defining a train of causal growth events or avalanche which can be ascribed a well defined dynamical exponent z loc 1 zeta c simeq 1 63 where zeta c is the roughness exponent of the interface we observe that the avalanche size distribution obeys a power law decay with an exponent kappa simeq 1 25

proportionate vs disproportionate distribution of wealth of two individuals in a tempered paretian ensemble abstract we study the distribution p of the random variable x 1 x 1 x 2 where x 1 and x 2 are the wealths of two individuals selected at random from the same tempered paretian ensemble characterized by the distribution x x x 1 where 0 is the pareto index and x is the cut off function we consider two forms of x a bounded function x 1 for l x h and zero otherwise and a smooth exponential function x exp l x x h in both cases x has moments of arbitrary order we show that for 1 p always has a unimodal form and is peaked at 1 2 so that most probably x 1 x 2 for 0 1 we observe a more complicated behavior which depends on the value of l h in particular for c a certain threshold value p has a three modal for a bounded x and a bimodal m shape for an exponential x form which signifies that in such ensembles the wealths x 1 and x 2 are disproportionately different

the discrete evolution model of bak and sneppen is conjugate to the classical contact process two fundamental models of critical phenomena are connected we show that the discrete bak sneppen evolution model is conjugate to the classical contact process this holds in discrete and continuous time on all graphs and for random as well as for deterministic choice of neighbors thus the extensive theory for the contact process applies to the discrete bak sneppen model too

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

levy nearest neighbors bak sneppen model we study a random neighbor version of the bak sneppen model where nearest neighbors are chosen according to a probability distribution decaying as a power law of the distance from the active site p x approximately x x ac omega all of the exponents characterizing the self organized critical state of this model depend on the exponent omega as omega 1 we recover the usual random nearest neighbor version of the model the pattern of results obtained for a range of values of omega is also compatible with the results of simulations of the original bs model in high dimensions moreover our results suggest a critical dimension dc 6 for the bak sneppen model in contrast with previous claims

measures of dependence for censored survival data in the linear regression model with normal errors the squared product moment correlation provides the standard measure of dependence between the explanatory variable and the response variable using the concept of information gain a measure of dependence can also be defined for more general regression models used in survival analysis such as the weibull regression model or cox s proportional hazards model further this measure of dependence can be conveniently estimated even when the response variable is subject to censoring and a simple approximation is available

extensions of unipotent characters and the inductive mckay condition abstract we prove that unipotent characters of groups of lie type extend to their inertia groups in the full automorphism group we then apply this in order to show that certain groups of lie type are good for the prime p 3 or p 2 this deals with all the exceptions not covered by our general approach for verifying the mckay conjecture for all groups of lie type in non defining characteristic

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

exact equations and scaling relations for f0 avalanche in the bak sneppen evolution model an infinite hierarchy of exact equations is derived for the newly observed f0 avalanche in the bak sneppen model by solving the first order exact equation we find that the critical exponent gamma governing the divergence of the average avalanche size is exactly 1 for all dimensions which has been confirmed by extensive simulations solution of the gap equation yields another universal result rho 1 rho is the exponent of relaxation to attractor scaling relations are established among the critical exponents gamma tau d sigma and nu for the f0 avalanche