topological fractal networks introduced by mixed degree distribution abstract several fundamental properties of real complex networks such as the small world effect the scale free degree distribution and recently discovered topological fractal structure have presented the possibility of a unique growth mechanism and allow for uncovering universal origins of collective behaviors however highly clustered scale free network with power law degree distribution or small world network models with exponential degree distribution are not self similarity we investigate networks growth mechanism of the branching deactivated geographical attachment preference that learned from certain empirical evidence of social behaviors it yields high clustering and spectrums of degree distribution ranging from algebraic to exponential average shortest path length ranging from linear to logarithmic we observe that the present networks fit well with small world graphs and scale free networks in both limit cases exponential and algebraic degree distribution respectively obviously lacking self similar property under a length scale transformation interestingly we find perfect topological fractal structure emerges by a mixture of both algebraic and exponential degree distributions in a wide range of parameter values the results present a reliable connection among small world graphs scale free networks and topological fractal networks and promise a natural way to investigate universal origins of collective behaviors

uncertainty measurement for interval valued information systems abstract interval valued information systems are generalized models of single valued information systems accuracy and roughness are employed to depict the uncertainty of a set under an attribute subset in a pawlak rough set model based on equivalence classes information theoretic measures of uncertainty for rough sets have also been proposed however there are few studies on uncertainty measurements for interval valued information systems this paper addresses the uncertainty measurement problem in interval valued information systems the concept of the similarity degree based on the possible degree is introduced consequently the similarity relation between two interval objects are constructed by a given similarity rate based on the similarity relation similarity classes are defined under this definition accuracy and roughness are given for interval valued information systems which are generalizations of the concepts accuracy and roughness for the equivalence relation based rough set model moreover an alternative uncertainty measure called the rough degree is proposed theoretical studies and numerical experiments show that the proposed measures are effective and suitable for interval valued information systems

growing scale free networks with tunable clustering we extend the standard scale free network model to include a triad formation step we analyze the geometric properties of networks generated by this algorithm both analytically and by numerical c

the similarity of weights on edges and discovering of community structure abstract in this paper we propose a weighted modularity q w based on the similarity of weights on edges and a threshold coefficient to evaluate the equivalence of edge weights simulations on benchmark networks and real networks show that optimization on the modularity enable us to obtain groups of nodes within which the edge weights are distributed uniformly but at random between them the communities can reveal the uniform connections stable relationships measured by the similarity of weights on edges between nodes or some similarity between nodes functions furthermore with the dynamical moving of we observe that optimization on the q w allows for the discovering of a special hierarchical organization which reveals different levels of uniform connections between nodes in networks the substructures revealed by the hierarchical organization enable us to obtain more information of networks and give a potential way for partly remedying the intrinsic resolution problem of modularity

uncertainty measures for rough formulae in rough logic an axiomatic approach rough set theory initiated by pawlak is a mathematical tool in dealing with inexact and incomplete information various types of uncertainty measure such as accuracy measure roughness measure etc which aim to quantify the imprecision of a rough set caused by its boundary region have been extensively studied in the existing literatures however a few of these uncertainty measures are explored from the viewpoint of formal rough set theory which however help to develop a kind of graded reasoning model in the framework of rough logic to solve such a problem a framework of uncertainty measure for formulae in rough logic is presented in this paper unlike the existing literatures we adopt an axiomatic approach to study the uncertainty measure in rough logic concretely we define the notion of rough truth degree by some axioms such a notion is demonstrated to be adequate for measuring the extent to which any formula is roughly true then based on this fundamental notion the notions of rough accuracy degree roughness degree for any formula and rough inclusion degree rough similarity degree between any two formulae are also proposed in addition their properties are investigated in detail these obtained results will be used to develop an approximate reasoning model in the framework of rough logic from the axiomatic viewpoint

comparisons of gaussian elimination algorithms on a cray y mp abstract the results of various implementations of gaussian elimination on full matrices on a single processor cray y mp are presented and discussed it is shown that when the manufacturer supplied blas kernels are used the difference between thebest versions of level 2 blas and level 3 blas blocked implementations is almost negligible it is suggested that to improve the performance of blocked gaussian elimination it is possible to utilize strassen s matrix multiplication algorithm

a topological phase transition between small worlds and fractal scaling in urban railway transportation networks abstract fractal and small worlds scaling laws are applied to study the growth of urban railway transportation networks using total length and total population as observational parameters in spite of the variety of populations and urban structures the variation of the total length of the railway network with the total population of conurbations follows similar patterns for large and middle metropolis diachronous analysis of data for urban transportation networks suggests that there is second order phase transition from small worlds behaviour to fractal scaling during their early stages of development

high performance solution of partial differential equations discretized using a chebyshev spectral collocation method when a chebyshev spectral collocation method is applied to a flow problem in a rectangularly decomposable domain it leads to the solution of a structured linear system since the linear system is solved at each step of a newton type iterative process an efficient method to solve it needs to be provided a level 3 blas based algorithm is presented and its efficiency is studied

systematic studies on three way decisions with interval valued decision theoretic rough sets decision theoretic rough sets dtrs are a representative rough set model the loss function is a pivotal ingredient of dtrs which is associated with the decision maker s evaluation considering the value of loss function with the imprecise evaluation interval valued dtrs ivdtrs and its mechanism in this paper are explored first we construct a basic model of ivdtrs the comparison between dtrs and ivdtrs is discussed in the frame of ivdtrs we then focus on deriving three way decisions with the aid of two conventional methods i e a certain ranking method and a degree of possibility ranking method respectively the certain ranking method converts an interval value into single and derives decision rules under a certain risk attitude of decision maker the degree of possibility ranking method assumes the flexibility of interval and utilizes the preference between interval values all the combinations and their prerequisites are summarized in which we obtain two types of decision rules based on the above analysis we further propose an optimization method for three way decisions with ivdtrs which is designed to minimize the overall uncertainty based on the shannon entropy we also compare these methods based on standard data sets finally the criteria for choosing a suitable method to three way decisions with ivdtrs are generated these results can support decision making in the uncertain environment

reduction target structure based hierarchical attribute reduction for two category decision theoretic rough sets abstract attribute reduction is an essential subject in rough set theory but because of quantitative extension it becomes a problem when considering probabilistic rough set prs approaches the decision theoretic rough set dtrs has a threshold semantics and decision feature and thus becomes a typical and fundamental prs based on reduction target structures this paper investigates hierarchical attribute reduction for a two category dtrs and is divided into five parts 1 the knowledge preservation property and reduct are explored by knowledge coarsening 2 the consistency preservation principle and reduct are constructed by a consistency mechanism 3 region preservation is analyzed and the separability between consistency preservation and region preservation is concluded thus the double preservation principle and reduct are studied 4 structure targets are proposed by knowledge structures and an attribute reduction is further described and simulated thus general reducts are defined to preserve the structure targets or optimal measures 5 the hierarchical relationships of the relevant four targets and reducts are analyzed and a decision table example is provided for illustration this study offers promotion rationality structure hierarchy and generalization and it establishes a fundamental reduction framework for two category dtrs the relevant results also provide some new insights into the attribute reduction problem for prs

double quantitative decision theoretic rough set the probabilistic rough set prs and the graded rough set grs are two quantification models that measure relative and absolute quantitative information between the equivalence class and a basic concept respectively as a special prs model the decision theoretic rough set dtrs mainly utilizes the conditional probability to express relative quantification however it ignores absolute quantitative information of the overlap between equivalence class and the basic set and it cannot reflect the distinctive degrees of information and extremely narrow their applications in real life in order to overcome these defects this paper proposes a framework of double quantitative decision theoretic rough set dq dtrs based on bayesian decision procedure and grs two kinds of dq dtrs model are constructed which essentially indicate the relative and absolute quantification after further studies to discuss decision rules and the inner relationship between these two models we introduce an illustrative case study about the medical diagnosis to interpret and express the theories which is valuable for applying these theories to deal with practical issues

attribute reduction in decision theoretic rough set models rough set theory can be applied to rule induction there are two different types of classification rules positive and boundary rules leading to different decisions and consequences they can be distinguished not only from the syntax measures such as confidence coverage and generality but also the semantic measures such as decision monotocity cost and risk the classification rules can be evaluated locally for each individual rule or globally for a set of rules both the two types of classification rules can be generated from and interpreted by a decision theoretic model which is a probabilistic extension of the pawlak rough set model as an important concept of rough set theory an attribute reduct is a subset of attributes that are jointly sufficient and individually necessary for preserving a particular property of the given information table this paper addresses attribute reduction in decision theoretic rough set models regarding different classification properties such as decision monotocity confidence coverage generality and cost it is important to note that many of these properties can be truthfully reflected by a single measure c in the pawlak rough set model on the other hand they need to be considered separately in probabilistic models a straightforward extension of the c measure is unable to evaluate these properties this study provides a new insight into the problem of attribute reduction

the further investigation of covering based rough sets uncertainty characterization similarity measure and generalized models the notion of rough sets was originally proposed by pawlak in pawlak s rough set theory the equivalence relation or partition plays an important role however the equivalence relation or partition is restrictive for many applications because it can only deal with complete information systems this limits the theory s application to a certain extent therefore covering based rough sets are derived by replacing the partitions of a universe with its coverings this paper focuses on the further investigation of covering based rough sets firstly we discuss the uncertainty of covering in the covering approximation space and show that it can be characterized by rough entropy and the granulation of covering secondly since it is necessary to measure the similarity between covering rough sets in practical applications such as pattern recognition image processing and fuzzy reasoning we present an approach which measures these similarities using a triangular norm we show that in a covering approximation space a triangular norm can induce an inclusion degree and that the similarity measure between covering rough sets can be given according to this triangular norm and inclusion degree thirdly two generalized covering based rough set models are proposed and we employ practical examples to illustrate their applications finally relationships between the proposed covering based rough set models and the existing rough set models are also made

using strassen s matrix multiplication in high performance solution of linear systems abstract performance characteristics of dense and structured blocked linear system solvers are studied when strassen s matrix multiplication is used in the update step results of experiments on a multiprocessor cray y mp are presented and discussed

highly clustered scale free networks we propose a model for growing networks based on a finite memory of the nodes the model shows stylized features of real world networks power law distribution of degree linear preferential attachment of new links and a negative correlation between the age of a node and its link attachment rate notably the degree distribution is conserved even though only the most recently grown part of the network is considered as the network grows the clustering reaches an asymptotic value larger than that for regular lattices of the same average connectivity and similar to the one observed in the networks of movie actors coauthorship in science and word synonyms these highly clustered scale free networks indicate that memory effects are crucial for a correct description of the dynamics of growing networks

rough set models in multigranulation spaces there exist several approaches to rough set approximations in a multigranulation space namely a family of equivalence relations in this paper we propose a unified framework to classify and compare existing studies an underlying principle is to explain rough sets in a multigranulation space through rough sets derived by using individual equivalence relations two basic models are suggested one model is based on a combination of a family of equivalence relations into an equivalence relation and the construction of approximations with respect to the combined relation by combining equivalence relations through set intersection and union respectively we construct two sub models the other model is based on the construction of a family of approximations from a set of equivalence relations and a combination of the family of approximations by using set intersection and union to combine a family of approximations respectively we again build two sub models as a result we have a total of four models we examine these models and give conditions under which some of them become the same

consistency measure inclusion degree and fuzzy measure in decision tables classical consistency degree has some limitations for measuring the consistency of a decision table in which the lower approximation of a target decision is only taken into consideration in this paper we focus on how to measure the consistencies of a target concept and a decision table and the fuzziness of a rough set and a rough decision in rough set theory for three types of decision tables complete incomplete and maximal consistent blocks the membership functions of an object are defined through using the equivalence class tolerance class and maximal consistent blocks including itself respectively based on these membership functions we introduce consistency measures to assess the consistencies of a target set and a decision table and define fuzziness measures to compute the fuzziness of a rough set and a rough decision in these three types of decision tables in addition the relationships among the consistency inclusion degree and fuzzy measure are established as well these results will be helpful for understanding the essence of the uncertainty in decision tables and can be applied for rule extraction and rough classification in practical decision issues

rise and fall of historic tram networks logistic approximation and discontinuous events abstract a logistic approximation was used to describe in terms of total length l and population h variables the growth and decay of historic transportation systems three successive stages separated for sharp discontinuities were detected for several european tramway and metro systems corresponding to a fast initial growth followed by an intermediate step of slow growth and a final stage of rapid decay a common generalized behaviour was obtained in the l h vs h variations relative to critical values of l and h parameters defined from the maximum in the l h ratio

parallel solution of almost block diagonal systems on the cray y mp using level 3 blas abstract in a recent publication 1992 the authors showed how efficient a new level 3 blas algorithm for almost block diagonal systems could be using just one processor of a cray y mp here they compare the corresponding results for up to eight processors using standard cray library parallel implementations of the level 3 blas

two bayesian approaches to rough sets abstract bayesian inference and probabilistic rough sets prss provide two methods for data analysis both of them use probabilities to express uncertainties and knowledge in data and to make inference about data many proposals have been made to combine bayesian inference and rough sets the main objective of this paper is to present a unified framework that enables us a to review and classify bayesian approaches to rough sets b to give proper perspectives of existing studies and c to examine basic ingredients and fundamental issues of bayesian approaches to rough sets by reviewing existing studies we identify two classes of bayesian approaches to prssand three fundamental issues one class is interpreted as bayesian classification rough sets which is built from decision theoretic rough set dtrs models proposed by yao wong and lingras the other class is interpreted as bayesian confirmation rough sets which is built from parameterized rough set models proposed by greco matarazzo and slowinski although the two classes share many similarities in terms of making use of bayes theorem and a pair of thresholds to produce three regions their semantic interpretations and hence intended applications are different the three fundamental issues are the computation and interpretation of thresholds the estimation of required conditional probabilities and the application of derived three regions dtrs models provide an interpretation and a method for computing a pair of thresholds according to bayesian decision theory naive bayesian rough set models give a practical technique for estimating probability based on bayes theorem and inference finally a theory of three way decisions offers a tool for building ternary classifiers the main contribution of the paper lies in weaving together existing results into a coherent study of bayesian approaches to rough sets rather than introducing new specific results

fractal small world dichotomy in real world networks we draw attention to a clear dichotomy between small world networks exhibiting exponential neighborhood growth and fractal like networks where neighborhoods grow according to a power law this distinction is observed in a number of real world networks and is related to the degree correlations and geographical constraints we conclude by pointing out that the status of human social networks in this dichotomy is far from clear

self similar scale free networks and disassortativity self similar networks with scale free degree distribution have recently attracted much attention since these apparently incompatible properties were reconciled in c song s havlin and h a makse nature 433 392 2005 by an appropriate box counting method that enters the measurement of the fractal dimension we study two genetic regulatory networks saccharomyces cerevisiae n m luscombe m m babu h yu m snyder s teichmann and m gerstein nature 431 308 2004 and escherichia coli http www ccg unam mx computationalgenomics regulondb datasets regulonnetdatasets html and http www gbf de systemsbiology and show their self similar and scale free features in extension to the datasets studied by c song s havlin and h a makse nature 433 392 2005 moreover by a number of numerical results we support the conjecture that self similar scale free networks are not assortative from our simulations so far these networks seem to be disassortative instead we also find that the qualitative feature of disassortativity is scale invariant under renormalization but it appears as an intrinsic feature of the renormalization prescription as even assortative networks become disassortative after a sufficient number of renormalization steps

decision region distribution preservation reduction in decision theoretic rough set model abstract in the pawlak rough set model the positive region the boundary region and the non negative region are monotonic with respect to the set inclusion of attributes however the monotonicity property of the decision regions positive region boundary region or non negative region with respect to the set inclusion of attributes does not hold in the decision theoretic rough set model therefore the decision regions may be changed after attribute reduction based on quantitative preservation or qualitative preservation of decision regions this effect is observed partly because three decision regions are defined by introducing the probabilistic threshold values in addition heuristic reduction algorithms based on decision regions may find super reducts because of the non monotonicity of decision regions to address the above issues this paper proposes solutions to the attribute reduction problem based on decision region preservation in the decision theoretic rough set model first the positive region distribution preservation reduct the boundary region distribution preservation reduct and the negative region distribution preservation reduct are introduced into the decision theoretic rough set model second three new monotonic measures are constructed by considering variants of the conditional information entropy from which we can obtain the heuristic reduction algorithms the results of the experimental analysis validate the monotonicity of new measures and verify the effectiveness of decision region distribution preservation reducts

large scale topological and dynamical properties of the internet we study the large scale topological and dynamical properties of real internet maps at the autonomous system level collected in a 3 yr time interval we find that the connectivity structure of the internet presents statistical distributions settled in a well defined stationary state the large scale properties are characterized by a scale free topology consistent with previous observations correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the internet the study of the internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models we compare the properties of growing network models with the present real internet data analysis

quantitative qualitative region change uncertainty certainty in attribute reduction knowledge coarsening is investigated to describe attribute deletion granule merging and its region distribution are used to gain region change functions region change with certainty monotonicity is analyzed in qualitative pawlak model region change with uncertainty drifting is analyzed in quantitative dtrs model attribute reduction is a fundamental research theme in rough sets and granular computing grc its scientific construction originally depends on the region change law at present only region change non monotonicity monotonicity is mined in the quantitative qualitative model the in depth region change truth and its grc mechanism have significance especially for follow up attribute reduction this paper commences probing region change essence mainly from a novel uncertainty certainty viewpoint concretely we make comparative region change analyses based on grc by resorting to the qualitative pawlak model and quantitative dtrs model the decision theoretic rough set model 1 knowledge coarsening is investigated to describe attribute deletion 2 granule merging and its region distribution are studied to probe region change functions 3 region change is analyzed in pawlak model to mine qualitative region change certainty and its relevant properties 4 region change is analyzed in dtrs model to mine quantitative region change uncertainty and its relevant properties 5 comparative region change analyses are summarized and further experiment verification is provided knowledge coarsening and granule merging establish grc mechanisms for extensive region change analyses quantitative qualitative region change uncertainty certainty and relevant principles are discovered via dtrs model pawlak model by virtue of the grc technology and comparative strategy this study reveals region change uncertainty certainty to deepen region change non monotonicity monotonicity furthermore it underlies attribute reduction especially with regard to quantitative models

quantitative information architecture granular computing and rough set models in the double quantitative approximation space of precision and grade because precision and grade act as fundamental quantitative information in approximation space they are used in relative and absolute quantifications respectively at present the double quantification regarding precision and grade is a novel and valuable subject but quantitative information fusion has become a key problem thus this paper constructs the double quantitative approximation space of precision and grade pg approx space and tackles the fusion problem using normal logical operations it further conducts double quantification studies on granular computing and rough set models 1 first for quantitative information organization and storage we construct space and plane forms of pg approx space using the cartesian product and for quantitative information extraction and fusion we establish semantics construction and semantics granules of pg approx space 2 second by granular computing we investigate three primary granular issues quantitative semantics complete system and optimal calculation accordingly six types of fundamental granules are proposed based on the semantic microscopic and macroscopic descriptions their semantics forms structures calculations and relationships are studied and the granular hierarchical structure is achieved 3 finally we investigate rough set models in pg approx space accordingly model regions are proposed by developing the classical regions model expansion is systematically analyzed some models are constructed as their structures are obtained and a concrete model is provided based on the quantitative information architecture this paper systematically conducts and investigates double quantification and establishes a fundamental and general exploration framework

decision theoretic three way approximations of fuzzy sets abstract a three way three valued or three region approximation of a fuzzy set is constructed from a pair of thresholds on the fuzzy membership function an element whose membership grade equals to or is greater than is put into the positive region an element whose membership grade equals to or is less than is put into the negative region and an element whose membership grade is between and is put into the boundary region a fundamental issue is the determination and interpretation of the required pair of thresholds in the framework of shadowed sets i e an example of three way approximations of fuzzy sets pedrycz provides an analytic solution to computing the thresholds by searching for a balance of uncertainty introduced by the three regions to gain further insights into three way approximations of fuzzy sets we introduce an alternative decision theoretic formulation in which the required thresholds are computed by minimizing decision cost

decision theoretic rough fuzzy set model and application this article investigates the rough approximation of a fuzzy concept on a probabilistic approximation space we propose the probabilistic rough fuzzy set by defining the conditional probability of a fuzzy event then we establish the model of probabilistic rough fuzzy set and discuss several properties in detail furthermore three generalizations of probabilistic rough fuzzy set namely 0 5 probabilistic rough fuzzy set variable precision probabilistic rough fuzzy set and bayesian rough fuzzy set are reported in order to give a systematic method of selecting parameters for the probabilistic rough fuzzy set we propose a decision theoretic rough fuzzy set that is we formulate a non parametric definition of the probabilistic rough fuzzy set moreover we illustrate the motivation and verify the validity of the decision theoretic rough fuzzy set by using a credit card applicant decision making problem furthermore the interrelationship between the decision theoretic rough fuzzy set and the probabilistic rough fuzzy set is explained the main contribution of this paper is twofold one is to extend the probabilistic rough set to fuzzy environment i e the probabilistic rough fuzzy set model another is to present an approach to select parameters needed in probabilistic rough fuzzy set modeling by using the process of decision making under conditions of risk

topology and correlations in structured scale free networks we study a recently introduced class of scale free networks showing a high clustering coefficient and nontrivial connectivity correlations we find that the connectivity probability distribution strongly depends on the fine details of the model we solve exactly the case of low average connectivity providing also exact expressions for the clustering and degree correlation functions the model also exhibits a lack of small world properties in the whole parameter range we discuss the physical properties of these networks in the light of the present detailed analysis

the superiority of three way decisions in probabilistic rough set models three way decisions provide a means for trading off different types of classification error in order to obtain a minimum cost ternary classifier this paper compares probabilistic three way decisions probabilistic two way decisions and qualitative three way decisions of the standard rough set model it is shown that under certain conditions when considering the costs of different types of miss classifications probabilistic three way decisions are superior to the other two

modeling rough granular computing based on approximation spaces the results reported in this paper create a step toward the rough set based foundations of data mining and machine learning the approach is based on calculi of approximation spaces in this paper we present the summarization and extension of our results obtained since 2003 when we started investigations on foundations of approximation of partially defined concepts see e g 2 3 7 37 20 21 5 42 39 38 40 we discuss some important issues for modeling granular computations aimed at inducing compound granules relevant for solving problems such as approximation of complex concepts or selecting relevant actions plans for reaching target goals the problems discussed in this article are crucial for building computer systems that assist researchers in scientific discoveries in many areas such as biology in this paper we present foundations for modeling of granular computations inside of system that is based on granules called approximation spaces our approach is based on the rough set approach introduced by pawlak 24 25 approximation spaces are fundamental granules used in searching for relevant complex granules called as data models e g approximations of complex concepts functions or relations in particular we discuss some issues that are related to generalizations of the approximation space introduced in 33 34 we present examples of rough set based strategies for the extension of approximation spaces from samples of objects onto a whole universe of objects this makes it possible to present foundations for inducing data models such as approximations of concepts or classifications analogous to the approaches for inducing different types of classifiers known in machine learning and data mining searching for relevant approximation spaces and data models are formulated as complex optimization problems the proposed interactive granular computing systems should be equipped with efficient heuristics that support searching for semi optimal granules

approaches to knowledge reduction based on variable precision rough set model this paper deals with approaches to knowledge reduction based on variable precision rough set model the concepts of lower distribution reduct and upper distribution reduct based on variable precision rough sets vprs are first introduced their equivalent definitions are then given and the relationships among lower and upper distribution reducts and alternative types of knowledge reduction in inconsistent systems are investigated it is proved that for some special thresholds lower distribution reduct is equivalent to the maximum distribution reduct whereas upper distribution reduct is equivalent to the possible reduct the judgement theorems and discernibility matrices associated with the lower and upper distribution reducts are also established from which we can obtain the approaches to knowledge reduction in vprs