dual fuzzy neighbourhood spaces i abstract a fuzzy neighbourhood space fns is determined by means of a system of fuzzy neighbourhoods of the ordinary points of that space we provide a general method for constructing bases for all fuzzy neighbourhood systems in a natural way from the fuzzy topologies of a special type of fuzzy topological spaces fts s which we call the fuzzy neighbourhood base spaces this construction is shown to be functorial of those new fts s the full subcategory of dual fuzzy neighbourhood spaces is shown to be isomorphic to the category of fns s and a fns will coincide with its dual fts iff it is topologically generated

probabilistic metrization of fuzzy uniformities abstract modifying the structure of fuzzy uniform spaces introduced by lowen 5 we propose a further concept of fuzzy uniformities so called fuzzy t

predictable nonwandering localization of covariant lyapunov vectors and cluster synchronization in scale free networks of chaotic maps covariant lyapunov vectors for scale free networks of henon maps are highly localized we revealed two mechanisms of the localization related to full and phase cluster synchronization of network nodes in both cases the localization nodes remain unaltered in the course of the dynamics i e the localization is nonwandering moreover this is predictable the localization nodes are found to have specific dynamical and topological properties and they can be found without computing of the covariant vectors this is an example of explicit relations between the system topology its phase space dynamics and the associated tangent space dynamics of covariant lyapunov vectors

on regularity axioms in fuzzy neighbourhood spaces abstract two notions of regularity in fuzzy neighbourhood spaces fns s have been introduced one by t m g ahsanullah and the other by the authors we prove that those two notions coincide we then characterize a regular fns as a fns whose level topologies for all 0

the categorical topology approach to fuzzy topology and fuzzy convergence it is the purpose of this paper to have a look at fuzzy topology from the point of view of categorical topology we first look at interesting subcategories of fts and determine whether they are bireflective and or coreflective we also describe the set of all subcategories of fts which are at the same time bireflective and coreflective due to the fact that top is one of these subcategories of fts ffs itself can be neither cartesian closed nor extensional we therefore secondly construct a supercategory of bts which is a topological quasitopos the construction of this quasitopos is based on the theory of convergence in fts

fuzzy addition in the l fuzzy real line abstract under the hypothesis l is a chain we construct a binary operation on the l fuzzy real line r l which reduces to the usual addition on r if is restricted to the embedded image of r in r l which yields a partially ordered abelian cancellation semigroup with identity and which is jointly fuzzy continuous on r l we show is unique i e it is the only extension of addition to r l which is consistent we study the relationship between and other fuzzy continuous extensions of the usual addition we also show that fuzzy translation is a weak fuzzy homeomorphism and under certain conditions a fuzzy homeomorphism

on r l o we prove there exists a somewhat simpler description of l and its fuzzy topology than the one given by hutton 2 and a considerably simpler description of than the one given by rodabaugh 4 as a result of this simplification we are able to give a direct proof of the continuity of an open problem in 4

on fuzzy pseudo normed vector spaces abstract we provide a method for introducing fuzzy pseudo metric topologies on sets and fuzzy pseudo normed topologies on vector spaces over r or c which will be fuzzy linear topologies we define fuzzy pseudo metrics for pairs of crisp points and fuzzy pseudo norms for crisp points as fuzzy real numbers 0 as defined by hutton we define the associated fuzzy open balls with crisp points for their centres and fuzzy real numbers 0 for their radii these form a basis for the associated fuzzy pseudo metric topology the axioms governing fuzzy pseudo metric and fuzzy pseudo norm are straightforward extensions for the corresponding axioms in the crisp case the formulation conforms with zadeh s extension principle we show that katsaras concept of fuzzy seminorm fuzzy sets and systems 12 1984 143 154 is equivalent to ours in the sense that both concepts will result in same fuzzy linear topologies

some properties of fuzzy neighborhood normed linear spaces abstract the aim of this paper is to investigate some properties relating to continuity uniform continuity and boundedness in fuzzy neighborhood normed linear spaces

on fuzzy neighbourhood spaces abstract we study the behaviour of the fuzzy closure operator of a fuzzy neighbourhood space fns for short with regard to the two operations of sum and conjunction on fuzzy subsets we supply sufficient conditions on a cech fuzzy closure operator to be the fuzzy closure operator of a fns finally we study quotient fuzzy neighbourhood systems with the help of the notion of dual fuzzy neighbourhood spaces of 11

fuzzy metric neighbourhood spaces abstract hohle founded the first coherent topological study of menger s probabilistic metric spaces by endowing them with fuzzy t uniformities we continue hohle s work in the special case when the triangular norm t used is min that is when the resulting probabilistic metric uniformities are lowen fuzzy uniformities in this case we call the induced fuzzy topological spaces fuzzy pseudo metric neighbourhood spaces we introduce axioms of n regularity and n normality which are satisfied in fuzzy pseudo metric neighbourhood spaces we introduced a lowen fuzzy uniform topology on the fuzzy real line ri called the n euclidean topology by means of a fuzzy probabilistic metric on r i called the axiom of n complete regularity which characterizes the lowen fuzzy uniform topologies

power law relations in random networks with communities most random graph models are locally tree like do not contain short cycles rendering them unfit for modeling networks with a community structure we introduce the hierarchical configuration model hcm a generalization of the configuration model that includes community structures while properties such as the size of the giant component and the size of the giant percolating cluster under bond percolation can still be derived analytically viewing real world networks as realizations of hcm we observe two previously undiscovered power law relations between the number of edges inside a community and the community sizes and between the number of edges going out of a community and the community sizes we also relate the power law exponent of the degree distribution with the power law exponent of the community size distribution in the case of extremely dense communities e g complete graphs this relation takes the simple form 1

the asymptotic degree distribution of a kind of evolving random tree in this paper we introduce an evolving random tree the model is a combination of a preferential attachment model and a uniform model we show that the distribution of the number of leaves at time n is approximated by the poisson distribution with parameter the proportional degree sequence obeys a power law with parameter and is a constant number that describes the probability of choosing the connecting form in the evolving random tree model

geometrical structure of the neuronal network of caenorhabditis elegans the neuronal network of the soil nematode caenorhabditis elegans c elegans which is a good prototype for biological studies is investigated here the neuronal network is simpli4ed as a graph we use three indicators to characterize the graph vertex degree generalized eccentricity ge and complete subgraphs the graph has the central part and the strong clustering structure we present a simple model which shows that the neuronal network has a high dimensional geometrical structure c

stationary and nonstationary properties of evolving networks with preferential linkage networks evolving by preferential attachment of both external and internal links are investigated the rate of adding an external link is assumed to depend linearly on the degree of a preexisting node to which a new node is connected the process of creating an internal link between a pair of existing vertices is assumed to be controlled entirely by the vertex that has more links than the other vertex in the pair and the rate of creation of such a link is assumed to be in general nonlinear in the degree of the more strongly connected vertex it is shown that degree distributions of networks evolving only by creating internal links display for large degrees a nonstationary power law decay with a time dependent scaling exponent nonstationary power law behaviors are numerically shown to persist even when the number of nodes is not fixed and both external and internal connections are introduced provided that the rate of preferential attachment of internal connections is nonlinear it is argued that nonstationary effects are not unlikely in real networks although these effects may not be apparent especially in networks with a slowly varying mean degree

trees networks and hydrology this paper reviews theoretical and observational material on form and function of natural networks appeared in somewhat disparate contexts from physics to biology whose study is related to hydrologic research moving from the exact result that drainage network configurations minimizing total energy dissipation are stationary solutions of the general equation describing landscape evolution we discuss the properties and the dynamic origin of the scale invariant structure of river patterns and its relation to optimal selection we argue that at least in the fluvial landscape nature works through imperfect searches for dynamically accessible optimal configurations and that purely random or deterministic constructs are clearly unsuitable to properly describe natural network forms we also show that optimal networks are spanning loopless configurations only under precise physical requirements that arise under the constraints imposed by continuity in the case of rivers every spanning tree proves a local minimum of total energy dissipation this is stated in a theorem form applicable to generic networks suggesting that other branching structures occurring in nature e g scale free and looping may possibly arise through optimality to different selective pressures we thus conclude that one recurrent self organized mechanism for the dynamic origin of fractal forms is the robust strive for imperfect optimality that we see embedded in many natural patterns chief and foremost hydrologic ones copyright 2006 by the american geophysical union

semi induced fuzzy topologies abstract induced fuzzy topologies were first introduced by wong 15 and weiss 14 they were studied by lowen who called them topologically generated fuzzy topologies 7 an induced fuzzy topology includes its modification topology we generalize induced fuzzy topologies to semi induced fuzzy topologies viz those which include their modification topologies the yet more general fuzzy topologies of structure c are also introduced the fully stratified fuzzy topological spaces of structure c are precisely the fuzzy neighbourhood spaces of lowen 4 in subsequent papers this relationship will be established and benefited from the two axioms of semi induction and structure c hold trivially for crisp fuzzy topologies we call such axioms fuzzy structural axioms in fuzzy topology in this paper we analyze the axioms of structure c semi induction and induction in this context we show how these axioms are linked to the three structural asioms of structures a b and d introduced by us in 11 the basic notions and results pertaining to structures a b and d are compiled in the first section the behaviour of warren s fuzzy boundary in semi induced fuzzy topological spaces fts s for short is also explained

maximal planar networks with large clustering coefficient and power law degree distribution in this article we propose a simple rule that generates scale free networks with very large clustering coefficient and very small average distance these networks are called random apollonian networks rans as they can be considered as a variation of apollonian networks we obtain the analytic results of power law exponent ensuremath gamma 3 and clustering coefficient c frac 46 3 ensuremath 36 phantom rule 0 2em 0ex mathrm ln phantom rule 0 2em 0ex frac 3 2 ensuremath approx 0 74 which agree with the simulation results very well we prove that the increasing tendency of average distance of rans is a little slower than the logarithm of the number of nodes in rans since most real life networks are both scale free and small world networks rans may perform well in mimicking the reality the rans possess hierarchical structure as c k ensuremath sim k ensuremath 1 that are in accord with the observations of many real life networks in addition we prove that rans are maximal planar networks which are of particular practicability for layout of printed circuits and so on the percolation and epidemic spreading process are also studied and the comparisons between rans and barab asi albert ba as well as newman watts nw networks are shown we find that when the network order n the total number of nodes is relatively small as n ensuremath sim 10 4 the performance of rans under intentional attack is not sensitive to n while that of ba networks is much affected by n and the diseases spread slower in rans than ba networks in the early stage of the suseptible infected process indicating that the large clustering coefficient may slow the spreading velocity especially in the outbreaks

linear fuzzy neighborhood spaces some of the properties of linear fuzzy neighborhood spaces are studied a special class of these spaces is the family of all fuzzy linear spaces whose topology is given by probabilistic seminorms as in the classical case it is shown that a linear fuzzy neighborhood space is probabilistic normable iff it is hausdorff and it has an n bounded convex fuzzy neighborhood of zero

nearness concepts in fuzzy neighbourhood spaces and in their fuzzy proximity spaces abstract the nearness concept giving rise to a fuzzy topological space is a fuzzy relation of closeness between fuzzy subsets and crisp points of the space lowen 8 introduced the category of fuzzy neighbourhood spaces which we denote by fns his heuristic idea was that the fuzzy topologies in fns should be characterized by means of fuzzy neighbourhoods of crisp points it may be argued that the fuzziness of some concepts related to fns is rather restricted but besides the benefits gained from this restriction in that fns is a category of well behaved fuzzy topological spaces cf 1 5 8 9 14 17 18 we argue in this paper that this restriction acts to reduce the above mentioned concept of nearness to the following one the nearness concept giving rise to a fuzzy neighbourhood space is a fuzzy relation of closeness between the crisp subsets and the crisp points of the space such fuzzy relation may naturally arise when fuzziness is engrained in the definition of the points of the space for instance in the fuzzy unit interval of hutton 4 artico and moresco 2 introduced a concept of fuzzy proximities we show that this concept is particular to fuzzy neighbourhood spaces only we then argue that the nearness concept giving rise to a fuzzy proximity space is a fuzzy relation of proximity defined on all pairs of crisp subsets of the space

on the determination of fuzzy topological spaces and fuzzy neighbourhood spaces by their level topologies abstract in this paper we discuss the problem of the reconstruction of a fuzzy topological space or a fuzzy neighbourhood space from an a priori given family of level topologies necessary and sufficient conditions for the existence of a solution are given and it is proved that in the particular case of fuzzy neighbourhood spaces this solution is always unique

fuzzy t locality spaces abstract a lowen fuzzy topological space is said to be localizable at points if it is completely determined in some prescribed manner by a system of fuzzy neighbourhoods called here fuzzy localities of its ordinary points we introduce for each lower semicontinuous triangular norm t a new type of those spaces which we call fuzzy t locality spaces the particularity of a fuzzy t locality space is that its fuzzy interior operator is obtained from its t locality system through a definition of fuzzy containment of fuzzy sets that employs the t residuated implication operator the origins of this new theory lie in the theory of probabilistic metric spaces indeed fuzzy t locality spaces provide a fuzzy topological framework for a naturally defined notion of metric t continuity of functions between probabilistic t metric spaces in a manner described in the last two sections of this article before that we establish characterizations and aspects of good behaviour for fuzzy t locality spaces including that every t locality system has an open basis we briefly study their level topologies and we show that all topologically generated spaces are fuzzy t locality spaces we prove that the category t fls of fuzzy t locality spaces and continuous functions between them is a topological category we show that it is closed in fts under the formation of optimal lifts of sources we provide a condition on t under which t fls becomes closed in fts also under the formation of co optimal lifts of sinks

a markov chain based numerical method for calculating network degree distributions this paper establishes a relation between scale free networks and markov chains and proposes a computation framework for degree distributions of scale free networks we first find that under the ba model the degree evolution of individual nodes in a scale free network follows some non homogeneous markov chains exploring the special structure of these markov chains we are able to develop an efficient algorithm to compute the degree distribution numerically the complexity of our algorithm is o t 2 where t is the number of time steps for adding new nodes we use three examples to demonstrate the computation procedure and compare the results with those from the existing methods pacs 84 35 i 64 60 fr 87 23 ge

mean field level analysis of epidemics in directed networks the susceptible infected removed spreading model in a directed graph is studied the mean field level rate equations are built with the degree degree connectivity correlation element and the in out degree distribution and the outbreak threshold is obtained analytically it is determined by the combination of connectivity probability and the degree distribution furthermore the methods of calculating the degree degree correlations in directed networks are presented the numerical results of the discrete epidemic processes in networks verify our analyses

complex scale free networks with tunable power law exponent and clustering abstract we introduce a network evolution process motivated by the network of citations in the scientific literature in each iteration of the process a node is born and directed links are created from the new node to a set of target nodes already in the network this set includes m ambassador nodes and l of each ambassador s descendants where m and l are random variables selected from any choice of distributions p l and q m the process mimics the tendency of authors to cite varying numbers of papers included in the bibliographies of the other papers they cite we show that the degree distributions of the networks generated after a large number of iterations are scale free and derive an expression for the power law exponent in a particular case of the model where the number of ambassadors is always the constant m and the number of selected descendants from each ambassador is the constant l the power law exponent is 2 l 1 l for this example we derive expressions for the degree distribution and clustering coefficient in terms of l and m we conclude that the proposed model can be tuned to have the same power law exponent and clustering coefficient of a broad range of the scale free distributions that have been studied empirically

fuzzy topologies with specified level properties abstract recently the author and p wuyts have studied the problem of defining fuzzy topologies when one is given the family of closure operators or the family of level topologies this paper is concerned with the relationship between these two problems when the family of topologies is a chain with the lt property there is a natural family of closure operators that generates the wuyts maximum i e the largest fuzzy topology with the given level topologies as a corollary it is shown that the fuzzy neighborhood spaces are precisely the pseudo regular spaces in general the relationship between the two problems is weak for example given an lt family of topologies and an l ftp family of semi closure operators there is a fuzzy topology associated with both if and only if the 0 level closure generates the 0 level topology and the closure of a set is always closed in the level topology lastly it is shown that when a function s domain has a wuyts maximum as its topology fuzzy continuity is equivalent to level continuity

a fuzzy r 1 separation axiom abstract we introduce a new r 1 separation axiom for fuzzy topological spaces we identify from the literature which lower fuzzy separation axioms are compatible with our r 1 axiom we provide alternative descriptions for this r 1 axiom in the special setting of fuzzy neighborhood spaces this special case has standard higher fuzzy separation axioms in it we establish aspects of compatibility of our r 1 axiom with those higher separation axioms as well as with fuzzy compactness

invariant probabilistic metrizability of fuzzy neighbourhood groups we introduce the notions of n local compactness and n openess in fuzzy neighbourhood spaces and in fuzzy neighbourhood groups together with some characterizations for them in particular we prove that a fuzzy neighbourhood space is n locally compact if and only if its level spaces for all 0 1 are locally compact a similar criterion is established for n open functions we also introduce the notion of fuzzy absolute value functions on groups we show that there exists a 1 1 correspondence between invariant fuzzy probabilistic pseudo metrics on groups and fuzzy absolute value functions we establish theorems on fuzzy probabilistic pseudo metrizability of fuzzy neighbourhood groups finally fuzzy probabilistic metrizability of quotient fuzzy neighbourhood groups is also taken into account

a weighted network evolution with traffic flow in this paper we propose a simple weighted network model that generalizes the complex network model evolution with traffic flow previously presented to investigate the relationship between traffic flow and network structure in the model the nodes in the network are represented by the traffic flow states the links in the network are represented by the transform of the traffic flow states and the traffic flow transported when performing the transform of the traffic flow states is considered as the weight of the link several topological features of this generalized weighted model such as the degree distribution and strength distribution have been numerically studied a scaling behavior between the strength and degree s klogks klogk is obtained by introducing some constraints to the generalized weighted model we study its subnetworks and find that the scaling behavior between the strength and degree is conserved though the topology properties are quite sensitive to the constraints

reachability and recoverability of sink nodes in growing acyclic directed networks abstract we study the growth of networks from a set of isolated ground nodes by the addition of one new node per time step and also of a fixed number of directed edges leading from the new node to randomly selected nodes already in the network a fixed width time window is used so that in general only nodes that entered the network within the latest window may receive new incoming edges the resulting directed network is acyclic at all times and allows some of the ground nodes then called sinks to be reached from some of the non ground nodes we regard such networks as representative of abstract systems of partially ordered constituents for example in some of the domains related to technological evolution two properties of interest are the number of sinks that can be reached from a randomly chosen non ground node its reach and for a fixed sink the number of nonoverlapping directed paths through which the sink can be reached at a given time from some of the latest nodes to have entered the network we demonstrate by means of simulations and also of analytic characterizations that reaches are distributed according to a power law and that the desired directed paths are expected to occur in very small numbers perhaps indicating that recovering sinks late in the process of network growth is strongly sensitive to accidental path disruptions

continuity of fuzzy multiplication in the n euclidean space abstract the n euclidean space m i is a fuzzy normed vector space whose underlying set is the smallest real vector space including all nonnegative fuzzy real numbers the fuzzy multiplication on the fuzzy real line easily extends to a multiplication on m i we show that under that multiplication m i is a fuzzy normed algebra in consequence fuzzy multiplication is shown to be continuous in the n euclidean topology on m i

on product fuzzy topological spaces abstract we provide complete answers to the following three questions in fuzzy topology when is the fuzzy closure operator finitely productive on a given family of fuzzy topological spaces which need not be fully stratified when is the fuzzy interior operator finitely productive on such a family when are the coordinate projections open maps from arbitrary products of members of such a family the answer to the first question is the harder one to establish the second and the third question turn out to have same answer

nonlinear barabasi albert network abstract in recent years there has been considerable interest in the structure and dynamics of complex networks one of the most studied networks is the linear barabasi albert model here we investigate the nonlinear barabasi albert growing network in this model a new node connects to a vertex of degree k with a probability proportional to k real each vertex adds m new edges to the network we derive an analytic expression for the degree distribution p k which is valid for all values of m and 1 in the limit the network is homogeneous if 1 there is a gel phase with m super connected nodes it is proposed a formula for the clustering coefficient which is in good agreement with numerical simulations the assortativity coefficient r is determined and it is shown that the nonlinear barabasi albert network is assortative disassortative if 1 and no assortative only when 1 in the limit the assortativity coefficient can be exactly calculated we find r 7 13 when m 2 finally the minimum average shortest path length l min is numerically evaluated increasing the network size l min diverges for 1 and it is equal to 1 when 1

relationship between the in degree and out degree of www abstract in this paper the relationship between the in degree and out degree of world wide web is studies at each time step a new node with out degree k out is added where k out obeys the power law distribution and its mean value is m the analytical and simulation results suggest that the exponent of in degree distribution would be i 2 1 m depending on the average out degree this finding is supported by the empirical data which has not been emphasized by the previous studies on directed networks

scale free properties of weighted networks with connectivity driven topology abstract the rate equations are used to study the scale free behavior of the weight distribution in evolving networks whose topology is determined only by degrees of preexisting vertices an analysis of these equations shows that the degree distribution and thereby the weight distribution remain unchanged when the probability rate of attaching new nodes is replaced with an unnormalized rate determined by the ratio of the degree of a randomly selected old node to the maximal node degree at the current stage of the network evolution such a modification of the attachment rule is argued to accelerate considerable numerical simulations of both unweighted and weighted networks belonging to the class of investigated evolving systems it is also proved that the studied rate equations have a solution corresponding to the total weight concentrated at individual vertices distribution displaying the power law behavior for asymptotically large weights

on separation axioms in fuzzy topological spaces fuzzy neighborhood spaces and fuzzy uniform spaces submitted by l zadeh a certain number of separation axioms for fuzzy topological spaces are provided all of which are good extensions of the topological t t or t2 all valid implications between the different axioms are studied and counterexamples are given for the nonvalid ones in this paper we introduce and study a number of separation properties of fuzzy topological fuzzy neighborhood and fuzzy uniform spaces all of which are extensions of t t and t in topological spaces we started from the viewpoint that separation axioms at least t originated in the need to guarantee in some sense uniqueness of limits and we tried to generalize the topological t axiom in that direction it appears that a whole range of axioms from a fairly weak condition wt up to a rather strong one tj all give rise to acceptable extensions of the topological property moreover it seems likely that it will not be possible to single out one axiom in fuzzy topology as the best extension of topological t2 we showed in a preceding paper 181 on completions of fuzzy uniform spaces that tj as well as wt collapsing with wt in uniform spaces have a specific role to play in the same sense wt has already shown its usefulness in the study of hyperspaces 131 and t2 in its relation with compactness 191 as to our other axioms from t to wt they all seem to be natural generalizations of the corresponding properties in topological spaces and of the different t notions in fuzzy topological spaces they are moreover hereditary and t and ty excepted also productive as to the relations with other work in this field we first point out that our wt axiom is in fact the 1 hausforff separation property of hiihle 2 31 hutton and reilly in 4 and rodabaugh in 161 also studied separation current address is rijksuniversitair centrum 2020 antwerpen belgium

a level topologies criterion for lowen fuzzy uniformizability throughout x will be a nonempty set we consider a subset m of x as a crisp fuzzy set and denote this again by m we denote the strong cut of a fuzzy set e ix by and its weak cut by e i

compact probabilistic metrics on bounded closed intervals of distribution functions a menger space a special type of probabilistic metric spaces is said to be compact if its strong uniformity is compact we construct in a natural way menger t metrics on the set of distribution functions one for each copula t a special type of continuous t norms we show that on each bounded closed interval of distribution functions the strong uniformities of all our spaces are induced by the modified levy metric hence they are compact we establish an alternative description and a number of good properties of our menger spaces which may render them well behaved examples for workers in the field

fuzzy neighborhood syntopogenous structures abstract we introduce the notion of fuzzy neighbourhood syntopogenous structures as a unified theory of fuzzy neighbourhood spaces lowen fuzzy uniformities and artico moresco fuzzy proximities our primary idea is that a fuzzy neighbourhood syntopogenous structure on a set x consists of fuzzy neighbourhood of the ordinary subsets of x

new minimum distance bounds for certain binary linear codes let an n k d code denote a binary linear code of length n dimension k and minimum distance at least d define d n k as the maximum value of d for which there exists a binary linear n k d code t verhoeff 1989 has provided an updated table of bounds on d n k for 1

completeness of the standard fuzzy real line for fns abstract we consider a special class of fuzzy metric neighbourhood spaces where the fuzzy distance is induced by a fuzzy pseudo norm this class is contained in lowen s well known category fns of fuzzy neighbourhood spaces we define the fuzzy limit of a sequence a fuzzy cauchy sequence and fuzzy convergence in fuzzy metric neighbourhood spaces we prove that the set of all nonnegative fuzzy real numbers r i under the n euclidean metric the standard fuzzy real line for fns is complete but that the smallest fuzzy neighbourhood real algebra m i including r i is not complete we also characterize the fuzzy cauchy sequences in m i in terms of their corresponding real sequences of cuts

fuzzy inner product spaces abstract the n euclidean space m i is a fuzzy pseudo normed algebra whose underlying set is the smallest real vector space including all nonnegative fuzzy real numbers we introduce axioms for a fuzzy inner product on a unitary m i module with values in m i we establish the fuzzy schwarz inequality for fuzzy inner products we introduce the cartesian product of fuzzy inner product spaces and investigate the relations between their underlying topologies we conclude the research by supplying two interesting examples of fuzzy inner product spaces

clique relaxations in social network analysis the maximum k plex problem this paper introduces and studies the maximum k plex problem which arises in social network analysis and has wider applicability in several important areas employing graph based data mining after establishing np completeness of the decision version of the problem on arbitrary graphs an integer programming formulation is presented followed by a polyhedral study to identify combinatorial valid inequalities and facets a branch and cut algorithm is implemented and tested on proposed benchmark instances an algorithmic approach is developed exploiting the graph theoretic properties of a k plex that is effective in solving the problem to optimality on very large sparse graphs such as the power law graphs frequently encountered in the applications of interest

fuzzy proximities induced by functional fuzzy separation abstract mashhour and morsi introduced an axiom of n complete regularity to characterize the lowen fuzzy uniformizable fuzzy neighbourhood spaces fns s guided by that axiom we supply a sierpinski object for the artico moresco category fp of fuzzy proximity spaces which katsaras has shown to be a topological category we define the degree of functional fuzzy separatedness of a pair of subsets of a fns we use that notion to define the cech fuzzy proximity of an n completely regular fns x we establish that it is the finest fuzzy proximity that induces the fns x we study in some detail the cech fuzzy proximities obtained from n normal fns s

fuzzy uniformities induced by fuzzy proximities abstract given a fuzzy proximity we construct a fuzzy uniformity in the sense of lowen which turns out to be the coarse stone among those which induce we show that some basic theorems of general topology about precompact uniformities have a nice extension to fuzzy set topology

bounded linear transformations between probabilistic normed vector spaces we denote by r i the set of all nonnegative fuzzy real numbers that is functions n from r 0 to the unit interval i which are left continuous and nonascending and satisfy 0 1 and 0 this set r i canonically includes r in the obvious way it is ordered by the usual pointwise order of real functions

pseudo closure operators in fuzzy neighbourhood base spaces abstract fuzzy neighbourhood base spaces and dual fuzzy neighbourhood spaces are two special types of fuzzy topological spaces they were introduced by morsi as useful tools for the theory explanation and application of lowen s fuzzy neighbourhood systems the notion of the pseudoclosure operator of a fuzzy topological space was introduced by mashhour et al as a tool in establishing a necessary and sufficient condition for the productivity of fuzzy kuratowski operators pseudo closure operators carry partial information about the fuzzy closure operators of their respective spaces we establish relationships between the pseudo closure operators of fuzzy neighbourhood base spaces their fuzzy closure operators and their level topologies we provide counterexamples to disprove other nonvalid relationships we also study some categorical aspects of the operation of taking the pseudo closure operators of fuzzy topological spaces

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved

synchronized clusters in coupled map networks i numerical studies we study the synchronization of coupled maps on a variety of networks including regular one and two dimensional networks scale free networks small world networks tree networks and random networks for small coupling strengths nodes show turbulent behavior but form phase synchronized clusters as coupling increases when nodes show synchronized behavior we observe two interesting phenomena first there are some nodes of the floating type that show intermittent behavior between getting attached to some clusters and evolving independently second we identify two different ways of cluster formation namely self organized clusters which have mostly intracluster couplings and driven clusters which have mostly intercluster couplings the synchronized clusters may be of dominant self organized type dominant driven type or mixed type depending on the type of network and the parameters of the dynamics we define different states of the coupled dynamics by considering the number and type of synchronized clusters for the local dynamics governed by the logistic map we study the phase diagram in the plane of the coupling constant and the logistic map parameter ensuremath mu for large coupling strengths and nonlinear coupling we find that the scale free networks and the caley tree networks lead to better cluster formation than the other types of networks with the same average connectivity for most of our study we use the number of connections of the order of the number of nodes as the number of connections increases the number of nodes forming clusters and the size of the clusters in general increase

optimization of the robustness of multimodal networks dirac s delta function x we refer to this type of network as a scale free multimodal network for m 2 the network is a bimodal network in the limit m approaches infinity the network models a scale free network we calculate and optimize the robustness for given values of the number of modes m the total number of nodes n and the average degree k using analytical formulas for the random and targeted node removal thresholds for network collapse we find when n 1 that i the robustness against random and targeted node removal for this multimodal network is controlled by a single combination of variables n 1 m 1 ii the robustness of the multimodal network against targeted node removal decreases rapidly when the number of modes becomes larger than a critical value that is of the order of ln n and iii the values of exponent opt that characterizes the scale free degree distribution of the multimodal network that maximize the robustness against both random and targeted node removal fall between 2 5 and 3

detecting link failures in complex network processes using remote monitoring abstract we study whether local structural changes in a complex network can be distinguished from passive remote time course measurements of the network s dynamics specifically the detection of link failures in a network synchronization process from noisy measurements at a single network component is considered by phrasing the detection task as a maximum a posteriori probability hypothesis testing problem we are able to obtain conditions under which the detection is 1 improved over the a priori and 2 asymptotically perfect in terms of the network spectrum and graph we find that in the case where the detector has knowledge of the network s state perfect detection is possible under general connectivity conditions regardless of the measurement location when the detector does not have state knowledge a remote signature permits improved but not perfect detection under the same connectivity conditions at its essence detectability is achieved because of the close connection between a network s topology its eigenvalues and local response characteristics

iterative decomposition of barabasi albert scale free networks we study a decomposition process where all nodes with a targeted degree are removed from the network each removal step results in changes in the degrees of the remaining nodes and other nodes may attain the targeted degree the processes continue iteratively until no more nodes with the targeted degree are present in the decomposed network the network model used in our study is the well known barabasi albert network that is built with an iterative growth based on preferential attachment our results show an exponential decay of the number of nodes removed at each step the total number of nodes removed in the whole process depends on the targeted degree and decay with a power law controlled by the same exponent as the degree distribution of the network

time evolution of the degree distribution of model a of random attachment growing networks abstract for random growing networks barabas and albert proposed a kind of model in barabas et al physica a 272 1999 173 i e model a in this paper for model a we give the differential format of master equation of degree distribution and obtain its analytical solution the obtained result p k t is the time evolution of degree distribution p k t is composed of two terms at given finite time one term decays exponentially the other reflects size effect at infinite time the degree distribution is the same as that of barabas and albert in this paper we also discuss the normalization of degree distribution p k t in detail

scale free networks with exponent one a majority of studied models for scale free networks have degree distributions with exponents greater than two real networks however can demonstrate essentially more heavy tailed degree distributions we explore two models of scale free equilibrium networks that have the degree distribution exponent ensuremath gamma 1 p q ensuremath sim q ensuremath ensuremath gamma such degree distributions can be identified in empirical data only if the mean degree of a network is sufficiently high our models exploit a rewiring mechanism they are local in the sense that no knowledge of the network structure apart from the immediate neighborhood of the vertices is required these models generate uncorrelated networks in the infinite size limit where they are solved explicitly we investigate finite size effects by the use of simulations we find that both models exhibit disassortative degree degree correlations for finite network sizes in addition we observe a markedly degree dependent clustering in the finite networks we indicate a real world network with a similar degree distribution

fuzzy proximities compatible with lowen fuzzy uniformities abstract we introduce a notion of fuzzy proximity which behaves well with respect to the fuzzy uniformities introduced by r lowen and prove some related results we also provide some critical remarks about fuzzy topologies fuzzy proximities and fuzzy uniformities

the urysohn lemma for fuzzy neighbourhood spaces abstract the framework fns of lowen s fuzzy neighbourhood spaces is a special environment for the study of fuzzy topology it encompasses its own metrics uniformities and proximities among other interesting constructions this is a fresh and fast growing field of research it promises to develop quickly to a calculus which combines depth with applicability within this framework we establish the urysohn lemma for the mashhour morsi axiom of n normality the formulation employs the mashhour morsi n euclidean space as the suitable fuzzy analogue of the real line the proof makes use of the artico moresco fuzzy proximities the lowen fuzzy uniformities and the mashhour morsi axiom of n complete regularity

fuzzy neighborhood spaces abstract in this paper we introduce a subcategory of the category of fuzzy topological spaces and we prove basic properties of this subcategory the main idea behind these spaces is the notion of fuzzy neighborhoods which enables us to introduce a local theory of fuzzy topologies

fuzzy t neighbourhood spaces part 3 t separation axioms we address the problem of identifying a useful set of mutually compatible separation axioms for each category t fns of t neighbourhood spaces it should contain one of t complete regularity that characterizes t uniformizability as indeed we achieve here this seems to necessitate the parameterization of most separation axioms by the triangular norm t with the exception of the two axioms t0 and r0 which have categorical definitions valid in any topological category we introduce t separation axioms in terms of t and fuzzy closure operators only making them meaningful for the whole of fts we then characterize them within t fns in terms of t fuzzy closures of crisp fuzzy subsets and t neighbourhoods we study our t axioms and establish some entailments between them our t complete regularity commits us to investigate a t real line which is a t neighbourhood space induced by a certain probabilistic t metric due to hohle fuzzy sets and systems 1 1978 311

a short note on fuzzy neighbourhood spaces abstract lowen introduced the concept of fuzzy neighbourhood spaces fns s and proved that the family of level topologies of a fns is an antichain wuyts characterized fns s as being those fully stratified fuzzy topological spaces fts s whose level topologies form antichains and whose fuzzy topologies are the finest possible for thier level topologies mashhour et al introduced the axiom of structure a on fts s and derived for it several characterizations and fuzzy topological consequences we use those characterizations of structure a and wuyts characterization of fns s to prove that a fts is a fns iff it is of structure a and fully stratified

scale free characteristics of random networks the topology of the world wide web the world wide web forms a large directed graph whose vertices are documents and edges are links pointing from one document to another here we demonstrate that despite its apparent random character the topology of this graph has a number of universal scale free characteristics we introduce a model that leads to a scale free network capturing in a minimal fashion the self organization processes governing the world wide web c 2000 elsevier science b v all rights reserved pacs 84 35 i 64 60 fr 87 23 ge

fuzzy uniform spaces in this paper we introduce the notion of fuzzy uniform spaces in section 2 we give some basic results and show how a fuzzy topology is derived from a fuzzy uniformity in sections 3 and 4 we prove that the notions which are introduced are good extensions and we prove that the category of uniform spaces is nicely injected in the category of fuzzy uniform spaces finally in section 5 we prove some basic results

fuzzy neighborhood convergence spaces we introduce a supercategory of the category nfs of fuzzy neighborhood spaces this supercategory is defined by means of local axioms similar to those of fns but the objects can also nicely be characterized internally via a notion of convergence in future work it shall be shown that this new category allows for a good treatment of compactifications

directed cycles and related structures in random graphs i static properties we study directed random graphs random graphs whose edges are directed and present new results on the so called strong components of those graphs we provide analytic and simulation results on two special classes of strong component called cycle components and knots which are important in random networks that represent certain computational systems