thresholds for epidemic spreading in networks we study the threshold of epidemic models in quenched networks with degree distribution given by a power law forthesusceptible infected susceptiblemodeltheactivitythresholdc vanishesinthelargesize limitonanynetworkwhosemaximumdegree kmax divergeswiththesystemsize atoddswithheterogeneous mean field hmf theory the vanishing of the threshold has nothing to do with the scale free nature of the networkbutstemsinsteadfromthelargesthubinthesystembeingactiveforanyspreadingrate 1 ffiffiffiffiffiffiffiffiffimax p and playing the role of a self sustained source that spreads the infection to the rest of the system the susceptible infected removedmodel displays insteadagreement with hmftheory and afinite threshold for scale rich networks we conjecture that on quenched scale rich networks the threshold of generic epidemic models is vanishing or finite depending on the presence or absence of a steady state

evaluation of an analytic approximate formula for the time varying sis prevalence in different networks abstract one of the most important quantities of the exact markovian sis epidemic process is the time dependent prevalence which is the average fraction of infected nodes unfortunately the markovian sis epidemic model features an exponentially increasing computational complexity with growing network size n in this paper we evaluate a recently proposed analytic approximate prevalence function introduced in van mieghem 2016 we compare the approximate function with the n intertwined mean field approximation nimfa and with simulation of the markovian sis epidemic process the results show that the new analytic prevalence function is comparable with other approximate methods

component models for fuzzy data the fuzzy perspective in statistical analysis is first illustrated with reference to the informational paradigm allowing us to deal with different types of uncertainties related to the various informational ingredients data model assumptions the fuzzy empirical data are then introduced referring to j lr fuzzy variables as observed on i observation units each observation is characterized by its center and its left and right spreads lr1 fuzzy number or by its left and right centers and its left and right spreads lr2 fuzzy number two types of component models for lr1 and lr2 fuzzy data are proposed the estimation of the parameters of these models is based on a least squares approach exploiting an appropriately introduced distance measure for fuzzy data a simulation study is carried out in order to assess the efficacy of the suggested models as compared with traditional principal component analysis on the centers and with existing methods for fuzzy and interval valued data an application to real fuzzy data is finally performed

note on fuzzy regression abstract regression analysis has been widely used in several areas for many years fuzzy linear regression was introduced in 1980 by h tanaka s uejima and k asai in this paper the general form of regression equations is developed for fuzzy numbers it is showed that the problem of fuzzy regression can be formulated as a mathematical programming problem the special case of linear regression yields a linear programming problem different measures of vagueness are introduced to define the best estimate two simple numerical examples illustrate the methodology and allow a partial comparison among the different measures of vagueness

approximate formula and bounds for the time varying susceptible infected susceptible prevalence in networks based on a recent exact differential equation the time dependence of the sis prevalence the average fraction of infected nodes in any graph is first studied and then upper and lower bounded by an explicit analytic function of time that new approximate tanh formula obeys a riccati differential equation and bears resemblance to the classical expression in epidemiology of kermack and mckendrick proc r soc london a 115 700 1927 1364 502110 1098 rspa 1927 0118 but enhanced with graph specific properties such as the algebraic connectivity the second smallest eigenvalue of the laplacian of the graph we further revisit the challenge of finding tight upper bounds for the sis and sir epidemic threshold for all graphs we propose two new upper bounds and show the importance of the variance of the number of infected nodes finally a formula for the epidemic threshold in the cycle or ring graph is presented

a possibilistic approach to latent component analysis for symmetric fuzzy data in many situations the available amount of data is huge and can be intractable when the data set is single valued latent component models are recognized techniques which provide a useful compression of the information this is done by considering a regression model between observed and unobserved latent variables in this paper an extension of latent component analysis to deal with fuzzy data is proposed our extension follows the possibilistic approach widely used both in the cluster and regression frameworks in this case the possibilistic approach involves the formulation of a latent component analysis for fuzzy data by optimization specifically a non linear programming problem in which the fuzziness of the model is minimized is introduced in order to show how our model works the results of two applications are proposed

possibilistic and probabilistic fuzzy clustering unification within the framework of the non extensive thermostatistics fuzzy clustering algorithms are becoming the major technique in cluster analysis in this paper we consider the fuzzy clustering based on objective functions they can be divided into two categories possibilistic and probabilistic approaches leading to two different function families depending on the conditions required to state that fuzzy clusters are a fuzzy c partition of the input data recently we have presented in menard and eboueya fuzzy sets and systems 27 to be published an axiomatic derivation of the possibilistic and maximum entropy inference mei clustering approaches based upon an unifying principle of physics that of extreme physical information epi defined by frieden physics from fisher information a unification cambridge university press cambridge 1999 here using the same formalism we explicitly give a new criterion in order to provide a theoretical justification of the objective functions constraint terms membership functions and weighting exponent m used in the probabilistic and possibilistic fuzzy clustering moreover we propose an unified framework including the two procedures this approach is inspired by the work of frieden and plastino and plastino and miller physics a 235 577 extending the principle of extremal information in the framework of the non extensive thermostatistics then we show how with the help of epi one can propose extensions of the fcm and possibilistic algorithms

the possibilistic c means algorithm insights and recommendations recently the possibilistic c means algorithm pcm was proposed to address the drawbacks associated with the constrained memberships used in algorithms such as the fuzzy c means fcm in this issue barni et al 1996 report a difficulty they faced while applying the pcm and note that it exhibits an undesirable tendency to converge to coincidental clusters the purpose of this paper is not just to address the issues raised by barni et al but to go further and analytically examines the underlying principles of the pcm and the possibilistic approach in general we analyze the data sets used by barni et al and interpret the results reported by them in the light of our findings

prevalence expansion in nimfa abstract the n intertwined mean field approximation nimfa is a reasonably accurate approximation of the exact sis epidemic process on a network the average fraction of infected nodes in the nimfa steady state also called the steady state prevalence in terms of the effective infection rate can be expanded into a power series around the nimfa epidemic threshold in this paper we investigate the convergence of the steady state prevalence taylor expansion we determine the radius of convergence in some special types of graphs we also show that the radius of convergence of the steady state prevalence expansion depends upon the network topology in particular the average degree of the network and the spectral gap of the adjacency matrix play a role

a numerical investigation of sign changing solutions to superlinear elliptic equations on symmetric domains in this paper we investigate numerically sign changing solutions of superlinear elliptic equations on symmetric domains based upon the symmetric criticality principle of palais the existence of sign changing solutions which re2ect the symmetry

least squares model fitting to fuzzy vector data abstract we consider model fitting problems in which the mathematical model of an observed event is a system of equations involving model parameters and observables and in which the data are fuzzy vectors such problems naturally arise in applications when data are scarce and information is vague about distributions of variances that are contained in the observations then it may only be possible to obtain estimates of membership functions of the data a model can be fitted to such data by maximizing the membership values of the adjusted observations we achieve this by minimizing the sum of squares of the deviations of the membership values from one the optimization problem simplifies significantly if the membership functions of the fuzzy vectors belong to a class of conical functions which are defined in terms of an elliptic norm properties of these functions are discussed and a membership propagation formula derived which component wise obeys the extension principle the least squares model fitting to the described fuzzy vector data deviates in important aspects from ordinary least squares one obtains as a solution a fuzzy model parameter vector and a number of indicators which describe the quality of the fit viz the grade of compatibility between data and model their discord and dilators of model and data membership functions these new concepts are illustrated and discussed in the context of an example from a terminal ballistics problem

second order mean field susceptible infected susceptible epidemic threshold given the adjacency matrix a of a network we present a second order mean field expansion that improves on the first order n intertwined susceptible infected susceptible sis epidemic model unexpectedly we found that in contrast to first order second order mean field theory is not always possible the network size n should be large enough under the assumption of large n we show that the crucial and characterizing quantity the sis epidemic threshold ensuremath tau c obeys an eigenvalue equation more complex than the one in the first order n intertwined model however the resulting epidemic threshold is more accurate ensuremath tau c 2 ensuremath tau c 1 o frac ensuremath tau c 1 n where the first order epidemic threshold is ensuremath tau c 1 frac 1 ensuremath lambda 1 a and where ensuremath lambda 1 a is the spectral radius of the adjacency matrix a

outliers detection and confidence interval modification in fuzzy regression abstract the existence of outliers in a set of experimental data can cause incorrect interpretation of the fuzzy linear regression results peters fuzzy sets and systems 63 1994 45 55 considered this problem for the nonfuzzy input and nonfuzzy output data type the present investigation focuses on nonfuzzy input and fuzzy output data type and proposes approaches to handle the outlier problem the main idea is to introduce a pre assigned k limiting value whose value must be determined based on the conditions of the current problem the advantages and problems of the proposed approaches are compared and discussed

multiple solutions for a semilinear boundary value problem a computational multiplicity proof abstract we prove the first genuine partial differential equation result on a conjecture concerning the number of solutions of second order elliptic boundary value problems with a nonlinearity which grows superlinearly at the proof makes massive use of computer assistance after approximate solutions have been computed by a numerical mountain pass algorithm combined with a newton iteration to improve accuracy a fixed point argument is used to show the existence of exact solutions close to the approximations

principal component analysis of fuzzy data using autoassociative neural networks this paper describes an extension of principal component analysis pca allowing the extraction of a limited number of relevant features from high dimensional fuzzy data our approach exploits the ability of linear autoassociative neural networks to perform information compression in just the same way as pca without explicit matrix diagonalization fuzzy input values are propagated through the network using fuzzy arithmetics and the weights are adjusted to minimize a suitable error criterion the inputs being taken as target outputs the concept of correlation coefficient is extended to fuzzy numbers allowing the interpretation of the new features in terms of the original variables experiments with artificial and real sensory evaluation data demonstrate the ability of our method to provide concise representations of complex fuzzy data

susceptible infected susceptible epidemics on the complete graph and the star graph exact analysis since mean field approximations for susceptible infected susceptible sis epidemics do not always predict the correct scaling of the epidemic threshold of the sis metastable regime we propose two novel approaches a an sis generalized model and b a modified sis model that prevents the epidemic from dying out i e without the complicating absorbing sis state both adaptations of the sis model feature a precisely defined steady state that corresponds to the sis metastable state and allow an exact analysis in the complete and star graph consisting of a central node and n leaves the n intertwined mean field approximation nimfa is shown to be nearly exact for the complete graph but less accurate to predict the correct scaling of the epidemic threshold c in the star graph which is found as c c 1 where 1 2logn 3 2loglogn and where c 1 1 n c is the first order epidemic threshold for the star in nimfa and equal to the inverse of the spectral radius of the star s adjacency matrix

interval regression analysis by quadratic programming approach when we use linear programming in possibilistic regression analysis some coefficients tend to become crisp because of the characteristic of linear programming on the other hand a quadratic programming approach gives more diverse spread coefficients than a linear programming one therefore to overcome the crisp characteristic of linear programming we propose interval regression analysis based on a quadratic programming approach another advantage of adopting a quadratic programming approach is to be able to integrate both the property of central tendency in least squares and the possibilistic property in fuzzy regression by changing the weights of the quadratic function we can analyze the given data from different viewpoints for data with crisp inputs and interval outputs the possibility and necessity models can be considered therefore the unified quadratic programming approach obtaining the possibility and necessity regression models simultaneously is proposed even though there always exist possibility estimation models the existence of necessity estimation models is not guaranteed if we fail to assume a proper function fitting to the given data as a regression model thus we consider polynomials as regression models since any curve can be represented by the polynomial approximation using polynomials we discuss how to obtain approximation models which fit well to the given data where the measure of fitness is newly defined to gauge the similarity between the possibility and the necessity models furthermore from the obtained possibility and necessity regression models a trapezoidal fuzzy output can be constructed

comments on a possibilistic approach to clustering in this comment we report a difficulty with the application of the possibilistic approach to fuzzy clustering pcm proposed by keller and krishnapuram 1993 in applying this algorithm we found that it has the undesirable tendency to produce coincidental clusters results illustrating this tendency are reported and a possible explanation for the pcm behavior is suggested

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

two and three way component models for lr fuzzy data in a possibilistic framework abstract in this work we address the data reduction problem for fuzzy data in particular following a possibilistic approach several component models for handling two and three way fuzzy data sets are introduced the two way models are based on classical principal component analysis pca whereas the three way ones on three way generalizations of pca as tucker3 and candecomp parafac the here proposed models exploit the potentiality of the possibilistic regression in fact the component models for fuzzy data can be seen as particular regression analyses between a set of observed fuzzy variables response variables and a set of unobservable crisp variables explanatory variables in order to show how the models work the results of an application to a three way fuzzy data set are illustrated

constrained mountain pass algorithm for the numerical solution of semilinear elliptic problems a new numerical algorithm for solving semilinear elliptic problems is presented a variational formulation is used and critical points of a c1 functional subject to a constraint given by a level set of another c1 functional or an intersection of such level sets of finitely many functionals are sought first constrained local minima are looked for then constrained mountain pass points the approach is based on the deformation lemma and the mountain pass theorem in a constrained setting several examples are given showing new numerical solutions in various applications

epidemics in networks with nodal self infection and the epidemic threshold since the susceptible infected susceptible sis epidemic threshold is not precisely defined in spite of its practical importance the classical sis epidemic process has been generalized to the sis model where a node possesses a self infection rate in addition to a link infection rate and a curing rate the exact markov equations are derived from which the steady state can be computed the major advantage of the sis model is that its steady state is different from the absorbing or overall healthy state and approximates for a certain range of small 0 the in reality observed phase transition also called the metastable state that is characterized by the epidemic threshold the exact steady state analysis for the complete graph illustrates the effect of small and the quality of the first order mean field approximation the n intertwined model proposed earlier apart from duality principles often used in the mathematical literature we present an exact recursion relation for the markov infinitesimal generator

a mountain pass method for the numerical solution of semilinear wave equations it is well known that periodic solutions of semilinear wave equations can be obtained as critical points of related functionals in the situation that we studied there is usually an obvious solution obtained as a solution of linear problem we formulate a dual variational problem in such a way that the obvious solution is a local minimum we then find additional non obvious solutions via a numerical mountain pass algorithm based on the theorems of ambrosetti rabinowitz and ekeland numerical results are presented

phase transition of the susceptible infected susceptible dynamics on time varying configuration model networks we present a degree based theoretical framework to study the susceptible infected susceptible sis dynamics on time varying rewired configuration model networks using this framework on a given degree distribution we provide a detailed analysis of the stationary state using the rewiring rate to explore the whole range of the time variation of the structure relative to that of the sis process this analysis is suitable for the characterization of the phase transition and leads to three main contributions 1 we obtain a self consistent expression for the absorbing state threshold able to capture both collective and hub activation 2 we recover the predictions of a number of existing approaches as limiting cases of our analysis providing thereby a unifying point of view for the sis dynamics on random networks 3 we obtain bounds for the critical exponents of a number of quantities in the stationary state this allows us to reinterpret the concept of hub dominated phase transition within our framework it appears as a heterogeneous critical phenomenon observables for different degree classes have a different scaling with the infection rate this phenomenon is followed by the successive activation of the degree classes beyond the epidemic threshold

survival time of the susceptible infected susceptible infection process on a graph the survival time t is the longest time that a virus a meme or a failure can propagate in a network using the hitting time of the absorbing state in an uniformized embedded markov chain of the continuous time susceptible infected susceptible sis markov process we derive an exact expression for the average survival time e t of a virus in the complete graph kn and the star graph k1 n 1 by using the survival time instead of the average fraction of infected nodes we propose a new method to approximate the sis epidemic threshold c that at least for kn and k1 n 1 correctly scales with the number of nodes n and that is superior to the epidemic threshold 1 c 1 1 of the n intertwined mean field approximation where 1 is the spectral radius of the adjacency matrix of the graph g although this new approximation of the epidemic threshold offers a more intuitive understanding of the sis process it remains difficult to compare outbreaks in different graph types for example the survival in an arbitrary graph seems upper bounded by the complete graph and lower bounded by the star graph as a function of the normalized effective infection rate 1 c however when the average fraction of infected nodes is used as a basis for comparison the virus will survive in the star graph longer than in any other graph making the star graph the worst case graph instead of the complete graph finally in non markovian sis the distribution of the spreading attempts over the infectious period of a node influences the survival time even if the expected number of spreading attempts during an infectious period the non markovian equivalent of the effective infection rate is kept constant both early and late infection attempts lead to shorter survival times interestingly just as in markovian sis the survival times appear to be exponentially distributed regardless of the infection and curing time distributions

fuzzy regression model with fuzzy input and output data for manpower forecasting in modeling a fuzzy system with fuzzy linear functions the vagueness of the fuzzy output data may be caused by both the indefiniteness of model parameters and the vagueness of the input data this situation occurs as the input data are envisaged as facts or events of an observation which are uncontrollable or uninfluenced by the observer rather than as the controllable levels of factors in an experiment in this research we concentrate on such a situation and refer to it as a generalized fuzzy linear function using this generalized fuzzy linear function a generalized fuzzy regression model is formulated a nonlinear programming model is proposed to identify the fuzzy parameters and their vagueness for the generalized regression model a manpower forecasting problem is used to demonstrate the use of the proposed model

new solutions of the gelfand problem abstract we propose a computer assisted method for proving existence and obtaining enclosures of solutions of nonlinear elliptic boundary value problems which may be successful in cases where purely analytic methods are not this method is an extension of our previous work interval comput 3 1994 106 121 z angew math mech 76 1996 171 176 on the gelfand problem to nonconvex domains in r 2 here we are able to prove the existence of a solution branch which was unknown so far

non markovian infection spread dramatically alters the susceptible infected susceptible epidemic threshold in networks most studies on susceptible infected susceptible epidemics in networks implicitly assume markovian behavior the time to infect a direct neighbor is exponentially distributed much effort so far has been devoted to characterize and precisely compute the epidemic threshold in susceptible infected susceptible markovian epidemics on networks here we report the rather dramatic effect of a nonexponential infection time while still assuming an exponential curing time on the epidemic threshold by considering weibullean infection times with the same mean but different power exponent for three basic classes of graphs the erd s renyi random graph scale free graphs and lattices the average steady state fraction of infected nodes is simulated from which the epidemic threshold is deduced for all graph classes the epidemic threshold significantly increases with the power exponents hence real epidemics that violate the exponential or markovian assumption can behave seriously differently than anticipated based on markov theory

analytical and numerical results for the fucik spectrum of the laplacian nontrivial solutions of u u vu 0 in rn with zero dirichlet condition on are studied the collection of the pairs v is the so called fucik spectrum f a new variational formulation for parts of f is presented and analyzed based on this formulation a minimization algorithm for the computation of a part of f is developed alternatively an approach using an implicit function argument is discussed analytically and via newton s method also numerically by combining the variational minimization method with newton s method a new bifurcation result is first observed numerically and then proved rigorously by replacing the variational minimization method by the mountain pass algorithm higher curves in f are found numerically several numerical results are discussed including a further example of the previously recorded phenomenon of crossing of fucik curves originating from different eigenvalues

accuracy criterion for the mean field approximation in susceptible infected susceptible epidemics on networks mean field approximations mfas are frequently used in physics when a process such as an epidemic or a synchronization on a network is approximated by mfa a major hurdle is the determination of those graphs for which mfa is reasonably accurate here we present an accuracy criterion for markovian susceptible infected susceptible sis epidemics on any network based on the spectrum of the adjacency and sis covariance matrix we evaluate the mfa criterion for the complete and star graphs analytically and numerically for connected erd s renyi random graphs for small size n 14 the accuracy of mfa increases with average degree and with n precise simulations up to network sizes n 100 of the mfa accuracy criterion versus n for the complete graph star square lattice and path graphs lead us to conjecture that the worst mfa accuracy decreases for large n proportionally to the inverse of the spectral radius of the adjacency matrix of the graph

susceptible infected susceptible model a comparison of n intertwined and heterogeneous mean field approximations we introduce the susceptible infected susceptible sis spreading model which is taken as a benchmark for the comparison between the n intertwined approximation and the pastor satorras and vespignani heterogeneous mean field hmf approximation of the sis model the n intertwined approximation the hmf approximation and the sis spreading model are compared for different graph types we focus on the epidemic threshold and the steady state fraction of infected nodes in networks with different degree distributions overall the n intertwined approximation is superior to the hmf approximation the n intertwined approximation is exactly the same as the hmf approximation in regular graphs however for some special graph types such as the square lattice graph and the path graph the two mean field approximations are both very different from the sis spreading model

a high linking algorithm for sign changing solutions of semilinear elliptic equations where n is a bounded domain with regular boundary the operator can be replaced by a more general second order uniformly elliptical operator in divergence form the solutions we are interested in nding are of minimax type and sign changing nonlinear elliptic equations of type 1 1 arise naturally in physics engineering and mathematical biology it is known that many such equations have multiple solutions their study has attracted the attention of many pure and applied mathematicians during the past two decades let us assume that f satis es the following regularity and growth conditions a1 f x t is locally lipschitz continuous in