the dynamics of spreading and immune strategies of sexually transmitted diseases on scale free network abstract we examine epidemic threshold and dynamics for sexually transmitted diseases stds spread using a multiple susceptible infected removed susceptible ode model on scale free networks we derive the threshold for the epidemic to be zero in infinite scale free network for a hard cut off scale free network we also prove the stability of disease free equilibrium and the persistence of stds infection the effects of two immunization schemes including proportional scheme and targeted vaccination are studied and compared we find that targeted strategy compare favorably to a proportional scheme in terms of effectiveness theory and simulations both prove that an appropriate condom using has prominent effect to control stds spread on scale free networks

on the projected bases for sp 4 u 2 and the orthogonalization problem the mutual expansion and overlaps of the projected smirnov tolstoy and szpikowski berej bases for the irreducible representations irreps of sp 4 restricted to u 2 are considered the equivalence relation connecting the overlaps of both st and sb bases after the definite substitution of parameters up to an elementary factor is presented the overlaps of the szpikowski berej basis states are rearranged to double sums restricted by the parameters characterizing the multiplicities of the repeating irreps of subgroup and thus representable in polynomial forms the regge type symmetry of these new expressions allows the proof of the earlier conjectured symmetry of the orthogonalization coefficients oc for the family of the biorthogonal bases of sp 4 u 2 su 4 su 2 su 2 and su n so n for two parametric irreps the structure of the orthogonalization coefficients for the szpikowski berej basis is also considered the symmetry of the b c abde functions as the conjectured numerator polynomials

dynamical behavior of susceptible infected recovered susceptible epidemic model on weighted networks abstract we study susceptible infected recovered susceptible epidemic model in weighted regular and random complex networks we institute a pairwise type mathematical model with a general transmission rate to evaluate the influence of the link weight distribution on the spreading process furthermore we develop a dimensionality reduction approach to derive the condition for the contagion outbreak finally we analyze the influence of the heterogeneity of weight distribution on the outbreak condition for the scenario with a linear transmission rate our theoretical analysis is in agreement with stochastic simulations showing that the heterogeneity of link weight distribution can have a significant effect on the epidemic dynamics

the analysis of an epidemic model on networks abstract the paper consider an epidemic model with birth and death on networks we derive the epidemic threshold r0 dependent on birth rate b death rate d natural death and from the infectious disease and natural death and cure rate and the stability of the equilibriums the disease free equilibrium and endemic equilibrium are analysed finally the effects of various immunization schemes are studied and compared we show that both targeted and acquaintance immunization strategies compare favorably to a proportional scheme in terms of effectiveness for active immunization the threshold is easier to apply practically to illustrate our theoretical analysis some numerical simulations are also included

dynamical diversity induced by individual responsive immunization abstract a voluntary vaccination allows for a healthy individual to choose vaccination according to the individual s local information hence vaccination has the potential to provide a complex negative feedback non infection decreases propensity for vaccination hence increasing infection and vice versa in this paper we investigate a kind of sis epidemic model with a deterministic and voluntary vaccination scheme in scale free networks we first study a threshold model with no historical information by using the comparative method we confirm that under some conditions there exist two critical values of infection rates to determine three kinds of epidemic dynamical behaviors the epidemic spread the asymptotical decay and the exponential decay furthermore a mean field approximation model can predict the maximal infection level but cannot predict the existence of two critical infection rates in numerical simulations we observe a maximum in epidemic duration as a function of the model parameter a similar phenomenon has been found in the model with historical information finally we study a degree weighted model with a nonnegative exponent where 0 corresponds to the threshold model we find that at the steady state the infection density increases with while the variation of the vaccination fraction is less straightforward

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

sets of matrices all infinite products of which converge an infinite product iit lmi of matrices converges on the right if limi m mi exists a set z ai i l of n x n matrices is called an rcp set rightconvergent product set if all infinite products with each element drawn from z converge such sets of matrices arise in constructing self similar objects like von koch s snowflake curve in various interpolation schemes in constructing wavelets of compact support and in studying nonhomogeneous markov chains this paper gives necessary conditions and also some sufficient conditions for a set x to be an rcp set these are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices necessary and sufficient conditions are given for a finite set z to be an rcp set having a limit function m d rit lad where d d d which is a continuous function on the space of all sequences d with the sequence topology finite rcp sets of column stochastic matrices are completely characterized some results are given on the problem of algorithmically deciding if a given set x is an rcp set

average receiving scaling of the weighted polygon koch networks with the weight dependent walk based on the weighted koch networks and the self similarity of fractals we present a family of weighted polygon koch networks with a weight factor r 0 r 1 we study the average receiving time art on weight dependent walk i e the walker moves to any of its neighbors with probability proportional to the weight of edge linking them whose key step is to calculate the sum of mean first passage times mfpts for all nodes absorpt at a hub node we use a recursive division method to divide the weighted polygon koch networks in order to calculate the art scaling more conveniently we show that the art scaling exhibits a sublinear or linear dependence on network order thus the weighted polygon koch networks are more efficient than expended koch networks in receiving information finally compared with other previous studies results i e koch networks weighted koch networks we find out that our models are more general

a note on global stability of the virose equilibrium for network based computer viruses epidemics abstract it has been recently conjectured in yang et al 2013 27 that the unique virose equilibrium of a network based model is globally asymptotically stable in the feasible region the aim of this paper is to investigate the global stability of the virose equilibrium by using the direct lyapunov method and appropriate grouping techniques for lyapunov function derivatives some sufficient conditions are established which improves the relevant results moreover we provide a new brief proof for global stability of the virus free equilibrium by using comparison method and discuss quantitatively the effect of various immunization schemes including uniform immunization targeted immunization and high risk immunization on the viruses spread which suggests promising strategy to halt viruses spread in computer networks

epidemic dynamics on scale free networks with piecewise linear infectivity and immunization we examine epidemic thresholds for disease spread using susceptible infected susceptible models on scale free networks with variable infectivity infectivity between nodes is modeled as a piecewise linear function of the node degree rather than the less realistic linear transformation considered previously with this nonlinear infectivity we derive conditions for the epidemic threshold to be positive the effects of various immunization schemes including ring and targeted vaccination are studied and compared we find that both targeted and ring immunization strategies compare favorably to a proportional scheme in terms of effectiveness

the generalized spectral radius and extremal norms the generalized spectral radius also known under the name of joint spectral radius or after taking logarithms maximal lyapunov exponent of a discrete inclusion is examined we present a new proof for a result of barabanov which states that for irreducible sets of matrices an extremal norm always exists this approach lends itself easily to the analysis of further properties of the generalized spectral radius we prove that the generalized spectral radius is locally lipschitz continuous on the space of compact irreducible sets of matrices and show a strict monotonicity property of the generalized spectral radius sufficient conditions for the existence of extremal norms are obtained

a small world and scale free network generated by sierpinski pentagon abstract the sierpinski pentagon is used to construct evolving networks whose nodes are all solid regular pentagons in the construction of the sierpinski pentagon up to the stage t and any two nodes are neighbors if and only if the intersection of corresponding pentagons is non empty and non singleton we show that such networks have the small world and scale free effects but are not fractal scaling

precise determination of the fractal dimensions of apollonian packing and space filling bearings the authors present high precision numerical values for the fractal dimensions of apollonian packing and various space filling bearings which might have application for mechanical gearworks for turbulence or for tectonic motion they find for the simplest apollonian packing df 1 305 684 or 0 000 010

dynamics for an seirs epidemic model with time delay on a scale free network abstract a new seirs epidemic model with time delay on a scale free network is introduced the basic reproduction number r 0 is defined and it does not depend on time delay by analyzing the characteristic equation of equilibria and constructing suitable lyapunov functionals the stability of steady states is proved system does not occur hopf bifurcation around disease free equilibrium e 0 uniform immunization and targeted immunization are investigated numerical simulations and sensitivity analysis are performed results indicate that time delay plays a significant part in propagation process of epidemics on a scale free network

the generalized spectral radius theorem an analytic geometric proof abstract let be a bounded set of complex matrices m a 1 a m a i the generalized spectral radius theorem states that where and are defined as follows lim sup m 1 m where m sup a a m with a the spectral radius lim sup m 1 m where m sup a a m with any matrix norm we give an elementary proof based on analytic and geometric tools which is in some ways simpler than the first proof by berger and wang

a new physical basis for the irreducible representations of the orthogonal group so 5 in the quasi spin formalism new physical state vectors for the irs of the group so 5 are constructed in an analytical form the state vectors are labelled by the quantum numbers of the isospin its third component by the number of particles and by a fourth properly defined quantum number the previously constructed bases have been compared with the new construction from the same point of view two examples illustrate the new construction

the sequence of radii of the apollonian packing we consider the distribution function n x of the curvatures of the disks in the apollonian packing of a curvilinear triangle that is n x counts the number of disks in the packing whose curvatures do not exceed x we show that log n x log x approaches the limit s as x tends to infinity where s is the exponent of the packing a numerical fit of a curve of the form y ans to the values of n 1000n for n 1 2 6400 produces the estimate s 1 305636 which is consistent with the known bounds 1 300197 s 1 314534

on the max version of the generalized spectral radius theorem abstract let be a bounded set of n n non negative matrices recently the max algebra version of the generalized spectral radius of was introduced we show that lim t i t 1 t where denotes the generalized spectral radius and t the hadamard power of this provides a description of that uses no max terminology as an application we give a short proof of the max version of the generalized spectral radius theorem

on the power method in max algebra abstract the eigenvalue problem for an irreducible nonnegative matrix a a ij in the max algebra system is a otimes x lambda x where a otimes x i mathop max j a ij x j and lambda turns out to be the maximum circuit geometric mean mu a a power method algorithm is given to compute mu a and eigenvector x the algorithm is developed by using results on the convergence of max powers of a which are proved using nonnegative matrix theory in contrast to an algorithm developed in 4 this new method works for any irreducible nonnegative a and calculates eigenvectors in a simpler and more efficient way some asymptotic formulas relating mu a the spectral radius and norms are also given

generalized spectral radius and its max algebra version abstract let c n n and r n n be bounded subsets and let and denote the generalized spectral radius of and the max algebra version of the generalized spectral radius of respectively we apply a single matrix description of to give a new elementary and straightforward proof of the berger wang formula in max algebra and consequently a new short proof of the original berger wang formula in the case of bounded subsets of n n non negative matrices we also obtain a new description of in terms of the schur hadamard product and prove new trace and max trace descriptions of and in particular we show that limsup m sup a m tr a 1 m limsup m sup a m tr a 1 m and limsup m sup b m tr b 1 m limsup m sup b m tr b 1 m where tr a max i 1 n a ii and b b ij

modulated scale free network in euclidean space a random network is grown by introducing at unit rate randomly selected nodes on the euclidean space a node is randomly connected to its ith predecessor of degree k i with a directed link of length l using a probability proportional to k i l alpha our numerical study indicates that the network is scale free for all values of alpha alpha c and the degree distribution decays stretched exponentially for the other values of alpha the link length distribution follows a power law d l approximately l delta where delta is calculated exactly for the whole range of values of alpha

max algebra and pairwise comparison matrices abstract this paper is a continuation of our 2004 paper max algebra and pairwise comparison matrices in which the max eigenvector of a symmetrically reciprocal matrix was used to approximate such a matrix by a transitive matrix this approximation was based on minimizing the maximal relative error in a later paper by dahl a different error measure was used and led to a slightly different approximating transitive matrix here some geometric properties of this approximation problem are discussed these lead among other results to a new characterization of a max eigenvector of an irreducible nonnegative matrix the case of toeplitz matrices is discussed in detail and an application to music theory that uses toeplitz symmetrically reciprocal matrices is given

generalization of space filling bearings to arbitrary loop size a model for space filling packing of discs rolling sliplessly on each other is presented this model is an extension of the model presented by herrmann et al to a basic loop size of an arbitrary even number of discs this model might have application in geophysics dense granular flows and turbulence we show the existence and uniqueness of such constructions analytically for loop size up to eight and numerically for bigger basic loop sizes and make precise the algorithm to construct them a classification of the solution is also given and extensive studies of the fractal dimension are performed showing that it varies considerably with the base loop size

limit eigenvalues of nonnegative matrices abstract in this paper we give a simple closed formula for a certain limit eigenvalue of a nonnegative matrix we bound the spectral radius of a nonnegative matrix in terms of this limit eigenvalue

standard random walks and trapping on the koch network with scale free behavior and small world effect a vast variety of real life networks display the ubiquitous presence of scale free phenomenon and small world effect both of which play a significant role in the dynamical processes running on networks although various dynamical processes have been investigated in scale free small world networks analytical research about random walks on such networks is much less in this paper we will study analytically the scaling of the mean first passage time mfpt for random walks on scale free small world networks to this end we first map the classical koch fractal to a network called koch network according to this proposed mapping we present an iterative algorithm for generating the koch network based on which we derive closed form expressions for the relevant topological features such as degree distribution clustering coefficient average path length and degree correlations the obtained solutions show that the koch network exhibits scale free behavior and small world effect then we investigate the standard random walks and trapping issue on the koch network through the recurrence relations derived from the structure of the koch network we obtain the exact scaling for the mfpt we show that in the infinite network order limit the mfpt grows linearly with the number of all nodes in the network the obtained analytical results are corroborated by direct extensive numerical calculations in addition we also determine the scaling efficiency exponents characterizing random walks on the koch network

epidemic threshold of node weighted susceptible infected susceptible models on networks in this paper we investigate the epidemic spreading on random and regular networks through a pairwise type model with a general transmission rate to evaluate the influence of the node weight distribution by using block matrix theory an epidemic threshold index is formulated to predict the epidemic outbreak an upper bound of the epidemic threshold is obtained by analyzing the monotonicity of spectral radius for nonnegative matrices theoretical results suggest that the epidemic threshold is dependent on both matrices and with the first matrix being related to the mean field model while the second one reflecting the heterogeneous transmission rates in particular for a linear transmission rate this study shows the negative correlation between the heterogeneity of weight distribution and the epidemic threshold which is different from the results for existing results from the edge weighted networks

dual bases and some coupling coefficients for su4 su2 su2 sun son and sp4 u2 general analytical expressions for isoscalar factors reduced wigner coefficients for the couplings p0 0q to 0 and p10 p20 to 0 in the chains su4 su2 su2 and sun son n 3 are derived by using the algebra of complementary groups and methods of analytical continuation a review of alternative approaches to the problem grounded on the concept of biorthogonal systems of non canonical bases is given it is demonstrated that the construction of dual bases polynomial and stretched or related ones is associated with the use of certain bilinear combinations of isoscalar factors and solutions of the boundary value problem for isoscalar factors overlaps are given for different non canonical basis states of the two parametric irreducible representations for the chains su4 su2 su2 sun son and sp4 u2 as well as the expansions of the projected basis states for sp4 u2 in terms of canonical basis states a new expression for some special su2 clebsch gordan coefficients is also given

self similar disk packings as model spatial scale free networks the network of contacts in space filling disk packings such as the apollonian packing is examined these networks provide an interesting example of spatial scale free networks where the topology reflects the broad distribution of disk areas a wide variety of topological and spatial properties of these systems is characterized their potential as models for networks of connected minima on energy landscapes is discussed

threshold conditions for sis epidemic models on edge weighted networks abstract we consider the disease dynamics of a susceptible infected susceptible model in weighted random and regular networks by using the pairwise approximation we build an edge based compartment model from which the condition of epidemic outbreak is obtained our results suggest that there exists a remarkable difference between the linear and nonlinear transmission rate for a linear transmission rate the epidemic threshold is completely determined by the mean weight which is different from the susceptible infected recovered model framework while for a nonlinear transmission rate the epidemic threshold is not only related to the mean weight but also closely related to the heterogeneity of weight distribution

generation of clusters in complex dynamical networks via pinning control many real world networks show community structure i e groups or clusters of nodes that have a high density of links within them but with a lower density of links between them in this paper by applying feedback injections to a fraction of network nodes various clusters are synchronized independently according to the community structure generated by the group partition of the network cluster synchronization this control is achieved by pinning i e applying linear feedback control to a subset of the network nodes those pinned nodes are selected not randomly but according to the topological structure of communities of a given network specifically for a given group partition of a network those nodes with direct connections between groups must be pinned in order to achieve cluster synchronization both the local stability and global stability of cluster synchronization are investigated taking the tree shaped network and the most modular network as two particular examples we illustrate in detail how the pinning strategy influences the generation of clusters the simulations verify the efficiency of the pinning schemes used in this paper

simultaneous schur stability let be a set of n n complex matrices denote by the multiplicative semi group generated by for an n n complex matrix a the spectrum of a and spectral radius of a are denoted by a and r a respectively motivated by a phenomenon in the closed unit disc of the complex plane we give a rigorous definition of simultaneous schur stability as follows is said to be simultaneously schur stable if r a 1 a and 1 a a is said to be asymptotically stable if there exists a norm on cn such that sup a a e 1 it is proved that for a bounded set of n n complex matrices is asymptotically stable if and only if it is simultaneously schur stable by way of simultaneous schur stability some applications are illustrated especially an analytic combinatorial proof of a recent result of considerable depth generalized gelfand spectral radius formula

connectivity distribution of spatial networks we study spatial networks constructed by randomly placing nodes on a manifold and joining two nodes with an edge whenever their distance is less than a certain cutoff we derive the general expression for the connectivity distribution of such networks as a functional of the distribution of the nodes we show that for regular spatial densities the corresponding spatial network has a connectivity distribution decreasing faster than an exponential in contrast we also show that scale free networks with a power law decreasing connectivity distribution are obtained when a certain information measure of the node distribution integral of higher powers of the distribution diverges we illustrate our results on a simple example for which we present simulation results finally we speculate on the role played by the limiting case p k proportional variant k 1 which appears empirically to be relevant to spatial networks of biological origin such as the ones constructed from gene expression data

on nilpotent fuzzy matrices abstract in this paper we study the issue of nilpotent fuzzy matrices we first provide some properties of nilpotent fuzzy matrices in terms of eigenvalues when a finite number of fuzzy matrices are simultaneously considered we establish some characterizations of the simultaneous nilpotence for a finite number of fuzzy matrices on the other hand it is well known that the max min algebra is one kind of lattice we shall extend the results on simultaneous nilpotence to matrices on a bounded distributive lattice

computational complexity arising from degree correlations in networks we apply a bethe peierls approach to statistical mechanics models defined on random networks of arbitrary degree distribution and arbitrary correlations between the degrees of neighboring vertices using the nondeterministic polynomial time hard optimization problem of finding minimal vertex covers on these graphs we show that such correlations may lead to a qualitatively different solution structure as compared to uncorrelated networks this results in a higher complexity of the network in a computational sense simple heuristic algorithms fail to find a minimal vertex cover in the highly correlated case whereas uncorrelated networks seem to be simple from the point of view of combinatorial optimization

a max version of the generalized spectral radius theorem abstract let be a bounded set of n n nonnegative matrices in max algebra in this paper we propose the notions of the max algebra version of the generalized spectral radius of and the max algebra version of the joint spectral radius of the max algebra version of the generalized spectral radius theorem is established we propose the relationship between the generalized spectral radius of in the sense of daubechies and lagarias and its max algebra version moreover a generalization of elsner and van den driessche s lemma is presented as well

on the stability of takagi sugeno fuzzy systems with time varying uncertainties this paper studies the problems of stability analysis of takagi sugeno free fuzzy systems with time varying uncertainties in our prior study we represented the time varying uncertainty incurred in characteristic interval matrices in terms of the stability of takagi sugeno free fuzzy systems with consequent parameter uncertainties based on mayer s convergent theorem for powers of single interval matrix and its generalization we further proposed some sufficient conditions for the takagi sugeno free fuzzy system with time varying uncertainties to be globally asymptotically stable in this paper we propose the notion of simultaneously nilpotent interval matrices to investigate the takagi sugeno free fuzzy system with time varying uncertainties to be strongly stable within steps where relates to the dimension of interval matrices moreover a unique situation for the deterministic takagi sugeno free fuzzy system to be strongly stable within steps is derived as well where relates to the dimension of characteristic matrices for the deterministic takagi sugeno free fuzzy system

epidemic threshold determined by the first moments of network with alternating degree distributions abstract during the alternating day night cycle people have differing behavior and hence different connection patterns such as going to work or home shopping and so on hence the true topological structure of human contact networks are not only time varying but also exhibit certain distribution regularity in this paper we will investigate epidemic spreading on time varying human contact networks which follow one degree distribution during daytime but another at night based on sis susceptible infected susceptible propagation mechanism we study the epidemic threshold of this network with alternating distributions a surprising result is that for the discrete time case the epidemic threshold is determined only by the first moments of the two alternating degree distributions if the degree of each node is constant for all nights a similar result is valid for the continuous time case if the duration is sufficiently small this work shows that the spreading dynamics of time varying networks with alternating distributions is completely different from the widely studied case of static spreading networks

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

on the asymptotic stability of nonnegative matrices in max algebra abstract in the max algebra system for an n n nonnegative matrix a a ij the eigenequation for max eigenvalue and corresponding max eigenvector x is a x x where a x i max 1 j n a ij x j and a is the maximum circuit geometric mean it is shown that the following conditions are mutually equivalent i a r n ii a 1 iii a lim k a k 0 where a max x 1 x 0 a x and a limsup k a k 1 k

bounded semigroups of matrices in this note are proved two conjectures of daubechies and lagarias the first asserts that if z is a bounded set of matrices such that all left infinite products converge then 8 generates a bounded semigroup the second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices one definition involves the conventional spectral radius and one involves the operator norm

asymptotic stability and generalized gelfand spectral radius formula let be a set of n n complex matrices for m 1 2 let m be the set of all products of matrices in of length m denote by the multiplicative semigroup generated by is said to be asymptotically stable in the sense of dynamical systems if there is 0 1 such that there are bounded neighborhoods u v cn of the origin for which av mu for all a m m 1 2 for a bounded set of n n complex matrices it is shown that the following conditions are mutually equivalent i is asymptotically stable ii g9 lim supm supa m a 1 m 1 iii lim supm supa m a 1 m 1 where a stands for the spectral radius of a and iv there exists a positive number such that a 1 for all a this fact answers an open question raised by brayton and tong the generalized gelfand spectral radius formula that is g9 conjectured by daubechies and lagarias and proved by berger and wang using advanced tools from ring theory and then by elsner using analytic geometric tools follows immediately form the above asymptotic stability theorem

analysis of epidemic models with demographics in metapopulation networks abstract in this paper two susceptible infected susceptible sis epidemic models are presented and analyzed by reaction diffusion processes with demographics in metapopulation networks firstly an sis model with constant inputting is discussed the model has a disease free equilibrium which is locally asymptotically stable when the basic reproduction number is less than unity otherwise it is unstable it has an endemic equilibrium which is globally asymptotically stable secondly in another sis model the birth rate is the form of logistic similarly the stability of disease free equilibrium and endemic equilibrium is also proved finally numerical simulations are performed to illustrate the analytical results

effects of degree biased transmission rate and nonlinear infectivity on rumor spreading in complex social networks we introduce a generalized rumor spreading model and analytically investigate the spreading of rumors on scale free sf networks in the standard rumor spreading model each node has an infectivity equal to its degree and connectivity is uniform across all links to generalize this model we introduce an infectivity function that determines the number of simultaneous contacts that a given node individual may establish with its connected neighbors and a connectivity strength function csf for the direct link between two connected nodes these lead to a degree biased propagation of rumors for nonlinear functions this generalization is reflected in the infectivity s exponent ensuremath alpha and the csf s exponent ensuremath beta we show that by adjusting exponents ensuremath alpha and ensuremath beta the epidemic threshold can be controlled this feature is absent in the standard rumor spreading model in addition we obtain a critical threshold we show that the critical threshold for our generalized model is greater than that of the standard model on a finite sf network theoretically we show that ensuremath beta ensuremath 1 leads to a maximum spreading of rumors and computation results on different networks verify our theoretical prediction also we show that a smaller ensuremath alpha leads to a larger spreading of rumors our results are interesting since we obtain these results regardless of the network topology and configuration

on simultaneously nilpotent fuzzy matrices abstract nilpotent fuzzy matrices play a crucial role in the study of fuzzy matrices in this paper we shall extend the nilpotence to the notion of simultaneous nilpotence for a finite set of fuzzy matrices the notion of simultaneous nilpotence relates to the infinite products of a finite number of fuzzy matrices which converge to the zero matrix properties of the simultaneous nilpotence will be established in the study of consecutive powers of a fuzzy matrix a controllable fuzzy matrix can be characterized by an associated nilpotent fuzzy matrix in this paper we propose the notion of simultaneously controllable fuzzy matrices which can be thought of as a generalization of the notion of controllable fuzzy matrices similar to the nilpotent characterization for a controllable fuzzy matrix the simultaneously controllable fuzzy matrices can be characterized by a finite set of associated simultaneously nilpotent fuzzy matrices

dynamics of a competing two strain sis epidemic model on complex networks with a saturating incidence rate this paper studies a two strain sis epidemic model with a competing mechanism and a saturating incidence rate on complex networks this type of incidence rate can be used to reflect the crowding effect of the infective individuals we first obtain the associated reproduction numbers for each of the two strains which determine the existence of the boundary equilibria the stability of the disease free and boundary equilibria are further examined besides this we also show that the two competing strains can coexist under certain conditions interestingly the saturating incidence rate can have specific effects on not only the stability of the boundary equilibria but also the existence of the coexistence equilibrium numerical simulations are presented to support the theoretical results

barabanov norms lipschitz continuity and monotonicity for the max algebraic joint spectral radius abstract we present several results describing the interplay between the max algebraic joint spectral radius jsr for compact sets of matrices and suitably defined matrix norms in particular we extend a classical result for the conventional algebra showing that the max algebraic jsr can be described in terms of induced norms of the matrices in the set we also show that for a set generating an irreducible semigroup in a cone theoretic sense a monotone barabanov norm always exists this fact is then used to show that the max algebraic jsr is locally lipschitz continuous on the space of compact irreducible sets of matrices with respect to the hausdorff distance we then prove that the max algebraic jsr is locally hoelder continuous on the space of compact sets of nonnegative matrices finally we prove a strict monotonicity property for the max algebraic jsr that echoes a fact for the classical jsr the single matrix characterisation of the max algebraic jsr plays a vital role in our proofs

on topological properties of the octahedral koch network abstract in this article we propose an octahedral koch network exhibiting abundant new properties compared to the triangular koch network analytical expressions for the degree distribution clustering coefficient and average path length are presented the scale free feature and small world property of the octahedral koch network are obtained via numerical analysis furthermore we show that the octahedral koch network is assortative finally we show that the projection of the octahedral koch network on the plane is the nearest neighbor coupled koch network and the critical exponents of degree distribution in the octahedral koch network is greater than three

origin of degree correlations in the internet and other networks it has been argued that the observed anticorrelation between the degrees of adjacent vertices in the network representation of the internet has its origin in the restriction that no two vertices have more than one edge connecting them here we propose a formalism for modeling ensembles of graphs with single edges only and derive values for the exponents and correlation coefficients characterizing them our results confirm that the conjectured mechanism does indeed give rise to correlations of the kind seen in the internet although only a part of the measured correlation can be accounted for in this way

on the continuity of the generalized spectral radius in max algebra abstract given a bounded set of n n non negative matrices let and denote the generalized spectral radius of and its max version respectively we show that sup t 0 n 1 t 1 t where t denotes the hadamard power of we apply this result to give a new short proof of a known fact that is continuous on the hausdorff metric space h of all nonempty compact collections of n n non negative matrices

geographical embedding of scale free networks a method for embedding graphs in euclidean space is suggested the method connects nodes to their geographically closest neighbors and economizes on the total physical length of links the topological and geometrical properties of scalefree networks embedded by the suggested algorithm are studied both analytically and through simulations our findings indicate dramatic changes in the embedded networks in comparison to their off lattice counterparts and call into question the applicability of off lattice scale free models to realistic everyday life networks

spreading of epidemics on scale free networks with nonlinear infectivity abstract in this paper we study the spreading of epidemics on scale free networks with infectivity which is nonlinear in the connectivity of nodes we will show that the nonlinear infectivity is more appropriate than constant or linear ones and give the epidemic threshold of the sis model on a scale free network with nonlinear infectivity in addition we compare the effects of nonlinear infectivity on the epidemic threshold with two other cases on infinite and finite scale free networks and find some new results such as with unit recovery rate and nonlinear irrational infectivity the epidemic threshold is always positive and the epidemic threshold can increase with network size on finite networks contrary to the findings in all previous work

clustering properties of a generalized critical euclidean network many real world networks exhibit a scale free feature have a small diameter and a high clustering tendency we study the properties of a growing network which has all these features in which an incoming node is connected to its ith predecessor of degree k i with a link of length l using a probability proportional to k beta i l alpha for alpha 0 5 the network is scale free at beta 1 with the degree distribution p k proportional to k gamma and gamma 3 0 as in the barabasi albert model alpha 0 beta 1 we find a phase boundary in the alpha beta plane along which the network is scale free interestingly we find a scale free behavior even for beta 1 for alpha 0 5 where the existence of a different universality class is indicated from the behavior of the degree distribution and the clustering coefficients the network has a small diameter in the entire scale free region the clustering coefficients emulate the behavior of most real networks for increasing negative values of alpha on the phase boundary

global stability of an sir model with differential infectivity on complex networks abstract in this paper an sir model with birth and death on complex networks is analyzed where infected individuals are divided into m groups according to their infection and contact between human is treated as a scale free social network we obtain the basic reproduction number r 0 as well as the effects of various immunization schemes the results indicate that the disease free equilibrium is locally and globally asymptotically stable in some conditions otherwise disease free equilibrium is unstable and exists an unique endemic equilibrium that is globally asymptotically stable our theoretical results are confirmed by numerical simulations and a promising way for infectious diseases control is suggested

evolving networks with disadvantaged long range connections we consider a growing network whose growth algorithm is based on the preferential attachment typical for scale free constructions but where the long range bonds are disadvantaged thus the probability of getting connected to a site at distance d is proportional to d alpha where alpha is a tunable parameter of the model we show that the properties of the networks grown with alpha 1 the structure of the network is quite different thus in this regime the node degree distribution is no longer a power law and it is well represented by a stretched exponential on the other hand the small world property of the growing networks is preserved at all values of alpha

approximating the spectral radius of sets of matrices in the max algebra is np hard the lower and average spectral radii measure respectively the minimal and average growth rates of long products of matrices taken from a finite set the logarithm of the average spectral radius is traditionally called the lyapunov exponent when one performs these products in the max algebra we obtain quantities that measure the performance of discrete event systems we show that approximating the lower and average max algebraic spectral radii is np hard

geography in a scale free network model we offer an example of a network model with a power law degree distribution p k approximately k alpha for nodes but which nevertheless has a well defined geography and a nonzero threshold percolation probability for alpha 2 the range of real world contact networks this is different from p c 0 for alpha 3 results for the original well mixed scale free networks in our lattice based scale free network individuals link to nearby neighbors on a lattice even considerable additional small world links do not change our conclusion of nonzero thresholds when applied to disease propagation these results suggest that random immunization may be more successful in controlling human epidemics than previously suggested if there is geographical clustering

geometric fractal growth model for scale free networks we introduce a deterministic model for scale free networks whose degree distribution follows a power law with the exponent ensuremath gamma at each time step each vertex generates its offspring whose number is proportional to the degree of that vertex with proportionality constant m ensuremath 1 mg1 we consider the two cases first each offspring is connected to its parent vertex only forming a tree structure second it is connected to both its parent and grandparent vertices forming a loop structure we find that both models exhibit power law behaviors in their degree distributions with the exponent ensuremath gamma 1 mathrm ln 2m ensuremath 1 mathrm ln m thus by tuning m the degree exponent can be adjusted in the range 2l ensuremath gamma l3 we also solve analytically a mean shortest path distance d between two vertices for the tree structure showing the small world behavior that is d ensuremath sim mathrm ln n mathrm ln overline k where n is system size and overline k is the mean degree finally we consider the case that the number of offspring is the same for all vertices and find that the degree distribution exhibits an exponential decay behavior

an analysis of transmission dynamics of drug resistant disease on scale free networks in this paper we present an sir drug resistant model on scale free networks the model has a disease free equilibrium which is globally asymptotically stable when a threshold quantity is less than one the model has a wild strain equilibrium and a resistant strain equilibrium which are asymptotically stable when their respective reproduction number is great than one when treatment leads to resistance development the model can have a unique coexistence equilibrium and we also prove the persistence of infection the effects of two immunization schemes including proportional immunization scheme and targeted immunization scheme are studied and compared we find that targeted strategy compare favorably to a proportional immunization has prominent effect to control the disease spread on scale free networks

complex networks modeled on the sierpinski gasket abstract in this paper we use the sierpinski gasket to construct evolving networks g t whose node set is the solid regular triangles in the construction of the sierpinski gasket up to the stage t and any two nodes are neighbors if and only if the corresponding solid triangles are in contact with each other on boundary using the encoding method we show that our evolving networks are scale free power law degree distribution and have the small world effect small average path length and high clustering coefficient

multifractality of space filling bearings and apollonian packings we study the structure of space filling bearings sfb and apollonian packings ap by determining the scaling behavior of the density distribution of points where the circles touch each other application of the sand box method to the average of theqth power of the number of touching points reveals the multifractal aspects of the structure of sfb and ap

bose realization for non canonical representations of the symplectic group sp 4 su 2 u 1 a new method is formulated for the construction of arbitrary unitary irreducible representations of the compact symplectic group sp 4 o 5 in orthonormal bases which are reduced with respect to the non canonical group chain sp 4 su 2 u 1 the method is based on a realization of the algebra of generators and basis states by means of a system of bose creation and annihilation operators as an illustration some series of representations with multiplicities equal to or less than three are given in explicit algebraic form

modifying the power method in max algebra in the max algebra system the eigenequation for an n n irreducible nonnegative matrix a aij is a x a x here a x i maxjaijxj and a is the maximum circuit geometric mean the complexity of the power method given in l elsner p van den driessche linear algebra appl to appear to compute a and x is considered under some assumptions on the critical matrix it is shown that the algorithm may have time complexity o n4 a modified power method based on karp s formula is presented for this new algorithm with no assumptions on the critical matrix a and x can be computed in o n3 time furthermore this algorithm can be used to compute all linearly independent eigenvectors corresponding to a

sis epidemic spreading with correlated heterogeneous infection rates abstract the epidemic spreading has been widely studied in homogeneous cases where each node may get infected by an infected neighbor with the same rate however the infection rate between a pair of nodes which may depend on e g their interaction frequency is usually heterogeneous and even correlated with their nodal degrees in the contact network in this paper we aim to understand how such correlated heterogeneous infection rates influence the epidemic spreading on different network topologies motivated by real world datasets we propose a correlated heterogeneous susceptible infected susceptible csis model which assumes that the infection rate i j j i between node i and j is correlated with the degree of the two end nodes i j c d i d j where indicates the strength of the correlation between the infection rates and nodal degrees and c is selected so that the average infection rate is 1 in this work the recovery rate is the same for all the nodes in order to understand the effect of such correlation on epidemic spreading we consider as well the corresponding uncorrected but still heterogeneous infection rate scenario as a reference where the original correlated infection rates in our csis model are shuffled and reallocated to the links of the same network topology we compare these two scenarios in the average fraction of infected nodes in the metastable state on erdos renyi er and scale free sf networks with a similar average degree through the continuous time simulations we find that when the recovery rate is small the negative correlation is more likely to help the epidemic spread and the positive correlation prohibits the spreading as the recovery rate increases to be larger than a critical value the positive but not negative correlation tends to help the spreading our findings are further analytically proved in a wheel network one central node connects with each of the nodes in a ring and validated on real world networks with correlated heterogeneous interaction frequencies

dynamics of a network based sis epidemic model with nonmonotone incidence rate abstract this paper studies the dynamics of a network based sis epidemic model with nonmonotone incidence rate this type of nonlinear incidence can be used to describe the psychological effect of certain diseases spread in a contact network at high infective levels we first find a threshold value for the transmission rate this value completely determines the dynamics of the model and interestingly the threshold is not dependent on the functional form of the nonlinear incidence rate furthermore if the transmission rate is less than or equal to the threshold value the disease will die out otherwise it will be permanent numerical experiments are given to illustrate the theoretical results we also consider the effect of the nonlinear incidence on the epidemic dynamics

asymptotic formula on average path length of fractal networks modeled on sierpinski gasket abstract in this paper we introduce a new method to construct evolving networks based on the construction of the sierpinski gasket using self similarity and renewal theorem we obtain the asymptotic formula for average path length of our evolving networks

recursive graphs with small world scale free properties we discuss a category of graphs recursive clique trees which have small world and scale free properties and allow a fine tuning of the clustering and the power law exponent of their discrete degree distribution we determine relevant characteristics of those graphs the diameter degree distribution and clustering parameter the graphs have also an interesting recursive property and generalize recent constructions with fixed degree distributions

the hausdorf dimension of the apollonian packing of circles we formulate the problem of determining the hausdorf dimension df of the apollonian packing of circles as an eigenvalue problem of a linear integral equation we show that solving a finite dimensional approximation to this infinite order matrix equation and extrapolating the results provides a fast algorithm for obtaining high precision numerical estimates for df we find that df 1 305 686 729 10 this is consistent with the rigorously known bounds on df and improves the precision of the existing estimate by three orders of magnitude

continuity of the generalized spectral radius in max algebra abstract let be an induced matrix norm associated with a monotone norm on r n and be the collection of all nonempty closed and bounded subsets of n n nonnegative matrices under this matrix norm for the hausdorff metric h between and is given by h max sup a inf b a b sup b inf a a b the max algebra system consists of the set of nonnegative numbers with sum a b max a b and the standard product ab for a b 0 for n n nonnegative matrices a b their product is denoted by a b where a b ij max 1 k n a ik b kj for each the max algebra version of the generalized spectral radius of is limsup m sup a m a 1 m where m a 1 a 2 a m a i here a is the maximum circuit geometric mean in this paper we prove that the max algebra version of the generalized spectral radius is continuous on the hausdorff metric space h the notion of the max algebra version of simultaneous nilpotence of matrices is also proposed necessary and sufficient conditions for the max algebra version of simultaneous nilpotence of matrices are presented as well

noncommutative pfaffians and classification of states of five dimensional quasi spin noncommutative pfaffians associated with an orthogonal algebra on are some special elements of the universal enveloping algebra u on using pfaffians we construct the fourth quantum number which together with the naturally defined three quantum numbers allows to classify the states of a five dimensional quasi spin the pfaffians are treated as creation operators for the new quantum number

complex networks on hyperbolic surfaces abstract we explore a novel method to generate and characterize complex networks by means of their embedding on hyperbolic surfaces evolution through local elementary moves allows the exploration of the ensemble of networks which share common embeddings and consequently share similar hierarchical properties this method provides a new perspective to classify network complexity both on local and global scale we demonstrate by means of several examples that there is a strong relation between the network properties and the embedding surface

sis model on homogeneous networks with threshold type delayed contact reduction we study the dynamics of a delayed sis epidemic model on homogeneous networks where it is assumed that individuals modify their contact patterns upon realizing the risk of infection this decision is made with some time delay and it is threshold type when the density of infected nodes reaches a critical value the number of links is reduced by a given factor such assumptions lead to a delay differential equation with discontinuous right hand side we show that if the basic reproduction number r 0 1 then the disease will be eradicated while it persists for r 0 1 in the latter case there is a globally asymptotically stable endemic equilibrium except for a crucial interval of reproduction numbers where the system shows oscillations we construct explicitly the unique slowly oscillatory periodic solution which has strong attractivity properties and show the existence of rapidly oscillatory periodic solutions with any frequency the amplitude of the oscillations is determined by the time delay our results indicate that with such information delays the link density of a network has an important effect on the qualitative dynamics of infectious diseases

scale free and small world properties of sierpinski networks abstract in this paper we construct the evolving networks from sierpinski carpet using the encoding approach in fractal geometry we consider the small similar copies of unit square as nodes of network where two nodes are neighbors if and only if their corresponding copies have common surface for our networks we check their scale free and small world effect by the self similar structures the exponent of power law on degree distribution is log 3 8 which is the hausdorff dimension of the carpet

dynamical analysis of a fractional sir model with birth and death on heterogeneous complex networks abstract in this paper a fractional sir model with birth and death rates on heterogeneous complex networks is proposed firstly we obtain a threshold value r 0 based on the existence of endemic equilibrium point e which completely determines the dynamics of the model secondly by using lyapunov function and kirchhoff s matrix tree theorem the globally asymptotical stability of the disease free equilibrium point e 0 and the endemic equilibrium point e of the model are investigated that is when r 0 1 the disease free equilibrium point e 0 is globally asymptotically stable and the disease always dies out when r 0 1 the disease free equilibrium point e 0 becomes unstable and in the meantime there exists a unique endemic equilibrium point e which is globally asymptotically stable and the disease is uniformly persistent finally the effects of various immunization schemes are studied and compared numerical simulations are given to demonstrate the main results

epidemic spreading with nonlinear infectivity in weighted scale free networks abstract in this paper we investigate the epidemic spreading for the sir model in weighted scale free networks with the nonlinear infectivity and weighted transmission rate concretely we introduce the infectivity exponent and the weight exponent into the epidemic system then examine the impact of and on the epidemic spreading we show that one can adjust the values of and to rebuild a nonzero finite epidemic threshold furthermore we also find the infectivity exponent has a stronger effect not only on the epidemic threshold but also on the epidemic prevalence in addition it is also interesting to see that the absence of the epidemic threshold appears not very dejected since the prevalence grows much more slowly as the transmission rate increases

a max version of the perron frobenius theorem abstract if a an n n nonnegative irreducible matrix then there exists a 0 and a positive vector x such that maxjaijxj a xi i 1 2 n furthermore a is the maximum geometric mean of a circuit in the weighted directed graph corresponding to a this theorem which we refer to as the max version of the perron frobenius theorem is well known in the context of matrices over the max algebra and also in the context of matrix scalings in the present work which is partly expository we bring out the intimate connection between this result and the perron frobenius theory we present several proofs of the result some of which use the perron frobenius theorem structure of max eigenvalues and max eigenvectors is described possible ways to unify the perron frobenius theorem and its max version are indicated some inequalities for a are proved

epidemic spread in adaptive networks with multitype agents in this paper we study the epidemic spread in adaptive networks with multitype agents we assume that the infection probability as well as the rewiring probability between agents is type dependent under this assumption we analyze a system of m type networks with mean field approximation these general expressions apply to networks with various choices of infection mechanisms and rewiring rules we explicitly evaluate the infection level for two type networks with intratype intertype infecting and rewiring and investigate their impacts on the epidemic threshold by plotting the bifurcation diagram for various parameters we find a bistability region where both the disease free state and the endemic state co exist for appropriate rewiring dynamics the area of each phase depends on the corresponding interaction modes we show that consistency between infecting and rewiring modes speeds up the disease spread while inconsistency contributes to halting the outbreak

the analysis of an epidemic model with time delay on scale free networks abstract a new epidemic s i s model with time delay on scale free networks is presented we give the formula of the basic reproductive number for the model and prove that the disease dies out when the basic reproductive number is less than unity and the disease is uniformly persistent when the basic reproductive number is more than unity the effects of various immunization schemes are studied numerical simulations are given to demonstrate the main results

the power algorithm in max algebra abstract it is proved that under certain conditions an algorithm resembling the power algorithm in conventional linear algebra can be used to find the eigenvector and eigenvalue of a max algebra system if the conditions are not satisfied the eigenvector can be found using an extension of the power algorithm

on the spectrum in max algebra abstract we give new proofs of several fundamental results of spectral theory in max algebra this includes the description of the spectrum in max algebra of a given non negative matrix via local spectral radii the spectral theorem and the spectral mapping theorem in max algebra the latter result is also generalized to the setting of power series in max algebra by applying certain continuity properties of the spectrum in max algebra our methods enable us to obtain some related results for the usual spectrum of complex matrices and distinguished spectrum for non negative matrices