bipartitioning of random graphs of fixed extensive valence the procedure of fu and anderson ibid vol 19 p 1605 1986 for the application of statistical mechanics to the problem of bipartitioning random graphs is extended to graphs of fixed extensive valence the cost function is shown to be independent of whether the valence is locally or globally constrained

global stability of endemic equilibrium of an epidemic model with birth and death on complex networks abstract we study global stability of endemic equilibrium of an epidemic model with birth and death on complex networks under some conditions the local asymptotic stability of the endemic equilibrium was established by zhang and jin 2011 for correlated networks and the global asymptotic stability was obtained by chen and sun 2014 for uncorrelated networks in this work we remove those conditions and prove by constructing a lyapunov function that the endemic equilibrium is globally asymptotically stable numerical simulations are also presented to illustrate the feasibility of the result

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

epidemic thresholds of the susceptible infected susceptible model on networks a comparison of numerical and theoretical results recent work has shown that different theoretical approaches to the dynamics of the susceptible infectedsusceptible sis model for epidemics lead to qualitatively different estimates for the position of the epidemic threshold in networks here we present large scale numerical simulations of the sis dynamics on various types of networks allowing the precise determination of the effective threshold for systems of finite size n we compare quantitatively the numerical thresholds with theoretical predictions of the heterogeneous mean field theory and of the quenched mean field theory we show that the latter is in general more accurate scaling with n with the correct exponent but often failing to capture the correct prefactor

global stability of virus spreading in complex heterogeneous networks various networks are possessed of an obvious heterogeneity in the connectivity properties and it is of practical significance to study epidemic spreading in networks of this kind pastor satorras and vespignani established the dynamical mean field reaction rate equations for the spreading of infections in complex heterogeneous networks based on the well known sis model and figured out an epidemic threshold lambda c such that if lambda effective spreading rate is above lambda c the infection spreads and becomes endemic the significance of this result is far reaching however the authors have not found a strict mathematical proof of their conclusion in the literature in this paper we approach this problem by proving that if lambda is above lambda c the infection spreads and approaches the unique positive stationary point of the reaction rate equations as long as there exist infected nodes in the network initially i e the virus infection process is globally stable

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

the distribution function of the effective field of the ising spin glass on the bethe lattice for the coordination number z 4 5 6 abstract the integral equation for the effective fields of the spin glass is solved analytically for the j ising model on the bether lattice with coordination number z 4 5 and 6 and symmetric distribution functions are obtained the paramagnetic distribution spin glass distribution with three delta functions and spin glass distributions with a continuous distribution are obtained the effect of the truncation of the algebraic equation for solving the integral equation is discussed

epidemic dynamics in finite size scale free networks many real networks present a bounded scale free behavior with a connectivity cutoff due to physical constraints or a finite network size we study epidemic dynamics in bounded scale free networks with soft and hard connectivity cutoffs the finite size effects introduced by the cutoff induce an epidemic threshold that approaches zero at increasing sizes the induced epidemic threshold is very small even at a relatively small cutoff showing that the neglection of connectivity fluctuations in bounded scale free networks leads to a strong overestimation of the epidemic threshold we provide the expression for the infection prevalence and discuss its finite size corrections the present paper shows that the highly heterogeneous nature of scale free networks does not allow the use of homogeneous approximations even for systems of a relatively small number of nodes

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

behaviors of susceptible infected epidemics on scale free networks with identical infectivity in this paper we propose a susceptible infected model with identical infectivity in which at every time step each node can only contact a constant number of neighbors we implemented this model on scale free networks and found that the infected population grows in an exponential form with the time scale proportional to the spreading rate furthermore by numerical simulation we demonstrated that the targeted immunization of the present model is much less efficient than that of the standard susceptible infected model finally we investigate a fast spreading strategy when only local information is available different from the extensively studied path finding strategy the strategy preferring small degree nodes is more efficient than that preferring large degree nodes our results indicate the existence of an essential relationship between network traffic and network epidemic on scale free networks

relaxation in graph coloring and satisfiability problems using t 0 monte carlo simulation we study the relaxation of graph coloring k col and satisfiability k sat two hard problems that have recently been shown to possess a phase transition in solvability as a parameter is varied a change from exponentially fast to power law relaxation and a transition to freezing behavior are found these changes take place for smaller values of the parameter than the solvability transition results for the coloring problem for colorable and clustered graphs and for the fraction of persistent spins for satisfiability are also presented

approximation in normal linear spaces abstract a historical account is given of the development of methods for solving approximation problems set in normed linear spaces approximation of both real functions and real data is considered with particular reference to lp or lp and chebyshev norms as well as coverage of methods for the usual linear problems an account is given of the development of methods for approximation by functions which are nonlinear in the free parameters and special attention is paid to some particular nonlinear approximating families

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

global stability of endemic equilibrium for a siqrs epidemic model on complex networks abstract we study the global stability of endemic equilibrium of a siqrs epidemic model on complex networks this model was proposed by li et al 2014 the local asymptotic stability of disease free equilibrium and the permanence of the disease were proved by them by use of a combination of lyapunov function method and a monotone iterative technique we prove that the endemic equilibrium is globally asymptotically stable numerical simulations indicate that quarantine strategy is an effective measure in preventing and controlling epidemic spreading

a mean field spin glass with short range interactions we formulate and study a spin glass model on the bethe lattice appropriate boundary fields replace the traditional self consistent methods they give our model well defined thermodynamic properties we establish that there is a spin glass transition temperature above which the single site magnetizations vanish and below which the edwards anderson order parameter is strictly positive in a neighborhood below the transition temperature we use bifurcation theory to establish the existence of a nontrivial distribution of single site magnetizations two properties of this distribution are studied the leading perturbative correction to the gaussian scaling form at the transition and the nonperturbative behavior of the tails

optimization of output load in a junction model an optimization problem of minimizing the output load of a junction model with multiple input and output channels is solved by the replica method simulation results show that for the case of extensive connectivity the replica solution obtained by assuming a vanishing solution space is satisfactory however for the case of intensive connectivity the degeneracy of the solution space causes the simulation results to deviate from the replica solution of vanishing volume

graph partitioning and dilute spin glasses the minimum cost solution the authors calculate the ground state cost function energy for graph bipartitioning and spin glasses on networks with fixed finite valences allowing a continuous distribution for the effective field compared with previous calculations using a field distribution in integral multiples of the coupling strength j the ground state energy is higher by less than one per cent although the distribution itself is altered drastically

stability analysis of an sis epidemic model with feedback mechanism on networks abstract in this paper an epidemic model with feedback mechanism on networks is investigated we obtain the basic reproductive number r 0 and analyze the stability behaviors of disease free equilibrium e 0 and endemic equilibrium e when r 0 1 the disease free equilibrium is globally asymptotically stable and when r 0 1 the endemic equilibrium is asymptotically stable although the feedback mechanism cannot change the basic reproductive number r 0 as we prove it should be indicated that it can weaken the spreading of diseases and reduce the endemic level finally the results of the stability and the effectiveness of the feedback mechanism are illustrated by some numerical simulations

global stability and attractivity of a network based sis epidemic model with nonmonotone incidence rate abstract in this paper we study the global stability and attractivity of the endemic equilibrium for a network based sis epidemic model with nonmonotone incidence rate the model was introduced in li 2015 we prove that the endemic equilibrium is globally asymptotically stable if a parameter of this model is sufficiently large and is globally attractive if the transmission rate satisfies c 1 2 where c is the epidemic threshold some numerical experiments are also presented to illustrate the theoretical results

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

approximation with one or few parameters nonlinear abstract in the case of one or few nonlinear parameters minimax approximation can be done by applying minimization with respect to the nonlinear parameters and reduction to a linear subproblem only the case of a linear subproblem with a haar subspace is considered in which case a linear remez algorithm is used

behavior of susceptible infected susceptible epidemics on heterogeneous networks with saturation we investigate saturation effects in susceptible infected susceptible models of the spread of epidemics in heterogeneous populations the structure of interactions in the population is represented by networks with connectivity distribution p k including scale free sf networks with power law distributions p k approximately k gamma considering cases where the transmission of infection between nodes depends on their connectivity we introduce a saturation function c k which reduces the infection transmission rate lambda across an edge going from a node with high connectivity k a mean field approximation with the neglect of degree degree correlation then leads to a finite threshold lambda c 0 for sf networks with 2 gamma 3 we also find in this approximation the fraction of infected individuals among those with degree k for lambda close to lambda c we investigate via computer simulation the contact process on a heterogeneous regular lattice and compare the results with those obtained from mean field theory with and without neglect of degree degree correlations

the free energy at t 0 of the spin glass in the case of the continuous distribution of the effective field on the bethe lattice for z 4 5 6 abstract we evaluated the free energy at t 0 of the j random ising model on the bethe lattice of the coordination number z 4 5 6 in the case where the distribution function of the effective field has a continuous distribution by using the method of the integral equation of the effective field

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

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

infection dynamics on scale free networks we discuss properties of infection processes on scale free networks relating them to the node connectivity distribution that characterizes the network considering the epidemiologically important case of a disease that confers permanent immunity upon recovery we derive analytic expressions for the final size of an epidemic in an infinite closed population and for the dependence of infection probability on an individual s degree of connectivity within the population as in an earlier study r pastor satorras and a vesipignani phys rev lett 86 3200 2001 phys rev e 63 006117 2001 for an infection that did not confer immunity upon recovery the epidemic process in contrast with many traditional epidemiological models does not exhibit threshold behavior and we demonstrate that this is a consequence of the extreme heterogeneity in the connectivity distribution of a scale free network finally we discuss effects that arise from finite population sizes showing that networks of finite size do exhibit threshold effects infections cannot spread for arbitrarily low transmission probabilities

graph bipartitioning and spin glasses on a random network of fixed finite valence the authors study the problem of bipartitioning a random graph of fixed finite valence using a mean field replica symmetric theory of an ising ferromagnet with zero magnetisation constraint the thermodynamics is determined by the probability distribution of an auxiliary field the expression for the ground state energy agrees with that proposed by mezard and parisi 1986 using a cavity field method but their expression for the fraction of crazy spins is reinterpreted

minimizing unsatisfaction in colourful neighbourhoods colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance here we consider the problem of maximizing the number of different colours available at the nodes and their neighbourhoods given a predetermined number of colours in the analytical framework of a tree approximation carried out at both zero and finite temperatures solutions obtained by population dynamics give rise to estimates of the threshold connectivity for the incomplete to complete transition which are consistent with those of existing algorithms the nature of the transition as well as the validity of the tree approximation are investigated

solving nonlinear systems of equations with only one nonlinear variable abstract we describe a method for solving systems of n 1 nonlinear equations in n 1 unknowns y te r and z te r n of the form a y z b y 0 where the n 1 n matrix a y and vector b y are functions of y alone such equations arise in minimax approximation we reduce the problem to one equation in y only an efficient quadratically convergent numerical technique based on newton s method in one variable is used to solve this equation computational details and results are provided and two generalizations are discussed

dynamics of an saits alcoholism model on unweighted and weighted networks abstract a novel saits alcoholism model on networks is introduced in which alcoholics are divided into light problem alcoholics and heavy problem alcoholics susceptible individuals can enter into the compartment of heavy problem alcoholics directly by contacting with light problem alcoholics or heavy problem alcoholics and the heavy problem alcoholics who receive treatment can relapse into the compartment of heavy problem alcoholics are also considered first the dynamics of our model on unweighted networks including the basic reproduction number existence and stability of equilibria are studied second the models with fixed weighted and adaptive weighted networks are introduced and investigated at last some simulations are presented to illustrate and extend our results our results show that it is very important to treat alcoholics to quit the drinking

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

zero temperature hysteresis in the random field ising model on a bethe lattice we consider the single spin flip dynamics of the random field ising model on a bethe lattice at zero temperature in the presence of a uniform external field we determine the average magnetization as the external field is varied from to by setting up the self consistent field equations which we show are exact in this case the qualitative behaviour of magnetization as a function of the external field unexpectedly depends on the coordination number z of the bethe lattice for z 3 with a gaussian distribution of the quenched random fields we find no jump in magnetization for any non zero strength of disorder for for weak disorder the magnetization shows a jump discontinuity as a function of the external uniform field which disappears for a larger variance of the quenched field we determine exactly the critical point separating smooth hysteresis curves from those with a jump we have checked our results by monte carlo simulations of the model on three and four coordinated random graphs which for large system sizes give the same results as on the bethe lattice but avoid surface effects altogether

graph bipartitioning and the bethe spin glass the problem of the equipartitioning of a random graph of fixed finite valence is studied by comparison with a ferromagnetic bethe lattice with random boundary conditions the simplicity of recursion relations for effective fields due to descendents on bethe lattices provides simple approximations for the optimal cost in quite good agreement with simulations

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

q partitioning of graphs with finite coordination number the np complete optimisation problem of q partitioning of graphs with finite connectivity is discussed using the effective local field method the authors obtain the local field distributions with and without the continuous part and use them to calculate the ground state energies of the three state potts spin glass on lattices with finite coordination number equal to three this in turn gives analytic results for the optimal cost of the 3 partitioning of graphs they perform simulations of actual 3 partitions of random graphs to compare with their theoretical results and obtain very good agreement ways for further improvement of the estimates are discussed

graph bipartitioning and statistical mechanics the problem of bipartitioning a random graph of fixed finite local valence connectivity so as to minimise the number of cross edges is studied by the application of monte carlo simulation of thermodynamics and annealing this problem is np complete and is relevant as an idealisation of several practical organisation problems evidence is presented for the existence of a multiplicity of metastable states and for non self averaging and ultrametricity in the space of overlaps albeit with a possible critical valence an empirical formula is presented for the optimal cost in excellent accord with values obtained by simulation and those known exactly

a fractional order model for ebola virus infection with delayed immune response on heterogeneous complex networks abstract most of the biological systems have long range temporal memory and modeling of such systems by fractional order differential equations has more advantages than classical models with integer orders in this paper we provide a fractional order ebola virus epidemic model with delayed immune response on heterogeneous complex networks the time delay is introduced in the cytotoxic t lymphocyte ctls term based on fractional laplace transform some conditions on stability are derived for the model the analysis shows that the fractional order time delay can effectively enrich the dynamics and strengthen the stability condition of fractional order infection model finally the derived theoretical results are justified by some numerical simulations

global stability analysis of an sir epidemic model with demographics and time delay on networks abstract in this paper a susceptible infected recovery sir epidemic model is governed with demographics and time delay on networks firstly the basic reproduction number r 0 is derived dependent on birth rate death rate recovery rate and transmission rate the disease free equilibrium of the model is stable when r 0 1 and unstable when r 0 1 secondly based on a jacobian matrix calculated along with the disease free equilibrium we find that the system does not occur hopf branch under the disease free equilibrium thirdly the global asymptotic stability of a disease free equilibrium and a unique endemic equilibrium are proved by structuring two lyapunov functions finally numerical simulations are performed to illustrate the analysis results

the graph partitioning and the spin glass comments on papers by sherrington and his group the recent theory of graph partitioning proposed by sherrington and his group 1987 1988 used the integral equation to determine the field distribution this integral equation and its solution are pointed out to be the same as those used by katsura and co workers 1979 1980 for spin glasses

extremal optimization of graph partitioning at the percolation threshold the benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem the performance of this new method called extremal optimization eo is compared with simulated annealing sa in extensive numerical simulations while generally a complex np hard problem the optimization of the graph partitions is particularly difficult for sparse graphs with average connectivities near the percolation threshold at this threshold the relative error of sa for large graphs is found to diverge relative to eo at equalized runtime on the other hand eo based on the extremal dynamics of self organized critical systems reproduces known results about optimal partitions at this critical point quite well