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

sets of conductivity and elasticity tensors stable under lamination a complete characterization is obtained of any set of constitutive conductivity or elasticity tensors that is stable under lamination between any two of the elements of the set the conditions are local and are expressed in terms of the curvature and tangent plane at points on the boundary of the set they are checked to hold for several well known examples of sets stable under a more general process of homogenization a companion paper investigates the link between these conditions and quasiconvexity 1994 john wiley sons inc

on the g closure in the polycrystalline problem this work is concerned with the g closure problem for electrical conductivity in three space dimensions in the context of polycrystalline composites its specific goal is to explore the attainability of a certain lower bound this bound was proved by avellaneda et al j appl phys 63 1988 pp 4989 5003 in the case of uniaxial basic crystal and isotropic composites the bound was conjectured by schulgasser j appl phys 54 1983 p 1380 the former work established the attainability of this bound only under very special circumstances e g when the basic crystal is uniaxial the main conclusion of the present work is that in the general case when the conductivity tensor of the basic crystal has distinct eigenvalues the lower bound is always attained by a two dimensional set ofrank one laminates

optimality conditions for nonconvex multistate control problems in the coefficients the purpose of this paper is to attain some optimality conditions for the identification of a diffusion matrix material under several restrictions assuming that the set of such diffusion matrices is closed for the h convergence we give a method to obtain admissible directions which applies to a not necessarily convex control set our results permit obtaining the diffusion matrix from the state functions

importance of small degree nodes in assortative networks with degree weight correlations it has been known that assortative network structure plays an important role in spreading dynamics for unweighted networks yet its influence on weighted networks is not clear in particular when weight is strongly correlated with the degrees of the nodes as we empirically observed in twitter here we use the self consistent probability method and revised nonperturbative heterogenous mean field theory method to investigate this influence on both susceptible infective recovered sir and susceptible infective susceptible sis spreading dynamics both our simulation and theoretical results show that while the critical threshold is not significantly influenced by the assortativity the prevalence in the supercritical regime shows a crossover under different degree weight correlations in particular unlike the case of random mixing networks in assortative networks the negative degree weight correlation leads to higher prevalence in their spreading beyond the critical transmissivity than that of the positively correlated in addition the previously observed inhibition effect on spreading velocity by assortative structure is not apparent in negatively degree weight correlated networks while it is enhanced for that of the positively correlated detailed investigation into the degree distribution of the infected nodes reveals that small degree nodes play essential roles in the supercritical phase of both sir and sis spreadings our results have direct implications in understanding viral information spreading over online social networks and epidemic spreading over contact networks

rank one plus a null lagrangian is an inherited property of two dimensional compliance tensors under homogenisation assume that the local compliance tensor of an elastic composite in two space dimensions is equal to a rank one tensor plus a null lagrangian there is only one symmetric one in 2d the purpose of this paper is to prove that the effective compliance tensor has the same representation rank one plus the null lagrangian this statement generalizes the well known result of hill 14 15 that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropic and shares the same shear modulus it also generalizes the surprising discovery in 1 that under a certain condition on the pure crystal moduli the shear modulus of an isotropic polycrystal is uniquely determined the present paper sheds light on this effect by placing it in a more general framework and using some elliptic pde theory rather than the translation method our results allow us to calculate the polycrystalline g closures of the special class of crystals under consideration our analysis is contrasted with a two dimensional model problem for shape memory polycrystals we show that the two problems can be thought of as elastic percolation problems one elliptic one hyperbolic

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

generation of uncorrelated random scale free networks uncorrelated random scale free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks we propose and analyze a model capable of generating random uncorrelated scale free networks with no multiple and self connections the model is based on the classical configuration model with an additional restriction on the maximum possible degree of the vertices we check numerically that the proposed model indeed generates scale free networks with no two and three vertex correlations as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree k respectively complex networks constitute a general framework for the topological characterization of many natural and technological systems whose complexity prevents a more detailed microscopic description f1 3g within this framework these systems are represented in terms of networks or graphs f4g in which vertices stand for the units composing the system while edges among vertices represent the interactions or relations between pairs of units the focus is thus shift to the topological characterization of the representative network a task which is largely more feasible and yields nevertheless a noticeable amount of information on the structure and properties of the original system the empirical analysis of many real complex networks has unveiled the presence of several typical properties widely found in systems belonging to a large variety of realms one of the most relevant is given by the scale free nature of the degree distribution pskd f1 3 5g defined as the probability that a randomly chosen vertex has degree k si e it is connected to other k verticesd in mathematical terms the scale free property translates into a power law function of the form psk d k g

optimal g closure bounds via stability under lamination optimal bounds for a two dimensional g closure problem are proved they are obtained by combining together bounds derived using fine properties of quasiconformal mappings and a stability property of the g closure which is known as stability under lamination

asymptotic properties of degree correlated scale free networks the possible correlation profiles of networks with a given scale free degree distribution are restricted and bounded by maximally correlated configurations dissortative networks consist of nested bilayers in which low degree vertices are connected to high degree vertices the number of these bilayers attains a constant value for large network size n assortative networks exhibit monolayers of low degree vertices the number of which grows monotonously with n analytical relations for the pearson correlation coefficient r of these extremal configurations are derived and shown to provide lower and upper bounds on the possible r values both bounds are found to vanish for large networks

a link between sets of tensors stable under lamination and quasiconvexity a link is found between quasiconvexity and the conditions for a set l of conductivity or elasticity tensors to be stable under lamination these conditions derived in the companion paper are shown here to be equivalent to the condition that for every point b on the boundary of the set l an operator tb dependent on the tangent plane and curvature of the set at b is a quasiconvex translation operator a separate class of quasiconvex translation operators is obtained which are candidates for proving that l is stable under homogenization the region stable under homogenization associated with any one of these operators shares a common boundary point and tangent plane with the set l and has curvature at that point not greater than the curvature of l the conditions under which there exists a representative subclass of these operators such that the associated regions stable under homogenization wrap around l remains unresolved it is proved that l can be characterized by minimizations of sums and dual energies in much the same way that convex sets can be characterized by their legendre transforms 1994 john wiley sons inc

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

modeling the evolution of weighted networks we present a general model for the growth of weighted networks in which the structural growth is coupled with the edges weight dynamical evolution the model is based on a simple weight driven dynamics and a weights reinforcement mechanism coupled to the local network growth that coupling can be generalized in order to include the effect of additional randomness and nonlinearities which can be present in real world networks the model generates weighted graphs exhibiting the statistical properties observed in several real world systems in particular the model yields a nontrivial time evolution of vertices properties and scale free behavior with exponents depending on the microscopic parameters characterizing the coupling rules very interestingly the generated graphs spontaneously achieve a complex hierarchical architecture characterized by clustering and connectivity correlations varying as a function of the vertices degree

reshuffling scale free networks from random to assortative to analyze the role of assortativity in networks we introduce an algorithm which produces assortative mixing to a desired degree this degree is governed by one parameter p changing this parameter one can construct networks ranging from fully random p 0 to totally assortative p 1 we apply the algorithm to a barabasi albert scale free network and show that the degree of assortativity is an important parameter governing the geometrical and transport properties of networks thus the average path length of the network increases dramatically with the degree of assortativity moreover the concentration dependences of the size of the giant component in the node percolation problem for uncorrelated and assortative networks are strongly different the behavior of the clustering coefficient is also discussed

restrictions on microstructure we consider the following question given a set of matrices with no rank one connections does it support a nontrivial young measure limit of gradients our main results are these a a young measure can be supported on four incompatible matrices b in two space dimensions a young measure cannot be supported on finitely many incompatible elastic wells c in three or more space dimensions a young measure can be supported on three incompatible elastic wells and d if supports a nontrivial young measure with mean value 0 then the linear span of must contain a matrix of rank one

on characterizing the set of possible effective tensors of composites the variational method and the translation method a general algebraic framework is developed for characterizing the set of possible effective tensors of composites a transformation to the polarization problem simplifies the derivation of the hashin shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials a general connection is established between two methods for bounding effective tensors of composites the first method is based on the variational principles of hashin and shtrikman the second method due to tartar murat lurie and cherkaev uses translation operators or equivalently quadratic quasiconvex functions a correspondence is established between these translation operators and bounding operators on the relevant non local projection operator t1 an important class of bounds namely trace bounds on the effective tensors of two component media are given a geometrical interpretation after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors a wide class of translation operators that generate these bounds is found the extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors these extremal translations generate attainable trace bounds even when the tensors of the two components are not well ordered in particular they generate the bounds of walpole on the effective bulk modulus the variational principles of gibiansky and cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by golden and papanicolaou an isomorphism is shown to underlie their variational principles this isomorphism is used to obtain dirichlet type variational principles and bounds for the effective tensors of general non selfadjoint problems it is anticipated that these variational principles which stem from the work of gibiansky and cherkaev will have applications in many fields of science

rank one convexity does not imply quasiconvexity we consider variational integrals defined for sufficiently regular functions u r m here is a bounded open subset of r n du x denotes the gradient matrix of u at x and f is a continuous function on the space of all real m n matrices m m n one of the important problems in the calculus of variations is to characterise the functions f for which the integral i is lower semicontinuous in this connection the following notions were introduced see 3 9 10

optimal bounds for conduction in two dimensional multiphase polycrystalline media a complete geometrical characterization is given of the set of all possible effective conductivity tensors for two dimensional composites made of an arbitrary number of given anisotropic phases it is rigorously established that any polycrystalline composite formed from an arbitrary possibly infinite number of phases can be replaced by a composite formed from only two of the given phases without altering the effective conductivity tensor

decelerated spreading in degree correlated networks while degree correlations are known to play a crucial role for spreading phenomena in networks their impact on the propagation speed has hardly been understood here we investigate a tunable spreading model on scale free networks and show that the propagation becomes slow in positively negatively correlated networks if nodes with a high connectivity locally accelerate decelerate the propagation examining the efficient paths offers a coherent explanation for this result while the k core decomposition reveals the dependence of the nodal spreading efficiency on the correlation our findings should open new pathways to delicately control real world spreading processes

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

theory of rumour spreading in complex social networks we introduce a general stochastic model for the spread of rumours and derive mean field equations that describe the dynamics of the model on complex social networks in particular those mediated by the internet we use analytical and numerical solutions of these equations to examine the threshold behaviour and dynamics of the model on several models of such networks random graphs uncorrelated scale free networks and scale free networks with assortative degree correlations we show that in both homogeneous networks and random graphs the model exhibits a critical threshold in the rumour spreading rate below which a rumour cannot propagate in the system in the case of scale free networks on the other hand this threshold becomes vanishingly small in the limit of infinite system size we find that the initial rate at which a rumour spreads is much higher in scale free networks than in random graphs and that the rate at which the spreading proceeds on scale free networks is further increased when assortative degree correlations are introduced the impact of degree correlations on the final fraction of nodes that ever hears a rumour however depends on the interplay between network topology and the rumour spreading rate our results show that scale free social networks are prone to the spreading of rumours just as they are to the spreading of infections they are relevant to the spreading dynamics of chain emails viral advertising and large scale information dissemination algorithms on the internet

arzela s theorem and strong uniform convergence on bornologies abstract in 1883 arzela 1983 1984 2 gave a necessary and sufficient condition via quasi uniform convergence for the pointwise limit of a sequence of real valued continuous functions on a compact interval to be continuous arzela s work paved the way for several outstanding papers a milestone was the p s alexandroff convergence introduced in 1948 to tackle the question for a sequence of continuous functions from a topological space not necessarily compact to a metric space in 2009 in the realm of metric spaces beer and levi 2009 10 found another necessary and sufficient condition through the novel notion of strong uniform convergence on finite sets we offer a direct proof of the equivalence of arzela alexandroff and beer levi conditions the proof reveals the internal gear of these important convergences and sheds more light on the problem we also study the main properties of the topology of strong uniform convergence of functions on bornologies initiated in beer and levi 2009 10

exact relations for effective tensors of polycrystals i necessary conditions the set of all effective moduli of a polycrystal usually has a nonempty interior when it does not we say that there is an exact relation for effective moduli this can indeed happen as evidenced by recent results 4 10 12 on polycrystals in this paper we describe a general method for finding such relations for effective moduli of laminates the method is applicable to any physical setting that can be put into the hilbert space framework developed by milton 13 the idea is to use the w function of milton 13 that transforms a lamination formula into a convex combination the method reduces the problem of finding exact relations to a problem from representation theory of so d d 2 or 3 corresponding to a particular physical setting when this last problem is solved there is a finite amount of calculation required to be done in order to answer the question completely at present each candidate relation has to be examined separately in order to confirm the stability under homogenization we apply our general theory to the settings of conductivity and two dimensional elasticity