condensation transitions in a model for a directed network with weighted links an exactly solvable model for the rewiring dynamics of weighted directed networks is introduced simulations indicate that the model exhibits two types of condensation i a phase in which for each node a finite fraction of its total out strength condenses onto a single link ii a phase in which a finite fraction of the total weight in the system is directed into a single node a virtue of the model is that its dynamics can be mapped onto those of a zero range process with many species of interacting particles an exactly solvable model of particles hopping between the sites of a lattice this mapping which is described in detail guides the analysis of the steady state of the network model and leads to theoretical predictions for the conditions under which the different types of condensation may be observed a further advantage of the mapping is that by exploiting what is known about exactly solvable generalisations of the zerorange process one can infer a number of generalisations of the network model and dynamics which remain exactly solvable

uncorrelated random networks we define a statistical ensemble of nondegenerate graphs i e graphs without multiple connections and self connections between nodes the node degree distribution is arbitrary but the nodes are assumed to be uncorrelated this completes our earlier publication phys rev 64 046118 2001 where trees and degenerate graphs were considered an efficient algorithm generating nondegenerate graphs is constructed the corresponding computer code is available on request finite size effects in scale free graphs i e those where the tail of the degree distribution falls like n beta are carefully studied we find that in the absence of dynamical internode correlations the degree distribution is cut at a degree value scaling like n gamma with gamma min 1 2 1 beta 1 where n is the total number of nodes the consequence is that independently of any specific model the internode correlations seem to be a necessary ingredient of the physics of scale free networks observed in nature

average path length in random networks analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found we apply the approach to classical random graphs of erdos and renyi er evolving networks introduced by barabasi and albert as well as random networks with asymptotic scale free connectivity distributions characterized by an arbitrary scaling exponent alpha 2 our result for 2 infinity there is a saturation effect for the average path length

fluctuation dissipation relations in complex networks in this paper we study fluctuations over several ensembles of maximum entropy random networks we derive several fluctuation dissipation relations characterizing the susceptibilities of different networks to changes in external fields in the case of networks with a given degree sequence we argue that the scale free topologies of real world networks may arise as a result of the self organization of real systems into sparse structures with low susceptibility to random external disruptions we also show that the ensembles of networks with a given degree sequence and networks characterized by two point correlations are equivalent to random networks with hidden variables

statistical mechanical load balancer for the web the maximum entropy principle from statistical mechanics states that a closed system attains an equilibrium distribution that maximizes its entropy we first show that for graphs with fixed number of edges one can define a stochastic edge dynamic that can serve as an effective thermalization scheme and hence the underlying graphs are expected to attain their maximum entropy states which turn out to be erdos renyi er random graphs we next show that i a rate equation based analysis of node degree distribution does indeed confirm the maximum entropy principle and ii the edge dynamic can be effectively implemented using short random walks on the underlying graphs leading to a local algorithm for the generation of er random graphs the resulting statistical mechanical system can be adapted to provide a distributed and local i e without any centralized monitoring mechanism for load balancing which can have a significant impact in increasing the efficiency and utilization of both the internet e g efficient web mirroring and large scale computing infrastructure e g cluster and grid computing

statistical mechanics of random graphs we discuss various aspects of the statistical formulation of the theory of random graphs with emphasis on results obtained in a series of our recent publications

variational asymptotic method for unit cell homogenization of periodically heterogeneous materials a new micromechanics model namely the variational asymptotic method for unit cell homogenization vamuch is developed to predict the effective properties of periodically heterogeneous materials and recover the local fields considering the periodicity as a small parameter we can formulate a variational statement of the unit cell through an asymptotic expansion of the energy functional it is shown that the governing differential equations and periodic boundary conditions of mathematical homogenization theories mht can be reproduced from this variational statement in comparison to other approaches vamuch does not rely on ad hoc assumptions has the same rigor as mht has a straightforward numerical implementation and can calculate the complete set of properties simultaneously without using multiple loadings this theory is implemented using the finite element method and an engineering program vamuch is developed for micromechanical analysis of unit cells many examples of binary composites fiber reinforced composites and particle reinforced composites are used to demonstrate the application power and accuracy of the theory and the code of vamuch

statistical thermodynamics for choice models on graphs abstract formalism based on equilibrium statistical thermodynamics is applied to communication networks of decision making individuals it is shown that in statistical ensembles for choice models properly defined disutility plays the same role as energy in statistical mechanics we demonstrate additivity and extensivity of disutility and build three types of equilibrium statistical ensembles the canonical the grand canonical and the super canonical using boltzmann probability measure one can reproduce the logit choice model we propose using q distributions for temperature evolution of moments of stochastic variables the formalism is applied to networks with fixed topologies of different degrees of symmetry for which analytic and numerical results are presented

spectral dimension of trees with a unique infinite spine using generating functions techniques we develop a relation between the hausdorff and spectral dimension of trees with a unique infinite spine furthermore it is shown that if the outgrowths along the spine are independent and identically distributed then both the hausdorff and spectral dimension can easily be determined from the probability generating function of the random variable describing the size of the outgrowths at a given vertex provided that the probability of the height of the outgrowths exceeding n falls off as the inverse of n we apply this new method to both critical non generic trees and the attachment and grafting model which is a special case of the vertex splitting model resulting in a simplified proof for the values of the hausdorff and spectral dimension for the former and novel results for the latter

effect of trap position on the efficiency of trapping in treelike scale free networks the conventional wisdom is that the role and impact of nodes on dynamical processes in scale free networks are not homogenous because of the presence of highly connected nodes at the tail of their power law degree distribution in this paper we explore the influence of different nodes as traps on the trapping efficiency of the trapping problem taking place on scale free networks to this end we study in detail the trapping problem in two families of deterministically growing scale free networks with treelike structure one family is non fractal the other is fractal in the first part of this work we attack a special case of random walks on the two network families with a perfect trap located at a hub i e node with the highest degree the second study addresses the case with trap distributed uniformly over all nodes in the networks for these two cases we compute analytically the mean trapping time mtt a quantitative indicator characterizing the trapping efficiency of the trapping process we show that in the non fractal scale free networks the mtt for both cases follows different scalings with the network order number of network nodes implying that trap s position has a significant effect on the trapping efficiency in contrast it is presented that for both cases in the fractal scale free networks the two leading scalings exhibit the same dependence on the network order suggesting that the location of trap has no essential impact on the trapping efficiency we also show that for both cases of the trapping problem the trapping efficiency is more efficient in the non fractal scale free networks than in their fractal counterparts

finiteness and fluctuations in growing networks we study the role of finiteness and fluctuations about average quantities for basic structural properties of growing networks we first determine the exact degree distribution of finite networks by generating function approaches the resulting distributions exhibit an unusual finite size scaling behaviour and they are also sensitive to the initial conditions we argue that fluctuations in the number of nodes of degree k become gaussian for fixed degree as the size of the network diverges we also characterize the fluctuations between different realizations of the network in terms of higher moments of the degree distribution

coincidence points generalized i nonexpansive multimaps and applications we establish coincidence point results for multimaps satisfying generalized i nonexpansive type conditions several invariant approximation results are obtained as applications a random coincidence point result is also proved and as applications random invariant approximation results are obtained our results unify extend and complement several well known results

a bending gradient model for thick plates part i theory abstract this is the first part of a two part paper dedicated to a new plate theory for out of plane loaded thick plates where the static unknowns are those of the kirchhoff love theory 3 in plane stresses and 3 bending moments to which six components are added representing the gradient of the bending moment the new theory called the bending gradient plate theory is described in the present paper it is an extension to arbitrarily layered plates of the reissner mindlin plate theory which appears as a special case of the bending gradient plate theory when the plate is homogeneous however we demonstrate also that in the general case the bending gradient model cannot be reduced to a reissner mindlin model in part two lebee and sab 2011 the bending gradient theory is applied to multilayered plates and its predictions are compared to those of the reissner mindlin theory and to full 3d pagano s exact solutions the main conclusion of the second part is that the bending gradient gives good predictions of both deflection and shear stress distributions in any material configuration moreover under some symmetry conditions the bending gradient model coincides with the second order approximation of the exact solution as the slenderness ratio l h goes to infinity

organization of complex networks without multiple connections we find a new structural feature of equilibrium complex random networks without multiple and self connections we show that if the number of connections is sufficiently high these networks contain a core of highly interconnected vertices the number of vertices in this core varies in the range between const x n1 2 and const x n2 3 where is the number of vertices in a network at the birth point of the core we obtain the size dependent cutoff of the distribution of the number of connections and find that its position differs from earlier estimates

cone compression and expansion fixed point theorems in frechet spaces with applications abstract new fixed point theory for multivalued maps between frechet spaces is presented applications are given to illustrate how the theory can be applied in practice

generation of arbitrarily two point correlated random networks random networks are intensively used as null models to investigate properties of complex networks we describe an efficient and accurate algorithm to generate arbitrarily two point degree degree correlated undirected random networks without self edges or multiple edges among vertices with the goal to systematically investigate the influence of two point correlations we furthermore develop a formalism to construct a joint degree distribution p j k which allows one to fix an arbitrary degree distribution p k and an arbitrary average nearest neighbor function k nn k simultaneously using the presented algorithm this formalism is demonstrated with scale free networks p k proportional variantk gamma and empirical complex networks p k taken from network as examples finally we generalize our algorithm to annealed networks which allows networks to be represented in a mean field like manner

a note on the existence of multiple fixed points for multivalued maps with applications abstract in this paper we discuss the existence of multiple fixed points for multivalued maps our fixed point results automatically lead to new existence principles for integral inclusions

asymptotic generalization of reissner mindlin theory accurate three dimensional recovery for composite shells abstract a rigorous and systematic dimensional reduction of a shell like structure is undertaken it starts with geometrically nonlinear three dimensional 3 d anisotropic elasticity theory and takes advantage of small parameters associated with the geometry this reduction is carried out using the variational asymptotic method and splits the 3 d problem into a linear one dimensional 1 d through the thickness analysis and a nonlinear two dimensional 2 d shell analysis the 2 d equations are put into the form of a nonlinear reissner mindlin shell theory details of which are dealt with in a separate paper the focus of this paper is on the through the thickness analysis which is solved by a 1 d finite element method and which provides two useful pieces of information a generalized 2 d constitutive law for the shell equations and a set of recovery relations that can be used to express the 3 d field variables through the thickness in terms of 2 d shell variables calculated in the shell analysis the resulting analysis can be incorporated into standard reissner mindlin shell finite element codes numerical results are compared with the exact solution and the excellent agreement validates the fidelity of this modeling approach

nonlinear urysohn discrete equations on the infinite interval a fixed point approach abstract this paper presents a variety of existence results for nonlinear urysohn discrete equations on the infinite interval in particular we consider for i 0 1 2 existence is established using fixed point arguments

network synchronization diffusion and the paradox of heterogeneity many complex networks display strong heterogeneity in the degree connectivity distribution heterogeneity in the degree distribution often reduces the average distance between nodes but paradoxically may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength here we offer a solution to this apparent paradox our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states we show that for a given degree distribution the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum this enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size numerical verification of the main results is provided for representative classes of small world and scale free networks

variational asymptotic micromechanics modeling of heterogeneous porous materials abstract this paper presents a numerical technique to predict the effective elastic properties of heterogeneous fluid filled porous media where the heterogeneity may result from dissimilar solid and fluid phase properties or due to mismatch in porous microstructure the technique is based on the variational asymptotic method of homogenization where finite element method is employed for discretization biot s theory of poroelasticity is used to describe porous media where both solid and fluid phase motions u u formulation are considered with associated strain measures the method estimates the poroelastic constitutive law in single analysis which makes it very efficient compared to other finite element based homogenization techniques the method is also general enough to compute all 28 elements of an anisotropic constitutive matrix other than estimating the effective properties the micro stress strain distribution is also obtained at no additional cost the method is successfully applied for homogenization of porous media fluid filled cavity and finally for effective property estimation of bone lamella in absence of any other direct method of porous media homogenization the present technique is compared with classical homogenization methods with fluid approximated as solid of very high poisson s ratio the suitability of this approximation and various other alternatives are also discussed it is shown that the present homogenization method can be an efficient tool for bone property estimation where fluid filled porous hierarchical micro nanostructure must be respected at all steps

exactly solvable scale free network model we study a deterministic scale free network recently proposed by barabasi ravasz and vicsek we find that there are two types of nodes the hub and rim nodes which form a bipartite structure of the network we first derive the exact numbers p k of nodes with degree k for the hub and rim nodes in each generation of the network respectively using this we obtain the exact exponents of the distribution function p k of nodes with k degree in the asymptotic limit of k infinity we show that the degree distribution for the hub nodes exhibits the scale free nature p k proportional to k gamma with gamma ln 3 ln 2 1 584 962 while the degree distribution for the rim nodes is given by p k proportional to e gamma k with gamma ln 3 2 0 405 465 second we analytically calculate the second order average degree of nodes d third we numerically as well as analytically calculate the spectra of the adjacency matrix a for representing topology of the network we also analytically obtain the exact number of degeneracies at each eigenvalue in the network the density of states i e the distribution function of eigenvalues exhibits the fractal nature with respect to the degeneracy fourth we study the mathematical structure of the determinant of the eigenequation for the adjacency matrix fifth we study hidden symmetry zero modes and its index theorem in the deterministic scale free network finally we study the nature of the maximum eigenvalue in the spectrum of the deterministic scale free network we will prove several theorems for it using some mathematical theorems thus we show that most of all important quantities in the network theory can be analytically obtained in the deterministic scale free network model of barabasi ravasz and vicsek therefore we may call this network model the exactly solvable scale free network

random fixed points of set valued maps some random fixed point theorems for set valued random operators under very mild conditions are established some recent results of o regan proc amer math soc 126 1998 3045 3053 and computers math applic 35 1998 27 34 are improved significantly the discussion in this paper underlines in addition to generality the unifying aspects of our result

mesoscopics and fluctuations in networks we describe fluctuations in finite size networks with a complex distribution of connections p k we show that the spectrum of fluctuations of the number of vertices with a given degree is poissonian these mesoscopic fluctuations are strong in the large degree region where p k less similar 1 n n is the total number of vertices in a network and are important in networks with fat tailed degree distributions

kinetic theory of random graphs from paths to cycles the structural properties of evolving random graphs are investigated treating linking as a dynamic aggregation process rate equations for the distribution of node to node distances paths and of cycles are formulated and solved analytically at the gelation point the typical length of paths and cycles l scales with the component size k as l approximately k 1 2 dynamic and finite size scaling laws for the behavior at and near the gelation point are obtained finite size scaling laws are verified using numerical simulations

finite size effects in barabasi albert growing networks we investigate the influence of the network s size on the degree distribution ensuremath pi k in barab asi albert model of growing network with initial attractiveness our approach based on moments of ensuremath pi k allows us to treat analytically several variants of the model and to calculate the cutoff function giving finite size corrections to ensuremath pi k we study the effect of initial configuration as well as of addition of more than one link per time step the results indicate that asymptotic properties of the cutoff depend only on the exponent ensuremath gamma in the power law describing the tail of the degree distribution the method presented in this paper is very general and can be applied to other growing networks

finite time fluctuations in the degree statistics of growing networks this paper presents a comprehensive analysis of the degree statistics in models for growing networks where new nodes enter one at a time and attach to one earlier node according to a stochastic rule the models with uniform attachment linear attachment the barabasi albert model and generalized preferential attachment with initial attractiveness are successively considered the main emphasis is on finite size i e finite time effects which are shown to exhibit different behaviors in three regimes of the size degree plane stationary finite size scaling large deviations

exact solution for the time evolution of network rewiring models we consider the rewiring of a bipartite graph using a mixture of random and preferential attachment the full mean field equations for the degree distribution and its generating function are given the exact solution of these equations for all finite parameter values at any time is found in terms of standard functions it is demonstrated that these solutions are an excellent fit to numerical simulations of the model we discuss the relationship between our model and several others in the literature including examples of urn backgammon and balls in boxes models the watts and strogatz rewiring problem and some models of zero range processes our model is also equivalent to those used in various applications including cultural transmission family name and gene frequencies glasses and wealth distributions finally some voter models and an example of a minority game also show features described by our model

topological phase transitions of random networks abstract to provide a phenomenological theory for the various interesting transitions in restructuring networks we employ a statistical mechanical approach with detailed balance satisfied for the transitions between topological states this enables us to establish an equivalence between the equilibrium rewiring problem we consider and the dynamics of a lattice gas on the edge dual graph of a fully connected network by assigning energies to the different network topologies and defining the appropriate order parameters we find a rich variety of topological phase transitions defined as singular changes in the essential feature s of the global connectivity as a function of a parameter playing the role of the temperature in the critical point scale free networks can be recovered

correlations in connected random graphs we study the properties of the giant connected component in random graphs with arbitrary degree distribution we concentrate on the degree degree correlations we show that the adjoining nodes in the giant connected component are correlated and derive analytic formulas for the joint nearest neighbor degree probability distribution using those results we describe correlations in maximal entropy connected random graphs we show that connected graphs are disassortative and that correlations are strongly related to the presence of one degree nodes leaves we propose an efficient algorithm for generating connected random graphs we illustrate our results with several examples

nonlinear bending analysis of radial stiffened annular laminated sector plates with dynamic relaxation method in this paper the nonlinear bending of laminated stiffened annular sector plates under mechanical loading with various boundary conditions is investigated the plates are made of layers with orthotropic properties and different fiber orientations which the aforementioned fibers are placed in a cartesian coordinate system based on first order shear deformation plate theory fsdt and von karman relations for large deflection nonlinear equilibrium equations are developed dynamic relaxation dr numerical method combined with the finite difference discretization technique is used to solve the plate nonlinear partial differential equations and fortran program is developed to generate the numerical results effects of the plate thickness to radius ratio boundary condition stiffener depth plate lay ups and the sector angle are discussed

on asymptotically correct linear laminated plate theory abstract the focus of this paper is the development of asymptotically correct theories for laminated plates the material properties of which vary through the thickness and for which each lamina is orthotropic this work is based on the variational asymptotical method a mathematical technique by which the three dimensional analysis of plate deformation can be split into two separate analyses a one dimensional through the thickness analysis and a two dimensional plate analysis the through the thickness analysis includes elastic constants for use in the plate theory and approximate closed form recovering relations for all three dimensional field variables expressed in terms of plate variables in general the specific type of plate theory that results from this procedure is determined by the procedure itself however in this paper only reissner like plate theories are considered often called first order shear deformation theories this paper makes three main contributions first it is shown that construction of an asymptotically correct reissner like theory for laminated plates of the type considered is not possible in general second a new point of view on the variational asymptotical method is presented leading to an optimization procedure that permits a derived theory to be as close to asymptotical correctness as possible third numerical results from such an optimum reissner like theory are presented these results include comparisons of plate displacement as well as of three dimensional field variables and are the best of all extant reissner like theories indeed they even surpass results from theories that carry many more generalized displacement variables

entropy of stochastic blockmodel ensembles stochastic blockmodels are generative network models where the vertices are separated into discrete groups and the probability of an edge existing between two vertices is determined solely by their group membership in this paper we derive expressions for the entropy of stochastic blockmodel ensembles we consider several ensemble variants including the traditional model as well as the newly introduced degree corrected version karrer et al phys rev e 83 016107 2011 which imposes a degree sequence on the vertices in addition to the block structure the imposed degree sequence is implemented both as soft constraints where only the expected degrees are imposed and as hard constraints where they are required to be the same on all samples of the ensemble we also consider generalizations to multigraphs and directed graphs we illustrate one of many applications of this measure by directly deriving a log likelihood function from the entropy expression and using it to infer latent block structure in observed data due to the general nature of the ensembles considered the method works well for ensembles with intrinsic degree correlations i e with entropic origin as well as extrinsic degree correlations which go beyond the block structure

asymptotic construction of reissner like composite plate theory with accurate strain recovery abstract the focus of this paper is to develop an asymptotically correct theory for composite laminated plates when each lamina exhibits monoclinic material symmetry the development starts with formulation of the three dimensional 3 d anisotropic elasticity problem in which the deformation of the reference surface is expressed in terms of intrinsic two dimensional 2 d variables the variational asymptotic method is then used to rigorously split this 3 d problem into a linear one dimensional normal line analysis and a nonlinear 2 d plate analysis accounting for classical as well as transverse shear deformation the normal line analysis provides a constitutive law between the generalized 2 d strains and stress resultants as well as recovering relations to approximately but accurately express the 3 d displacement strain and stress fields in terms of plate variables calculated in the plate analysis it is known that more than one theory may exist that is asymptotically correct to a given order this nonuniqueness is used to cast a strain energy functional that is asymptotically correct through the second order into a simple reissner like plate theory although it is not possible in general to construct an asymptotically correct reissner like composite plate theory an optimization procedure is used to drive the present theory as close to being asymptotically correct as possible while maintaining the beauty of the reissner like formulation numerical results are presented to compare with the exact solution as well as a previous similar yet very different theory the present theory has excellent agreement with the previous theory and exact results

appearance of vertices of infinite order in a model of random trees we study an equilibrium statistical mechanical model of tree graphs which are made up of a linear subgraph the spine to which leaves are attached we prove that the model has two phases a generic phase where the spine becomes infinitely long in the thermodynamic limit and all vertices have finite order and a condensed phase where the spine is finite with probability 1 and a single vertex of infinite order appears in the thermodynamic limit we calculate the spectral dimension of the graphs in both phases and prove the existence of a gibbs measure we discuss generalizations of this model and the relationship with models of nongeneric random trees

entropy of network ensembles in this paper we generalize the concept of random networks to describe network ensembles with nontrivial features by a statistical mechanics approach this framework is able to describe undirected and directed network ensembles as well as weighted network ensembles these networks might have nontrivial community structure or in the case of networks embedded in a given space they might have a link probability with a nontrivial dependence on the distance between the nodes these ensembles are characterized by their entropy which evaluates the cardinality of networks in the ensemble in particular in this paper we define and evaluate the structural entropy i e the entropy of the ensembles of undirected uncorrelated simple networks with given degree sequence we stress the apparent paradox that scale free degree distributions are characterized by having small structural entropy while they are so widely encountered in natural social and technological complex systems we propose a solution to the paradox by proving that scale free degree distributions are the most likely degree distribution with the corresponding value of the structural entropy finally the general framework we present in this paper is able to describe microcanonical ensembles of networks as well as canonical or hidden variable network ensembles with significant implications for the formulation of network constructing algorithms

condensation in nongeneric trees we study nongeneric planar trees and prove the existence of a gibbs measure on infinite trees obtained as a weak limit of the finite volume measures it is shown that in the infinite volume limit there arises exactly one vertex of infinite degree and the rest of the tree is distributed like a subcritical galton watson tree with mean offspring probability m 1 we calculate the rate of divergence of the degree of the highest order vertex of finite trees in the thermodynamic limit and show it goes like 1 m n where n is the size of the tree these trees have infinite spectral dimension with probability one but the spectral dimension calculated from the ensemble average of the generating function for return probabilities is given by 2 2 if the weight wn of a vertex of degree n is asymptotic to n

homogeneous complex networks abstract we discuss various ensembles of homogeneous complex networks and a monte carlo method of generating graphs from these ensembles the method is quite general and can be applied to simulate micro canonical canonical or grand canonical ensembles for systems with various statistical weights it can be used to construct homogeneous networks with desired properties or to construct a non trivial scoring function for problems of advanced motif searching

random fixed point theorems and approximation in cones a very general random fixed point theorem is first proved as applications some random approximation theorems and random fixed point theorems are obtained recent results due to itoh j math anal appl 67 1979 261 273 lin proc amer math soc 103 1988 502 506 and xu proc amer math soc 110 1990 495 500 either are improved or are direct consequences of our results the stochastic versions of the recent results of lin 1988 and li proc amer math soc 97 1986 277 280 are also given

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

existence of bounded solutions for multivalued differential systems on demontre l existence de solutions bornees pour le systeme x t a t x t f t x t x s sur un intervalle i reel non borne ou f est une application a valeur convexe semi continue superieurement a est une matrice n n avec conditions de caratheodory s est un sous ensemble d un espace de banach

statistical ensemble of scale free random graphs a thorough discussion of the statistical ensemble of scale free connected random tree graphs is presented methods borrowed from field theory are used to define the ensemble and to study analytically its properties the ensemble is characterized by two global parameters the fractal and the spectral dimensions which are explicitly calculated it is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified the stability of the scale free regime is also considered when it breaks down either a scale is spontaneously generated or else a singular node appears and the graphs become crumpled a new computer algorithm to generate these random graphs is proposed possible generalizations are also discussed in particular more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary degenerate scale free random graphs

geometrically exact intrinsic theory for dynamics of moving composite plates a geometrically exact and fully intrinsic theory is presented for dynamics of composite plates undergoing large deformation to say that the formulation is intrinsic means that it is without displacement and rotation variables although the equations are geometrically exact the highest degree nonlinearities are quadratic there are no singularities associated with finite rotation methods for posing problems in this framework along with advantages of the formulation are discussed

fractality in complex networks critical and supercritical skeletons fractal scaling char22 a power law behavior of the number of boxes needed to tile a given network with respect to the lateral size of the box char22 is studied we introduce a box covering algorithm that is a modified version of the original algorithm introduced by song et al nature london 433 392 2005 this algorithm enables easy implementation fractal networks are viewed as comprising a skeleton and shortcuts the skeleton embedded underneath the original network is a special type of spanning tree based on the edge betweenness centrality it provides a scaffold for the fractality of the network when the skeleton is regarded as a branching tree it exhibits a plateau in the mean branching number as a function of the distance from a root for nonfractal networks on the other hand the mean branching number decays to zero without forming a plateau based on these observations we construct a fractal network model by combining a random branching tree and local shortcuts the scaffold branching tree can be either critical or supercritical depending on the small worldness of a given network for the network constructed from the critical supercritical branching tree the average number of vertices within a given box grows with the lateral size of the box according to a power law an exponential form in the cluster growing method the critical and supercritical skeletons are observed in protein interaction networks and the world wide web respectively the distribution of box masses i e the number of vertices within each box follows a power law p m m ensuremath sim m ensuremath ensuremath eta the exponent ensuremath eta depends on the box lateral size ensuremath ell b for small values of ensuremath ell b ensuremath eta is equal to the degree exponent ensuremath gamma of a given scale free network whereas ensuremath eta approaches the exponent ensuremath tau ensuremath gamma ensuremath gamma ensuremath 1 as ensuremath ell b increases which is the exponent of the cluster size distribution of the random branching tree finally we study the perimeter h ensuremath alpha of a given box ensuremath alpha i e the number of edges connected to different boxes from a given box ensuremath alpha as a function of the box mass m b ensuremath alpha it is obtained that the average perimeter over the boxes with box mass m b is likely to scale as h m b ensuremath sim m b irrespective of the box size ensuremath ell b

reconstruction of first order shear deformation shell theory for laminated and sandwich shells a first order shear deformation shell theory fsdt based on general curvilinear coordinates has been reconstructed through a strain energy transformation based on the definition of the mean displacements the transverse shear strains that are constant through the thickness are improved to vary through the thickness it is assumed that the displacements and in surface strains of three dimensional theory are approximated by those of fsdt relationship between fsdt and three dimensional theory has been systematically established in the averaged least square sense this relationship provides the closed form recovering relations for three dimensional variables expressed in terms of fsdt relations the present approach has been applied to a higher order zigzag shell theory comparisons of displacements and stresses of laminated and sandwich shells using the present theory are made with the original fsdt and three dimensional exact solutions

scale free networks as entropy competition complex networks describe several and different real world systems consisting of a number of interacting elements a very important characteristic of such networks is the degree distribution that strongly controls their behavior based on statistical mechanics three classes of uncorrelated complex networks are identified here depending on the role played by the connectivities amongst elements in particular by identifying the connectivities of a node with the number of its nearest neighbors we show that the power law is the most probable degree distribution that both nodes and neighbors in a reciprocal competition assume when the respective entropy functions reach their maxima under mutual constraint as a result we obtain scaling exponent values as a function of the structural characteristics of the whole network moreover our approach sheds light on the exponential and poissonian degree distributions derived respectively when connectivities are thought of as degenerated connections or as half edges thus all three classes of degree distributions are derived starting from a common principle and leading to a general and unified framework for investigating the network structure

tree networks with causal structure a geometry of networks endowed with a causal structure is discussed using the conventional framework of the equilibrium statistical mechanics the popular growing network models appear as particular causal models we focus on a class of tree graphs an analytically solvable case general formulas are derived describing the degree distribution the ancestor descendant correlation and the probability that a randomly chosen node lives at a given geodesic distance from the root it is shown that the hausdorff dimension d h of the causal networks is generically infinite in contrast to the maximally random trees where it is generically finite

properties of random graphs with hidden color we investigate in some detail a recently suggested general class of ensembles of sparse undirected random graphs based on a hidden stub coloring with or without the restriction to nondegenerate graphs the calculability of local and global structural properties of graphs from the resulting ensembles is demonstrated cluster size statistics are derived with generating function techniques yielding a well defined percolation threshold explicit rules are derived for the enumeration of small subgraphs duality and redundancy is discussed and subclasses corresponding to commonly studied models are identified

anisotropic generalization of stinchcombe s solution for the conductivity of random resistor networks on a bethe lattice our study is based on the work of stinchcombe 1974 j phys c solid state phys 7 179 and is devoted to the calculations of average conductivity of random resistor networks placed on an anisotropic bethe lattice the structure of the bethe lattice is assumed to represent the normal directions of the regular lattice we calculate the anisotropic conductivity as an expansion in powers of the inverse coordination number of the bethe lattice the expansion terms retained deliver an accurate approximation of the conductivity at resistor concentrations above the percolation threshold we make a comparison of our analytical results with those of bernasconi 1974 phys rev b 9 4575 for the regular lattice

statistical mechanics of scale free networks at a critical point complexity without irreversibility based on a rigorous extension of classical statistical mechanics to networks we study a specific microscopic network hamiltonian the form of this hamiltonian is derived from the assumption that individual nodes increase or decrease their utility by linking to nodes with a higher or lower degree than their own we interpret utility as an equivalent to energy in physical systems and discuss the temperature dependence of the emerging networks we observe the existence of a critical temperature tc where total energy utility and network architecture undergo radical changes along this topological transition we obtain ensemble averages of scale free networks with complex hierarchical topology the scale free nature emerges strictly within equilibrium with a clearly defined microcanonical ensemble and the principle of detailed balance fulfilled this provides evidence that complex networks may arise without irreversibility the utility approach establishes a link between classical statistical physics and a wide variety of applications in socioeconomic statistical systems

the statistical geometry of scale free random trees the properties of random trees galton watson trees with scale free power like probability distribution of coordinations are investigated in the thermodynamic limit the scaling form of volume probability is found and the connectivity dimensions are determined and compared with other exponents which describe the growth the local spectral dimension is also determined through the study of the massless limit of the gaussian model on such trees

subgraphs in random networks understanding the subgraph distribution in random networks is important for modeling complex systems in classic erdos networks which exhibit a poissonian degree distribution the number of appearances of a subgraph g with n nodes and g edges scales with network size as approximately n n g however many natural networks have a non poissonian degree distribution here we present approximate equations for the average number of subgraphs in an ensemble of random sparse directed networks characterized by an arbitrary degree sequence we find scaling rules for the commonly occurring case of directed scale free networks in which the outgoing degree distribution scales as p k approximately k gamma considering the power exponent of the degree distribution gamma as a control parameter we show that random networks exhibit transitions between three regimes in each regime the subgraph number of appearances follows a different scaling law approximately nalpha where alpha n g s 1 for gamma gamma c where s is the maximal outdegree in the subgraph and gamma c s 1 we find that certain subgraphs appear much more frequently than in erdos networks these results are in very good agreement with numerical simulations this has implications for detecting network motifs subgraphs that occur in natural networks significantly more than in their randomized counterparts

fractal dimensions of percolating networks we use the generating function formalism to calculate the fractal dimensions for the percolating cluster at criticality in erdos renyi er and random scale free sf networks with degree distribution p k ck we show that the chemical dimension is dl 2 for er and sf networks with 4 as in percolation in d dc 6 dimensions for 3 4 and df 2 2 3 3 4 and dc 2 1 3 3 4 we discuss the meaning of these dimensions for networks

the spectral dimension of generic trees we define generic ensembles of infinite trees these are limits as n of ensembles of finite trees of fixed size n defined in terms of a set of branching weights among these ensembles are those supported on trees with vertices of a uniformly bounded order the associated probability measures are supported on trees with a single spine and hausdorff dimension dh 2 our main result is that the spectral dimension of the ensemble average is ds 4 3 and that the critical exponent of the mass defined as the exponential decay rate of the two point function along the spine is 1 3