spectra of complex networks we propose a general approach to the description of spectra of complex networks for the spectra of networks with uncorrelated vertices and a local treelike structure exact equations are derived these equations are generalized to the case of networks with correlations between neighboring vertices the tail of the density of eigenvalues ensuremath rho ensuremath lambda at large ensuremath lambda is related to the behavior of the vertex degree distribution p k at large k in particular as p k ensuremath sim k ensuremath ensuremath gamma ensuremath rho ensuremath lambda ensuremath sim ensuremath lambda 1 ensuremath 2 ensuremath gamma we propose a simple approximation which enables us to calculate spectra of various graphs analytically we analyze spectra of various complex networks and discuss the role of vertices of low degree we show that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real world networks e g of the internet

a small world of weak ties provides optimal global integration of self similar modules in functional brain networks the human brain is organized in functional modules such an organization presents a basic conundrum modules ought to be sufficiently independent to guarantee functional specialization and sufficiently connected to bind multiple processors for efficient information transfer it is commonly accepted that small world architecture of short paths and large local clustering may solve this problem however there is intrinsic tension between shortcuts generating small worlds and the persistence of modularity a global property unrelated to local clustering here we present a possible solution to this puzzle we first show that a modified percolation theory can define a set of hierarchically organized modules made of strong links in functional brain networks these modules are large world self similar structures and therefore are far from being small world however incorporating weaker ties to the network converts it into a small world preserving an underlying backbone of well defined modules remarkably weak ties are precisely organized as predicted by theory maximizing information transfer with minimal wiring cost this trade off architecture is reminiscent of the strength of weak ties crucial concept of social networks such a design suggests a natural solution to the paradox of efficient information flow in the highly modular structure of the brain

random walk on fractals numerical studies in two dimensions monte carlo calculations are used to investigate some statistical properties of random walks on fractal structures two kinds of lattices are used the sierpinski gasket and the infinite percolation cluster in two dimensions among other problems the authors study i the range rn of the walker number of distinct visited sites during n steps average value sn variance sigma n and asymptotic distribution ii renewal theory return to the original site probability of return p0 n mean number of returns nu n the probability distribution of the walker position p n r after n steps is discussed the asymptotic behaviour n 1 of these quantities exhibits power laws with associated exponents the numerical values of these exponents are in good agreement with recent theoretical predictions alexander and orbach 1982 rammal and toulouse 1982

random walks on graphs ideas techniques and results random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology in recent years relevant mathematical results have been obtained in this field and new ideas have been introduced which can be fruitfully extended to different areas and disciplines here we aim at giving a brief but comprehensive perspective of these progresses with a particular emphasis on physical aspects

evolving apollonian networks with small world scale free topologies we propose two types of evolving networks evolutionary apollonian networks eans and general deterministic apollonian networks gdans established by simple iteration algorithms we investigate the two networks by both simulation and theoretical prediction analytical results show that both networks follow power law degree distributions with distribution exponents continuously tuned from 2 to 3 the accurate expression of clustering coefficient is also given for both networks moreover the investigation of the average path length of ean and the diameter of gdan reveals that these two types of networks possess small world feature in addition we study the collective synchronization behavior on some limitations of the ean

operator valued frames we develop a natural generalization of vector valued frame theory we term operator valued frame theory using operator algebraic methods this extends work of the second author and d han which can be viewed as the mul tiplicity one case and extends to higher multiplicity their dilation approach we prove several results for operator valued frames concerning duality dis jointeness complementarity and composition of operator valued frames and the relationship between the two types of similarity left and right of such frames a key technical tool is the parametrization of parseval operator val ued frames in terms of a class of partial isometries in the hilbert space of the analysis operator we apply these notions to an analysis of multiframe gener ators for the action of a discrete group g on a hilbert space one of the main results of the han larson work was the parametrization of the parseval frame generators in terms of the unitary operators in the von neumann algebra gen erated by the group representation and the resulting norm path connectedness of the set of frame generators due to the connectedness of the group of unitary operators of an arbitrary von neumann algebra in this paper we general ize this multiplicity one result to operator valued frames however both the parameterization and the proof of norm path connectedness turn out to be necessarily more complicated and this is at least in part the rationale for this paper our parameterization involves a class of partial isometries of a different von neumann algebra these partial isometries are not path connected in the norm topology but only in the strong operator topology we prove that the set of operator frame generators is norm pathwise connected precisely when the von neumann algebra generated by the right representation of the group has no minimal projections as in the multiplicity one theory there are analogous results for general non parseval frames

a trust region algorithm for heterogeneous multiobjective optimization this paper presents a trust region method for multiobjective heterogeneous optimization problems one of the objective functions is an expensive black box function given for example by a time co

spectra and eigenvectors of scale free networks we study the spectra and eigenvectors of the adjacency matrices of scale free networks when bidirectional interaction is allowed so that the adjacency matrix is real and symmetric the spectral density shows an exponential decay around the center followed by power law long tails at both spectrum edges the largest eigenvalue ensuremath lambda 1 depends on system size n as ensuremath lambda 1 ensuremath sim n 1 4 for large n and the corresponding eigenfunction is strongly localized at the hub the vertex with largest degree the component of the normalized eigenfunction at the hub is of order unity we also find that the mass gap scales as n ensuremath 0 68

power control in wireless networks stability and delay independence for a general class of distributed algorithms we show that for a general class of distributed power control algorithms in wireless networks if a feasible steady state power allocation exists this is asymptotically stable for arbitrary gains and time varying heterogeneous delays the analysis exploits certain contraction properties of the interference in such algorithms and makes use of lyapunov razumikhin functions to address the infinite dimensional character of the problem

a deterministic small world network created by edge iterations abstract small world networks are ubiquitous in real life systems most previous models of small world networks are stochastic the randomness makes it more difficult to gain a visual understanding on how do different nodes of networks interact with each other and is not appropriate for communication networks that have fixed interconnections here we present a model that generates a small world network in a simple deterministic way our model has a discrete exponential degree distribution we solve the main characteristics of the model

infinite networks and variation of conductance functions in discrete laplacians for a given infinite connected graph g v e and an arbitrary but fixed conductance function c we study an associated graph laplacian c it is a generalized difference operator where the differences are measured across the edges e in g and the conductance function c represents the corresponding coefficients the graph laplacian a key tool in the study of infinite networks acts in an energy hilbert space e computed from c using a certain parseval frame we study the spectral theoretic properties of graph laplacians in fact for fixed c there are two versions of the graph laplacian one defined naturally in the l2 space of v and the other in e the first is automatically selfadjoint but the second involves a krein extension we prove that as sets the two spectra are the same aside from the point 0 the point zero may be in the spectrum of the second but not the first we further study the fine structure of the respective spectra as the conductance function varies showing now how the spect

synchronizability of complex networks recent interest in the study of networks associated with complex systems has led to a better understanding of the factors and parameters relating the topology of the associated networks with the dynamics of the systems and in particular their synchronization in this paper by using known and new results from spectral graph theory we characterize relevant factors which affect the synchronization of complex networks

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

eigenvalue spectra of complex networks we examine the eigenvalue spectrum of the adjacency matrix of a random scale free network with an average of p edges per vertex using the replica method we show how in the dense limit when p one can obtain two relatively simple coupled equations whose solution yields for an arbitrary complex network for scale free graphs with degree distribution exponent we obtain an exact expression for the eigenvalue spectrum when 3 and show that 1 2 1 for large in the limit we recover known results for the erdos renyi random graph

growth model for fractal scale free networks generated by a random walk abstract the diversity of networked systems with fractal structures suggests that knowing the underlying mechanism that generates the fractality is necessary for building a model of the development of complex networks in the present paper we propose a growth model of a network generated by a random walk and show that the evolving graph forms a fractal structure with various properties including the scale free property if the graph which provides a space where a random walk occurs by itself is formed by the random walk the proposed model is regulated by two parameters p v and p e which define the probability of either a roundabout path via a new vertex or a shortcut being formed by the random walk respectively the power law exponent describing the vertex degree distribution is determined by the ratio p e p v and is related to an internal factor f i via the relation 1 f i 1 where f i is a parameter that describes the local structure generated by the random walk a sufficiently small p v provides the small world property to the model network the small world property is usually considered to be incompatible with the fractal scaling property open m c l c d c where open m c is the average number of vertices which can be reached from a randomly chosen vertex in at most l c steps however we demonstrate that fractality can be reconciled with the small world property by introducing a size dependent fractal cluster dimension d c

brief paper a stability analysis with time delay of primal dual power control in optical links primal dual power control algorithms in optical networks must be designed properly to remain stable in the presence of time delays control algorithms at the signal transmitters continuously update their channel powers in response to dynamic pricing information from the network links the control algorithms optimize the optical signal to noise ratios osnrs of the channels in a distributive and non cooperative manner we consider the case of a single link with multiple channels and a constant time delay these results also apply to single sink general network configurations if the time delay represents the longest round trip time in the network we derive sufficient conditions for the stability of the closed loop system based on a tunable parameter in the control algorithm the work combines singular perturbation theory with lyapunov razumikhin time delay techniques we verify our results through simulations based on realistic network parameters

non monotone trust region algorithms for nonlinear optimization subject to convex constraints this paper presents two new trust region methods for solving nonlinear optimization problems over convex feasible domains these methods are distinguished by the fact that they do not enforce strict monotonicity of the objective function values at successive iterates the algorithms are proved to be convergent to critical points of the problem from any starting point extensive numerical experiments show that this approach is competitive with the lancelot package

griffiths singularities and algebraic order in the exact solution of an ising model on a fractal modular network we use an exact renormalization group transformation to study the ising model on a complex network composed of tightly knit communities nested hierarchically with the fractal scaling recently discovered in a variety of real world networks varying the ratio k j of intercommunity to intracommunity couplings we obtain an unusual phase diagram at high temperatures or small k j we have a disordered phase with a griffiths singularity in the free energy due to the presence of rare large clusters which we analyze through the yang lee zeros in the complex magnetic field plane as the temperature is lowered true long range order is not seen but there is a transition to algebraic order where pair correlations have power law decay with distance reminiscent of the xy model the transition is infinite order at small k j and becomes second order above a threshold value k j m the existence of such slowly decaying correlations is unexpected in a fat tailed scale free network where correlations longer than nearest neighbor are typically suppressed

erratum apollonian networks simultaneously scale free small world euclidean space filling and with matching graphs phys rev lett 94 018702 2005

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

flexible construction of hierarchical scale free networks with general exponent extensive studies have been done to understand the principles behind architectures of real networks recently evidence for hierarchical organization in many real networks has also been reported here we present a hierarchical model that reproduces the main experimental properties observed in real networks scale free of degree distribution p k frequency of the nodes that are connected to k other nodes decays as a power law p k ensuremath sim k ensuremath ensuremath gamma and power law scaling of the clustering coefficient c k ensuremath sim k ensuremath 1 the major points of our model can be summarized as follows a the model generates networks with scale free distribution for the degree of nodes with general exponent ensuremath gamma g2 and arbitrarily close to any specified value being able to reproduce most of the observed hierarchical scale free topologies in contrast previous models cannot obtain values of ensuremath gamma g2 58 b our model has structural flexibility because i it can incorporate various types of basic building blocks e g triangles tetrahedrons and in general fully connected clusters of n nodes and ii it allows a large variety of configurations i e the model can use more than n ensuremath 1 copies of basic blocks of n nodes the structural features of our proposed model might lead to a better understanding of architectures of biological and nonbiological networks

global convergence of a class of trust region algorithms for optimization with simple bounds this paper extends the known excellent global convergence properties of trust region algorithms for unconstrained optimization to the case where bounds on the variables are present weak conditions on the accuracy of the hessian approximations are considered it is also shown that when the strict complementarily condition holds the proposed algorithms reduce to an unconstrained calculation after finitely many iterations allowing a fast asymptotic rate of convergence

invariant recognition in potts glass neural networks the authors show that a potts glass neural network can be used for invariant recognition each neuron is modelled as a q state potts variable and p potts configurations are stored in the network the learning rule is a generalized hebb rule so that all pq permutations of the patterns are stabilized simultaneously they analyse the model for an extensive number of patterns and present results of numerical simulations to confirm the analytical results and to illustrate the relaxation to equilibrium the model is applied to the recognition of isomorphic graphs finally they show how to generalize the model such that its stable states are patterns were the permutational symmetry is broken in a hierarchical manner

building complex networks with platonic solids we propose a unified model to build planar graphs with diverse topological characteristics which are of relevance in real applications here convex regular polyhedra platonic solids are used as the building blocks for the construction of a variety of complex planar networks these networks are obtained by merging polyhedra face by face on a tree structure leading to planar graphs we investigate two different constructions 1 a fully deterministic construction where a self similar fractal structure is built by using a single kind of polyhedron which is iteratively attached to every face and 2 a stochastic construction where at each step a polyhedron is attached to a randomly chosen face these networks are scale free small world clustered and sparse sharing several characteristics of real world complex networks we derive analytical expressions for the degree distribution the clustering coefficient and the mean degree of nearest neighbors showing that these networks have power law degree distributions with tunable exponents associated with the building polyhedron and they possess a hierarchical organization that is determined by planarity

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

fractality and the small world effect in sierpinski graphs although some real networks exhibit self similarity there is no standard definition of fractality in graphs on the other hand the small world phenomenon is one of the most important common properties of real interconnection networks in this paper we relate these two properties in order to do so we focus on the family of sierpinski networks for the sierpinski gasket the sierpinski carpet and the sierpinski tetra we give the basic properties and we calculate the box counting dimension as a measure of their fractality we also define a deterministic family of graphs which we call small world sierpinski graphs we show that our construction preserves the structure of sierpinski graphs including its box counting dimension while the small world phenomenon arises thus in this family of graphs fractality and small world effect are simultaneously present

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

degree and component size distributions in the generalized uniform recursive tree we propose a generalized model for the famous uniform recursive tree urt by introducing an imperfect growth process which may generate disconnected components clusters the model undergoes an interesting phase transition from a singly connected network to a graph consisting of fully isolated nodes we investigate the distributions of degree and component sizes by both theoretical predictions and numerical simulations for the nontrivial cases we show that the network has an exponential degree distribution while its component size distribution follows a power law both of which are related to the imperfect growth process we also predict the growth dynamics of the individual components all analytical solutions are successfully contrasted with computer simulations

scale free networks generated by random walkers we present a simple mechanism for generating undirected scale free networks using random walkers where the network growth is determined by choosing parent vertices by sequential random walks we show that this mechanism produces scale free networks with degree exponent 3 and clustering coefficients depending on random walk length the mechanism can be interpreted in terms of preferential attachment without explicit knowledge of node degrees

the exact solution of the mean geodesic distance for vicsek fractals vicsek fractals are one of the most interesting classes of fractals and the study of their structural properties is important in this paper the exact formula for the mean geodesic distance of vicsek fractals is found the quantity is computed precisely through the recurrence relations derived from the self similar structure of the fractals considered the obtained exact solution exhibits that the mean geodesic distance approximately increases as a power law function of the number of nodes with the exponent equal to the reciprocal of the fractal dimension the closed form solution is confirmed by extensive numerical calculations

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

spectra of real world graphs beyond the semicircle law results on the spectra of adjacency matrices corresponding to models of real world graphs we find that when the number of links grows as the number of nodes the spectral density of uncorrelated random matrices does not converge to the semicircle law furthermore the spectra of real world graphs have specific features depending on the details of the corresponding models in particular scale free graphs develop a trianglelike spectral density with a power law tail while small world graphs have a complex spectral density consisting of several sharp peaks these and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks

a noncooperative game approach to osnr optimization in optical networks we consider a game theory framework for power control in optical networks channel optical signal to noise ratio osnr optimization is formulated as an m player noncooperative game based on a general network osnr model conditions for existence and uniqueness of the game equilibrium solution are given an iterative algorithm for power control is proposed that uses only channel specific feedback measurements and is shown to converge to the equilibrium solution

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

network formed by traces of random walks abstract we propose a model of time evolving networks in which a kind of transport between vertices generates new edges in the graph we call the model network formed by traces of random walks because the transports are represented abstractly by random walks our numerical calculations yield several important properties observed commonly in complex networks although the graph at initial time is only a one dimensional lattice for example the distribution of vertex degree exhibits various behaviors such as exponential power law like and bi modal distribution according to change of probability of extinction of edges another property such as strong clustering structure and small mean vertex vertex distance can also be found the transports represented by random walks in a framework of strong links between regular lattice is a new mechanisms which yields biased acquisition of links for vertices

an extension of duality to a game theoretic framework this paper extends some duality results from a standard optimization setup to a noncooperative nash game framework a nash game ng with coupled constraints is considered solving directly such a coupled ng requires coordination among possibly all players an alternative approach is proposed based on its relation to a special constrained optimization problem for the ng game cost function with respect to the second argument that admits a fixed point solution specific separability properties of the ng game cost are exploited and duality results are developed this duality extension leads naturally to a hierarchical decomposition into a lower level ng with no coupled constraints and a higher level system optimization problem in the second part of the paper these theoretical results are applied to a coupled ng with coupled constraints as encountered in optical networks

a trust region method for norm constrained problems a trust region method for the solution of nonlinear optimization problems with norm constraints is analyzed such optimization problems arise in parameter identification and nonlinear eigenvalue problems the characteristics of these applications motivate the formulation and analysis of the trust region method the algorithms studied here allow for inexact gradient information and the use of subspace methods for the approximate solution of subproblems characterizations and the descent properties of trust region steps are given criteria for the existence of successful iterations under inexact gradient information and under the use of subspace methods are established and global convergence of the method is proven

fractal small world dichotomy in real world networks we draw attention to a clear dichotomy between small world networks exhibiting exponential neighborhood growth and fractal like networks where neighborhoods grow according to a power law this distinction is observed in a number of real world networks and is related to the degree correlations and geographical constraints we conclude by pointing out that the status of human social networks in this dichotomy is far from clear

topology of growing networks accelerated by intermediary process abstract we examined the effect of accelerated growth increase in the mean vertex degree caused by internal links between pre existing nodes on the topology of growing networks and showed that a kind of local rule edge additions stimulated by randomly chosen intermediaries intermediary process not only guarantees a highly clustered structure but also induces the scale free property on growing networks without an explicit preferential rule with the aid of accelerating growth we relate the power law exponent describing the degree distribution to the rate of acceleration and an internal factor which expresses the contribution of internal links to the rate of increase in degrees the behavior of the internal factor generates different forms of time dependence of degrees according to whether 2 or 2 accelerating growth plays a dominant role in the structural formation of networks based on local rules in contrast to cases when non local rules are applied because similar studies showed that a simple preferential linking rule a non local rule easily induces power laws without accelerating growth

fractal networks induced by movements of random walkers on a tree graph abstract we show that a fractal scale free and small world network can grow on a non fractal cayley tree through the simple random process of shortcut creation by movements of random walkers the cayley tree provides a stage for the walkers to diffuse into a locally one dimensional structure with the small world property self organization of fractal graph by adding shortcut edges to the locally one dimensional structure generated by walkers flow from the root vertex to the outermost shell of the cayley tree is a novel mechanism for generating fractal graphs also we show numerically that the fractal box counting dimension and a power law exponent describing the degree distribution of the resulting graph are mainly determined by the bifurcation number of the original cayley tree in our model the fractal cluster dimension must change with the size of the evolving graph this is a general result that resolves an apparent contradiction between the fractality in the context of the cluster growing method and the small world property of the graph

large scale topological and dynamical properties of the internet we study the large scale topological and dynamical properties of real internet maps at the autonomous system level collected in a 3 yr time interval we find that the connectivity structure of the internet presents statistical distributions settled in a well defined stationary state the large scale properties are characterized by a scale free topology consistent with previous observations correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the internet the study of the internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models we compare the properties of growing network models with the present real internet data analysis

reciprocal relation between the fractal and the small world properties of complex networks the fractal and the small world properties of complex networks are systematically studied both in the box covering bc and the cluster growing cg measurements we elucidate that complex networks possessing the fractal small world nature in the bc measurement are always fractal small world even in the cg measurement and vice versa while the fractal dimensions d text b by the bc measurement and d text c by the cg measurement are generally different this implies that two structural properties of networks fractality and small worldness cannot coexist in the same length scale these properties can however crossover from one to the other by varying the length scale we show that the crossover behavior in a network near the percolation transition appears both in the bc and cg measurements and is scaled by a unique characteristic length ensuremath xi

effects of triad formations stimulated by intermediaries on network topology abstract triad formation of vertices is considered a significant mechanism in the emergence of highly clustered structures in real networks however the net effect of triad formations on network topology has yet to be understood completely since triad formations are usually studied with additional effects including the attachment of new vertices to prevent a saturation of the number of edges where almost all vertices are directly linked to each other in this paper we focus on the net effects of triad formations stimulated by randomly chosen intermediaries on network topologies such as local clustering and evolution of degrees we show that the local clustering of vertices with degree k is divided into an essential term 1 k which can be widely seen in real networks and additional terms depending on the initial network topology also we derive an equation which measures the influence of local structures of networks on the time evolution of vertex degrees according to which triad formations lead to the so called rich get richer phenomenon in the evolution of degrees local events like a triad formation stimulated by pre existing vertices leads not only to highly clustered structures but to a typical power law form in the degree distribution with a power law exponent of about 2

resistance distance in potting networks abstract on a resistor network the resistance distance is the effective resistance r u v between two nodes u v of the network in this paper we study a family of self similar and symmetric networks named potting networks and obtain a recursion formula to calculate resistance distances

analytical solution of average path length for apollonian networks with the help of recursion relations derived from the self similar structure we obtain the solution of average path length overline d t for apollonian networks in contrast to the well known numerical result overline d t ensuremath propto text ln text n t 3 4 j s andrade jr et al phys rev lett 94 018702 2005 our rigorous solution shows that the average path length grows logarithmically as overline d t ensuremath propto text ln text n t in the infinite limit of network size n t the extensive numerical calculations completely agree with our closed form solution

explicit determination of mean first passage time for random walks on deterministic uniform recursive trees the determination of mean first passage time mfpt for random walks in networks is a theoretical challenge and is a topic of considerable recent interest within the physics community in this paper according to the known connections between mfpt effective resistance and the eigenvalues of graph laplacian we first study analytically the mfpt between all node pairs of a class of growing treelike networks which we term deterministic uniform recursive trees durts since one of its particular cases is a deterministic version of the famous uniform recursive tree the interesting quantity is determined exactly through the recursive relation of the laplacian spectra obtained from the special construction of durts the analytical result shows that the mfpt between all couples of nodes in durts varies as n text text ln text n for large networks with node number n second we study trapping on a particular network of durts focusing on a special case with the immobile trap positioned at a node having largest degree we determine exactly the average trapping time att that is defined as the average of fpt from all nodes to the trap in contrast to the scaling of the mfpt the leading behavior of att is a linear function of n interestingly we show that the behavior for att of the trapping problem is related to the trapping location which is in comparison with the phenomenon of trapping on fractal t graph although both networks exhibit tree structure finally we believe that the methods could open the way to exactly calculate the mfpt and att in a wide range of deterministic media

structural and spectral properties of a family of deterministic recursive trees rigorous solutions as one of the most significant models the uniform recursive tree urt has found many applications in a variety of fields in this paper we study rigorously the structural features and spectral properties of the adjacency matrix for a family of deterministic uniform recursive trees durts that are deterministic versions of urt firstly from the perspective of complex networks we investigate analytically the main structural characteristics of durts and obtain the accurate solutions for these properties which include degree distribution average path length distribution of node betweenness and degree correlations then we determine the complete eigenvalues and their corresponding eigenvectors of the adjacency matrix for durts our research may shed light on a better understanding of the features for urt also the analytical methods used here are capable of being extended to many other deterministic networks making the precise computation of their properties especially the full spectrum characteristics possible

a family of small world network models built by complete graph and iteration function abstract small world networks are popular in real life complex systems in the past few decades researchers presented amounts of small world models in which some are stochastic and the rest are deterministic in comparison with random models it is not only convenient but also interesting to study the topological properties of deterministic models in some fields such as graph theory theorem computer sciences and so on as another concerned darling in current researches community structure modular topology is referred to as an useful statistical parameter to uncover the operating functions of network so building and studying such models with community structure and small world character will be a demanded task hence in this article we build a family of sparse network space n t which is different from those previous deterministic models even though our models are established in the same way as them iterative generation by randomly connecting manner in each time step every resulting member in n t has no absolutely self similar feature widely shared in a large number of previous models this makes our insight not into discussing a class certain model but into investigating a group various ones spanning a network space somewhat surprisingly our results prove all members of n t to possess some similar characters a sparsity b exponential scale feature p k k and c small world property here we must stress a very screming but intriguing phenomenon that the difference of average path length a p l between any two members in n t is quite small which indicates this random connecting way among members has no great effect on a p l at the end of this article as a new topological parameter correlated to reliability synchronization capability and diffusion properties of networks the number of spanning trees on a representative member n b t of n t is studied in detail then an exact analytical solution for its spanning trees entropy is also obtained

deterministic self similar models of complex networks based on very symmetric graphs abstract using very symmetric graphs we generalize several deterministic self similar models of complex networks and we calculate the main network parameters of our generalization more specifically we calculate the order size and the degree distribution and we give an upper bound for the diameter and a lower bound for the clustering coefficient these results yield conditions under which the network is a self similar and scale free small world network we remark that all these conditions are posed on a small base graph which is used in the construction as a consequence we can construct complex networks having prescribed properties we demonstrate this fact on the clustering coefficient we propose eight new infinite classes of complex networks one of these new classes is so rich that it is parametrized by three independent parameters

a general geometric growth model for pseudofractal scale free web abstract we propose a general geometric growth model for pseudofractal scale free web psw which is controlled by two tunable parameters we derive exactly the main characteristics of the networks degree distribution second moment of degree distribution degree correlations distribution of clustering coefficient as well as the diameter which are partially determined by the parameters analytical results show that the resulting networks are disassortative and follow power law degree distributions with a more general degree exponent tuned from 2 to 1 ln 3 ln 2 the clustering coefficient of each individual node is inversely proportional to its degree and the average clustering coefficient of all nodes approaches to a large nonzero value in the infinite network order the diameter grows logarithmically with the number of network nodes all these reveal that the networks described by our model have small world effect and scale free topology

a key heterogeneous structure of fractal networks based on inverse renormalization scheme abstract self similarity property of complex networks was found by the application of renormalization group theory based on this theory network topologies can be classified into universality classes in the space of configurations in return through inverse renormalization scheme a given primitive structure can grow into a pure fractal network then adding different types of shortcuts it exhibits different characteristics of complex networks however the effect of primitive structure on networks structural property has received less attention in this paper we introduce a degree variance index to measure the dispersion of nodes degree in the primitive structure and investigate the effect of the primitive structure on network structural property quantified by network efficiency numerical simulations and theoretical analysis show a primitive structure is a key heterogeneous structure of generated networks based on inverse renormalization scheme whether or not adding shortcuts and the network efficiency is positively correlated with degree variance of the primitive structure

convergence of quasi newton matrices generated by the symmetric rank one update quasi newton algorithms for unconstrained nonlinear minimization generate a sequence of matrices that can be considered as approximations of the objective function second derivatives this paper gives conditions under which these approximations can be proved to converge globally to the true hessian matrix in the case where the symmetric rank one update formula is used the rate of convergence is also examined and proven to be improving with the rate of convergence of the underlying iterates the theory is confirmed by some numerical experiments that also show the convergence of the hessian approximations to be substantially slower for other known quasi newton formulae

correction to the paper on global convergence of a class of trust region algorithms for optimization with simple bounds a correction is made to the paper entitled global convergence of a class of trust region algorithms for optimization with simple bounds corm gould and toint siam j numer anal 25 1988 pp 433 460 first an error is pointed out in the proof of the fact that the correct set of active bounds is determined after a finite number of iterations a new proof of the relevant theorem is given

shortest paths and load scaling in scale free trees the average node to node distance of scale free graphs depends logarithmically on n the number of nodes while the probability distribution function of the distances may take various forms here we analyze these by considering mean field arguments and by mapping the m 1 case of the barabasi albert model into a tree with a depth dependent branching ratio this shows the origins of the average distance scaling and allows one to demonstrate why the distribution approaches a gaussian in the limit of n large the load the number of the shortest distance paths passing through any node is discussed in the tree presentation

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

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

charge and current distribution in graphs we consider graphs made of one dimensional wires connected at vertices and on which may live a scalar potential we are interested in a scattering situation where such a network is connected to infinite leads we study the correlations of the charge in such graphs out of equilibrium as well as the distribution of the currents in the wires inside the graph these quantities are related to the scattering matrix of the graph we discuss the case where the graph is weakly connected to the wires

shortest paths and load scaling in scale free trees szabo alava and kertesz phys rev e 66 026101 2002 considered two questions about the scale free random tree given by the m 1 case of the barabasi albert ba model identical with a random tree model introduced by szymanski in 1987 what is the distribution of the node to node distances and what is the distribution of node loads where the load on a node is the number of shortest paths passing through it they gave heuristic answers to these questions using a mean field approximation replacing the random tree by a certain fixed tree with carefully chosen branching ratios by making use of our earlier results on scale free random graphs we shall analyze the random tree rigorously obtaining and proving very precise answers to these questions we shall show that after dividing by n the number of nodes the load distribution converges to an integer distribution x with pr x c 2 2c 1 2c 3 c 0 1 2 confirming the asymptotic power law with exponent 2 predicted by szabo alava and kertesz for the distribution of node node distances we show asymptotic normality and give a precise form for the far from normal large deviation law we note that the mean field methods used by szabo alava and kertesz give very good results for this model

correlations in random apollonian network abstract in this paper by both simulations and theoretical predictions we study two and three node or degree correlations in random apollonian network ran which have small world and scale free topologies using the rate equation approach under the assumption of continuous degree we first give the analytical solution for two node correlations expressed by average nearest neighbor degree annd then we revisit the degree distribution of ran using rate equation method and get the exact connection distribution based on which we derive a more accurate result for mean clustering coefficient as an average quantity of three degree correlations than the one previously reported analytical results reveal that annd has no correlations with respect to degree while clustering coefficient is dependent on degree showing a power law behavior as c k k 1 the obtained expressions are successfully contrasted with extensive numerical simulations

spectral reciprocity and matrix representations of unbounded operators abstract we study a family of unbounded hermitian operators in hilbert space which generalize the usual graph theoretic discrete laplacian for an infinite discrete set x we consider operators acting on hilbert spaces of functions on x and their representations as infinite matrices the focus is on l 2 x and the energy space h e in particular we prove that these operators are always essentially self adjoint on l 2 x but may fail to be essentially self adjoint on h e in the general case we examine the von neumann deficiency indices of these operators and explore their relevance in mathematical physics finally we study the spectra of the h e operators with the use of a new approximation scheme

degree dependent intervertex separation in complex networks we study the mean length l k of the shortest paths between a vertex of degree k and other vertices in growing networks where correlations are essential in a number of deterministic scale free networks we observe a power law correction to a logarithmic dependence l k a ln n k gamma 1 2 ck gamma 1 n in a wide range of network sizes here n is the number of vertices in the network gamma is the degree distribution exponent and the coefficients a and c depend on a network we compare this law with a corresponding l k dependence obtained for random scale free networks growing through the preferential attachment mechanism in stochastic and deterministic growing trees with an exponential degree distribution we observe a linear dependence on degree l k approximately a ln n ck we compare our findings for growing networks with those for uncorrelated graphs

hierarchical regular small world networks two new networks are introduced that resemble small world properties these networks are recursively constructed but retain a fixed regular degree they possess a unique one dimensional lattice backbone overlaid by a hierarchical sequence of long distance links mixing real space and small world features both networks one 3 regular and the other 4 regular lead to distinct behaviors as revealed by renormalization group studies the 3 regular network is planar has a diameter growing as with system size n and leads to super diffusion with an exact anomalous exponent dw 1 306 but possesses only a trivial fixed point tc 0 for the ising ferromagnet in turn the 4 regular network is non planar has a diameter growing as exhibits ballistic diffusion dw 1 and a non trivial ferromagnetic transition tc 0 it suggests that the 3 regular network is still quite geometric while the 4 regular network qualifies as a true small world with mean field properties as an engineering application we discuss synchronization of processors on these networks

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

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

impact of initial lattice structures on networks generated by traces of random walks abstract we show that the platform stage of network evolution plays a principal role in the topology of resulting networks generated by short cuts stimulated by the movements of a random walker the mechanism of which tends to produce power law degree distributions to examine the numerical results we have developed a statistical method which relates the power law exponent to random properties of the subgraph developed in the platform stage as a result we find that an important exponent in the network evolution is which characterizes the size of the subgraph in the form v t where v and t denote the number of vertices in the subgraph and the time variable respectively 2d lattices can impose specific limitations on the walker s diffusion which keeps the value of within a moderate range and provides typical properties of complex networks 1d and 3d cases correspond to different ends of the spectrum for with 2d cases in between especially for 2d square lattices a discontinuous change of the network structure is observed which varies according to whether is greater or less than 2 for 1d cases we show that emergence of nearly complete subgraphs is guaranteed by 1 2 although the transient power law is permitted at low increase rates of edges additionally the model exhibits a spontaneous emergence of highly clustered structures regardless of its initial structure

a fractal growth model exploring the connection pattern of hubs in complex networks abstract fractal is ubiquitous in many real world networks previous researches showed that the strong disassortativity between the hub nodes on all length scales was the key principle that gave rise to the fractal architecture of networks although fractal property emerged in some models there were few researches about the fractal growth model and quantitative analyses about the strength of the disassortativity for fractal model in this paper we proposed a novel inverse renormalization method named box based preferential attachment bpa to build the fractal growth models in which the preferential attachment was performed at box level the proposed models provided a new framework that demonstrated small world fractal transition also we firstly demonstrated the statistical characteristic of connection patterns of the hubs in fractal networks the experimental results showed that given proper growing scale and added edges the proposed models could clearly show pure small world or pure fractal or both of them it also showed that the hub connection ratio showed normal distribution in many real world networks at last the comparisons of connection pattern between the proposed models and the biological and technical networks were performed the results gave useful reference for exploring the growth principle and for modeling the connection patterns for real world networks

small world to fractal transition in complex networks a renormalization group approach we show that renormalization group rg theory applied to complex networks is useful to classify network topologies into universality classes in the space of configurations the rg flow readily identifies a small world fractal transition by finding i a trivial stable fixed point of a complete graph ii a nontrivial point of a pure fractal topology that is stable or unstable according to the amount of long range links in the network and iii another stable point of a fractal with shortcuts that exist exactly at the small world fractal transition as a collateral the rg technique explains the coexistence of the seemingly contradicting fractal and small world phases and allows us to extract information on the distribution of shortcuts in real world networks a problem of importance for information flow in the system

random walk statistics on fractal structures we consider some statistical properties of simple random walks on fractal structures viewed as networks of sites and bonds range renewal theory mean first passage time etc asymptotic behaviors are shown to be controlled by the fractal d and spectral d dimensionalities of the considered structure a simple decimation procedure giving the value of d is outlined and illustrated in the case of the sierpinski gaskets recent results for the trapping problem the self avoiding walk and the true self avoiding walk are briefly reviewed new numerical results for diffusion on percolation clusters are also presented

a review of fractality and self similarity in complex networks we review recent findings of self similarity in complex networks using the box covering technique it was shown that many networks present a fractal behavior which is seemingly in contrast to their small world property moreover even non fractal networks have been shown to present a self similar picture under renormalization of the length scale these results have an important effect in our understanding of the evolution and behavior of such systems a large number of network properties can now be described through a set of simple scaling exponents in analogy with traditional fractal theory

superlinear and quadratic convergence of affine scaling interior point newton methods for problems with simple bounds without strict complementarity assumption a class of affine scaling interior point methods for bound constrained optimization problems is introduced which are locally q superlinear or q quadratic convergent it is assumed that the strong second order sufficient optimality conditions at the solution are satisfied but strict complementarity is not required the methods are modifications of the affine scaling interior point newton methods introduced by t f coleman and y li math programming 67 189 224 1994 there are two modifications one is a modification of the scaling matrix the other one is the use of a projection of the step to maintain strict feasibility rather than a simple scaling of the step a comprehensive local convergence analysis is given a simple example is presented to illustrate the pitfalls of the original approach by coleman and li in the degenerate case and to demonstrate the performance of the fast converging modifications developed in this paper

a unified model for sierpinski networks with scale free scaling and small world effect abstract in this paper we propose an evolving sierpinski gasket based on which we establish a model of evolutionary sierpinski networks esns that unifies deterministic sierpinski network z z zhang s g zhou t zou l c chen j h guan eur phys j b 60 2007 259 and random sierpinski network z z zhang s g zhou z su t zou j h guan eur phys j b 65 2008 141 to the same framework we suggest an iterative algorithm generating the esns on the basis of the algorithm some relevant properties of presented networks are calculated or predicted analytically analytical solution shows that the networks under consideration follow a power law degree distribution with the distribution exponent continuously tuned in a wide range the obtained accurate expression of clustering coefficient together with the prediction of average path length reveals that the esns possess small world effect all our theoretical results are successfully contrasted by numerical simulations moreover the evolutionary prisoner s dilemma game is also studied on some limitations of the esns i e deterministic sierpinski network and random sierpinski network

global convergence of trust region interior point algorithms for infinite dimensional nonconvex minimization subject to pointwise bounds a class of interior point trust region algorithms for infinite dimensional nonlinear optimization subject to pointwise bounds in l p banach spaces 2 le p le infty is formulated and analyzed the problem formulation is motivated by optimal control problems with l p controls and pointwise control constraints the interior point trust region algorithms are generalizations of those recently introduced by coleman and li siam j optim 6 1996 pp 418 445 for finite dimensional problems many of the generalizations derived in this paper are also important in the finite dimensional context all first and second order global convergence results known for trust region methods in the finite dimensional setting are extended to the infinite dimensional framework of this paper

on the spectra of hypertrees in this paper we study the spectral properties of a family of trees characterized by two main features they are spanning subgraphs of the hypercube and their vertices bear a high degree of connectedness hierarchy such structures are here called binary hypertrees and they can be recursively defined as the so called hierarchical product of several complete graphs on two vertices ams classification 05c50 05c05

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved