retrieval properties of a hopfield model with random asymmetric interactions the process of pattern retrieval in a hopfield model in which a random antisymmetric component is added to the otherwise symmetric synaptic matrix is studied by computer simulations the introduction of the antisymmetric component is found to increase the fraction of random inputs that converge to the memory states however the size of the basin of attraction of a memory state does not show any significant change when asymmetry is introduced in the synaptic matrix we show that this is due to the fact that the spurious fixed points which are destabilized by the introduction of asymmetry have very small basins of attraction the convergence time to spurious fixed point attractors increases faster than that for the memory states as the asymmetry parameter is increased the possibility of convergence to spurious fixed points is greatly reduced if a suitable upper limit is set for the convergence time this prescription works better if the synaptic matrix has an antisymmetric component

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

nonlinear dynamics in coupled fuzzy control systems i coherence and chaos frustration in triangle configuration abstract nonlinear dynamics and chaos are studied in a system for which a complete set of equations of motion such as equations of newton navier stokes and van der pol is not available as a very general system as such we consider coupled classical spins pendulums each of which is under control by a fuzzy system that is designed to align the spin to an unstable fixed point the fuzzy system provides a deterministic procedure to control an object without use of a differential equation the positions and velocities of the spins are monitored periodically and each fuzzy control gives a momentum to its associated spin in the reverse directions if the monitoring is made with an interval short enough the spin spin interactions are overwhelmed by the fuzzy control and the system converges to a state as designed however a long interval monitoring induces dynamics of too late response and thereby results in chaos a great variety of dynamics are generated under very delicate balance between the fuzzy control and the spin spin interaction in which two independent mechanisms of creating negative and positive liapunov exponents interact with each other

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

chaos in a neural network circuit abstract we have constructed a neural network circuit of four clipped high grain integrating operational amplifiers coupled to each other through an array of digitally programmable resistor ladders mdacs in addition to fixed point and cyclic behavior the circuit exhibits chaotic behavior with complex strange attractors which are approached through period doubling intermittent attractor expansion and or quasiperiodic pathways couplings between the nonlinear circuit elements are controlled by a computer which can automatically search through the space of couplings for interesting phenomena we report some initial statistical results relating the behavior of the network to properties of its coupling matrix through these results and further research the circuit should help resolve fundamental issues concerning chaos in neural networks

the nature of attractors in an asymmetric spin glass with deterministic dynamics the authors study the attractors in an infinite range ising spin glass model with deterministic dynamics where the interactions have asymmetry specified by a parameter k they find a duality relation between the attractors for models with asymmetry parameters k and 1 k the attractors are fixed points or limit cycles of short length except for k 1 at which the average cycle length diverges reminiscent of a phase transition and the model has many similarities to the random map model as well as to the infinite range symmetric spin glass in thermal equilibrium including the fact that a few attractors dominate the weight the extent of this dominance varies from sample to sample and so is given by a non trivial probability distribution pi y which they compute numerically

self similar scaling of density in complex real world networks abstract despite their diverse origin networks of large real world systems reveal a number of common properties including small world phenomena scale free degree distributions and modularity recently network self similarity as a natural outcome of the evolution of real world systems has also attracted much attention within the physics literature here we investigate the scaling of density in complex networks under two classical box covering renormalizations network coarse graining and also different community based renormalizations the analysis on over 50 real world networks reveals a power law scaling of network density and size under adequate renormalization technique yet irrespective of network type and origin the results thus advance a recent discovery of a universal scaling of density among different real world networks p j laurienti k e joyce q k telesford j h burdette s hayasaka universal fractal scaling of self organized networks physica a 390 20 2011 3608 3613 and imply an existence of a scale free density also within among different self similar scales of complex real world networks the latter further improves the comprehension of self similar structure in large real world networks with several possible applications

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

a study of network dynamics the dynamics of a discrete time neural network model are investigated first a numerical survey of network power spectra is reported for networks of varying size with random weight matrices and initial states the steepness of the logistic function and a symmetry measure of the weight matrix are taken as control parameters summary statistics are presented to give gross measures of the model s temporal activity in parameter space second a detailed study of the dynamics of a particular network is described complex dynamical behavior is observed including hopf bifurcations the ruelle takens newhouse route to chaos showing mode locking at rational winding numbers and the destruction of an invariant torus and the period doubling route to chaos

emergent properties in random complex automata abstract studies of large randomly assembled binary boolean node automata have demonstrated that such systems can spontaneously exhibit enormously ordered dynamical behavior an important approach to characterizing these behaviors has consisted of studying ensembles of automata ensemble specifications have been based on 1 choice of boolean function regulating each node in the automaton 2 random or biased mappings of the 2 n automata states into themselves 3 the numbers of inputs k per node this article briefly reviews these approaches reports new results on the expected behavior of threshold automata and automata with a particular class of biased mappings in addition it discusses emergent highly ordered dynamical behavior in random automata rich in a specific class of canalizing boolean functions due to the crystallization of powerful subautomata called forcing structures the ordered behaviors may be of importance in biological evolution in physics and in the design of adaptive automata

synchronization multistability and basin crisis in coupled pendula abstract the synchronization dynamics of two linearly coupled pendula is studied in this paper based on the lyapunov stability theory and linear matrix inequality lmi some necessary and sufficient conditions for global asymptotic synchronization are derived from which an estimated threshold coupling k th for the on set of full synchronization is obtained the numerical value of k th determined from the average energies of the systems is in good agreement with theoretical analysis prior to the on set of synchronization the boundary crisis of the chaotic attractor is identified in the bistable states where two asymmetric periodic attractors co exist it is shown that the coupled pendula can attain multistable states via a new dynamical transition the basin crisis that occur prior to the on set of stable synchronization the essential feature of basin crisis is that the two co existing attractors are destroyed while new three or more co existing attractors of the same or different periodicity are created in addition the linear perturbation technique and the routh hurwitz criteria are employed to investigate the stability of steady states and clearly identify the different types of bifurcations likely to be encountered finally two parameter phase plots show various regions of chaos hyperchaos and periodicity

attractor properties of dynamical systems neural network models we propose tools to probe the nature of attractors in dynamical systems these include the activity distribution the evolutions of the state damage activity damage and temporal correlation damage when they are used to study the retrieval attractors in dilute asymmetric neural networks a transition from a partially frozen phase to an unfrozen phase is found for networks trained with sufficiently noisy data near storage saturation and this confirms that the retrieval attractors are more chaotic in this case we are also able to demonstrate that the retrieval attractors in dilute asymmetric neural networks are not clouds of attractors but consist of a single chaotic attractor for each stored pattern furthermore they facilitate the device of effective freezing procedures which significantly improve the quality of retrieval in neural networks

computing the measure of nonattracting chaotic sets abstract recently a numerical method called the pim triple algorithm has been formulated for generating long orbits on nonattracting chaotic sets we investigate under which conditions the measure for a nonattracting invariant set generated by the pim triple algorithm agrees with the natural measure

features and heterogeneities in growing network models many complex networks from the world wide web to biological networks grow taking into account the heterogeneous features of the nodes the feature of a node might be a discrete quantity such as a classification of a url document such as personal page thematic website news blog search engine social network etc or the classification of a gene in a functional module moreover the feature of a node can be a continuous variable such as the position of a node in the embedding space in order to account for these properties in this paper we provide a generalization of growing network models with preferential attachment that includes the effect of heterogeneous features of the nodes the main effect of heterogeneity is the emergence of an effective fitness for each class of nodes determining the rate at which nodes acquire new links the degree distribution exhibits a multiscaling behavior analogous to the the fitness model this property is robust with respect to variations in the model as long as links are assigned through effective preferential attachment beyond the degree distribution in this paper we give a full characterization of the other relevant properties of the model we evaluate the clustering coefficient and show that it disappears for large network size a property shared with the barabasi albert model negative degree correlations are also present in this class of models along with nontrivial mixing patterns among features we therefore conclude that both small clustering coefficients and disassortative mixing are outcomes of the preferential attachment mechanism in general growing networks

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

the roles of stability and symmetry in the dynamics of neural networks the authors study the retrieval phase of spin glass like neural networks considering that the dynamics should depend only on gauge invariant quantities they propose that two such parameters characterising the symmetry of the neural net s connections and the stabilities of the patterns are responsible for most of the dynamical effects this is supported by a numerical study of the shape of the basins of attraction for a one pattern neural network model the effects of stability and symmetry on the short time dynamics of this model are studied analytically and the full dynamics for vanishing symmetry is shown to be exactly solvable

new formulation of fem for deterministic and stochastic beams through generalization of fuchs approach this paper proposes an alternative way of constructing the global stiffness matrix of the finite element method for bending beams it also applies the new formulation to first and second moment analysis of stochastic beams which involve spatially uncertain bending stiffness originating from fuchs idea of decoupling the shear and bending components in the bending beam the element level stiffness matrix is diagonalized the generalized stress strain strain displacement and equilibrium relationships are assembled respectively and then are combined to form the global stiffness matrix the advantage of the new formulation is that the bending stiffness explicitly appears in the global stiffness matrix the mean vector and covariance matrix of the displacement of the beam are then obtained in terms of probabilistic characteristics of the uncertain bending stiffness this is in contrast to the conventional finite element method in stochastic setting which is based on the perturbation technique the example is given to illustrate the efficacy of the new formulation and its application to bending of stochastic beams

scaling laws for noise induced superpersistent chaotic transients a super persistent chaotic transient is characterized by the following scaling law for its average lifetime tau approximately exp c p p c alpha where c 0 and alpha 0 are constants p or p c is a bifurcation parameter and p c is its critical value as p approaches p c from above the exponent in the exponential dependence diverges leading to an extremely long transient lifetime historically the possibility of such transient raised the question of whether asymptotic attractors are relevant to turbulence here we investigate the phenomenon of noise induced super persistent chaotic transients in particular we construct a prototype model based on random maps to illustrate this phenomenon we then approximate the model by stochastic differential equations and derive the scaling laws for the transient lifetime versus the noise amplitude epsilon for both the subcritical p p c cases our results are the following in the subcritical case where a chaotic attractor exists in the absence of noise noise induced transients can be more persistent in the following sense of double exponential and algebraic scaling tau approximately exp k0 exp k1epsilon gamma for small noise amplitude epsilon where k0 0 k1 0 and gamma 0 are constants the longevity of the transient lifetime in this case is striking for the supercritical case where there is already a super persistent chaotic transient noise can significantly reduce the transient lifetime these results add to the understanding of the interplay between random and deterministic chaotic dynamics with surprising physical implications

explicit optimum design abstract the optimization of a trussed type structure of given geometry and material properties can be formulated as an exact and explicit mathematical programming problem in a mixed space of design variables and behaviour variables three techniques are presented corresponding to the three classical analysis methods of structural theory in the case of a single loading condition without variables linking the proposed method is very efficient since it eliminates the problem of multiple reanalysis without increasing the dimensionality of the problem and the number of constraints in the other cases the numerical efficiency of the technique depends on the specific problem to be solved

double precision errors in the logistic map statistical study and dynamical interpretation the nature of the round off errors that occur in the usual double precision computation of the logistic map is studied in detail different iterative regimes from the whole panoply of behaviors exhibited in the bifurcation diagram are examined histograms of errors in trajectories given and for the case of fully developed chaos an explicit formula is found it is shown that the statistics of the largest double precision error as a function of the map parameter is characterized by jumps whose location is determined by certain boundary crossings in the bifurcation diagram both jumps and locations seem to present geometric convergence characterized by the two first feigenbaum constants eventually a comparison with benford s law for the distribution of the leading digit of compilation of numbers is discussed

complex networks embedded in space dimension and scaling relations between mass topological distance and euclidean distance 2 andr exp a 2 suchthat m r d for d e m increases exponentially with as known for 0 while r is constant and independent of f or 2de we find the e

the bifurcation to homoclinic tori in the quasiperiodically forced duffing oscillator abstract using recently developed techniques of wiggins 1988 we study the bifurcation to homoclinic tori in the quasiperiodically forced duffing oscillator we show how homoclinic tori give rise to chaotic dynamics for single degree of freedom quasiperiodically forced oscillators in much the same way as smale horseshoes in the periodically forced case we give a complete description of the bifurcation set in the five dimensional parameter space for the two frequency forced duffing oscillator and then show how the two frequency results can be used to understand the effects of an arbitrary but finite number of frequencies

unimodal beam elements abstract a uniform prismatic elastic element in small displacements theory can be modeled as consisting of extension torsion and flexural elements acting in parallel the extension and torsion elements are unimodal whereas the flexural elements are bimodal due to the combined action of bending moments and shear forces this dissimilarity between the elements is the source of inconsistency between the analysis of a truss and a framed structure especially in a force approach it is shown that by an appropriate change of basis one can uncouple the flexural element into a unimodal moment element which carries the average moment of the element and a unimodal shear element which carries the shear force and related moment as a result any framed structure can be viewed as a generalized truss and analyzed accordingly in a standard form it is further shown that it is possible to physically construct a moment element and a shear element both exhibiting unimodal deformation patterns consequently one can replace a classical beam element by a moment element and a shear element assembled in parallel the approach is tested numerically in the case of beams of constant height subjected to several loading conditions preliminary results indicate that in theory substantial weight reductions can be obtained when designing structures composed of parallel unimodal elements

relaxation closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networks attractors in asymmetric neural networks with deterministic parallel dynamics present a chaotic regime at symmetry where the average length of the cycles increases exponentially with system size and an oscillatory regime at high symmetry where the average length of the cycles is 2 we show both with analytic arguments and numerically that there is a sharp transition at a critical symmetry between a phase where the typical cycles have length 2 and basins of attraction of vanishing weight and a phase where the typical cycles are exponentially long with system size and the weights of their attraction basins are distributed as in a random map with reversal symmetry the timescale after which cycles are reached grows exponentially with system size n and the exponent vanishes in the symmetric limit where the transition can be related to the dynamics of the infinite system where cycles are never reached using the closing probabilities as a tool we also study the relaxation of the function where is the local field experienced by the neuron i in the symmetric system it plays the role of a ljapunov function which drives the system towards its minima through steepest descent this interpretation survives even if only on the average also for small asymmetry this acts like an effective temperature the larger the asymmetry the faster the relaxation and the higher the asymptotic value reached e reaches very deep minima in the fixed points of the dynamics which are reached with vanishing probability and attains a larger value on the typical attractors which are cycles of length 2

dynamic proximity of spatio temporal sequences recurrent networks can generate spatio temporal neural sequences of very large cycles having an apparent random behavior nonetheless a proximity measure between these sequences may be defined through comparison of the synaptic weight matrices that generate them following the dynamic neural filter dnf formalism we demonstrate this concept by comparing teacher and student recurrent networks of binary neurons we show that large sequences providing a training set well exceeding the cover limit allow for good determination of the synaptic matrices alternatively assuming the matrices to be known very fast determination of the biases can be achieved thus a spatio temporal sequence may be regarded as spatio temporal encoding of the bias vector we introduce a linear support vector machine svm variant of the dnf in order to specify an optimal weight matrix this approach allows us to deal with noise spatio temporal sequences generated by different dnfs with the same number of neurons may be compared by calculating correlations of the synaptic matrices of the reconstructed dnfs other types of spatio temporal sequences need the introduction of hidden neurons and or the use of a kernel variant of the svm approach the latter is being defined as a recurrent support vector network rsvn

kuramoto model with frequency degree correlations on complex networks we study the kuramoto model on complex networks in which natural frequencies of phase oscillators and the vertex degrees are correlated using the annealed network approximation and numerical simulations we explore a special case in which the natural frequencies of the oscillators and the vertex degrees are linearly coupled we find that in uncorrelated scale free networks with the degree distribution exponent 2l ensuremath gamma l3 the model undergoes a first order phase transition while the transition becomes second order at ensuremath gamma g3 if ensuremath gamma 3 the phase synchronization emerges as a result of a hybrid phase transition that combines an abrupt emergence of synchronization as in first order phase transitions and a critical singularity as in second order phase transitions the critical fluctuations manifest themselves as avalanches in the synchronization process comparing our analytical calculations with numerical simulations for erd ifmmode mbox h o else h o fi s r enyi and scale free networks we demonstrate that the annealed network approach is accurate if the mean degree and size of the network are sufficiently large we also study analytically and numerically the kuramoto model on star graphs and find that if the natural frequency of the central oscillator is sufficiently large in comparison to the average frequency of its neighbors then synchronization emerges as a result of a first order phase transition this shows that oscillators sitting at hubs in a network may generate a discontinuous synchronization transition

structural preferential attachment stochastic process for the growth of scale free modular and self similar systems many complex systems have been shown to share universal properties of organization such as scale independence modularity and self similarity we borrow tools from statistical physics in order to study structural preferential attachment spa a recently proposed growth principle for the emergence of the aforementioned properties we study the corresponding stochastic process in terms of its time evolution its asymptotic behavior and the scaling properties of its statistical steady state moreover approximations are introduced to facilitate the modeling of real systems mainly complex networks using spa finally we investigate a particular behavior observed in the stochastic process the peloton dynamics and show how it predicts some features of real growing systems using prose samples as an example

the length of attractors in asymmetric random neural networks with deterministic dynamics the author has developed a method to detect attractors of any length in large neural networks with up to 1024 neurons within a reasonable period of cpu time in networks with symmetric couplings only stable states and in the case of parallel dynamics cycles of length 2 exist the presented simulations suggest that in sufficiently large systems this holds also for couplings up to a distinct value of asymmetry beyond this value extremely long cycles are detected and the average cycle length depends exponentially on system size

unfolding a chaotic bifurcation the differential equation of the sinusoidally forced pendulum is studied by digital simulation in a regime where two simple symmetrically related chaotic attractors grow and merge continuously as the forcing amplitude is increased by introducing a small constant bias in the forcing to break the symmetry two discontinuous bifurcations unfold from the single merging event considering both forcing amplitude and bias together as controls a codimension two bifurcation of chaotic attractors is defined whose geometric structure in control phase space is closely related to the elementary cusp catastrophe the chaotic bifurcations are explained in terms of homoclinic structures smale cycles in the poincare map

nonlinear dynamics and symbolic dynamics of neural networks a piecewise linear equation is proposed as a method of analysis of mathematical models of neural networks a symbolic representation of the dynamics in this equation is given as a directed graph on an n dimensional hypercube this provides a formal link with discrete neural networks such as the original hopfield models analytic criteria are given to establish steady states and limit cycle oscillations independent of network dimension model networks that display multiple stable limit cycles and chaotic dynamics are discussed the results show that such equations are a useful and efficient method of investigating the behavior of neural networks

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

the number of metastable states in the projection rule neural network we calculate the direct average n of the number of metastable states in the projection rule neural network by means of the saddle point method n is obtained as a function of the hamming distance g to a given pattern stability gamma and energy e the critical storage capacity of the model is calculated the examination of the energy dependence of n leads to lower bounds for the energies of the spin glass states

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

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

transition from regular to complex behaviour in a discrete deterministic asymmetric neural network model the authors study the long time behaviour of the transient before the collapse on the periodic attractors of a discrete deterministic asymmetric neural network model the system has a finite number of possible states so it is not possible to use the term chaos in the usual sense of sensitive dependence on the initial condition nevertheless on varying the asymmetry parameter k one observes a transition from ordered motion i e short transients and short periods on the attractors to complex temporal behaviour this transition takes place for the same value kc at which one has a change in behaviour for the mean transient length from a power law in the size of the system n to an exponential law in n the complex behaviour during the transient shows strong analogies with chaotic behaviour decay of temporal correlations positive shannon entropy nonconstant renyi entropies of different orders moreover the transition is very similar to the intermittent transition in chaotic systems a scaling law for the shannon entropy and strong fluctuations of the effective shannon entropy along the transient for k kc

scaling theory of transport in complex biological networks transport is an important function in many network systems and understanding its behavior on biological social and technological networks is crucial for a wide range of applications however it is a property that is not well understood in these systems probably because of the lack of a general theoretical framework here based on the finding that renormalization can be applied to bionetworks we develop a scaling theory of transport in self similar networks we demonstrate the networks invariance under length scale renormalization and we show that the problem of transport can be characterized in terms of a set of critical exponents the scaling theory allows us to determine the influence of the modular structure on transport in metabolic and protein interaction networks we also generalize our theory by presenting and verifying scaling arguments for the dependence of transport on microscopic features such as the degree of the nodes and the distance between them using transport concepts such as diffusion and resistance we exploit this invariance and we are able to explain based on the topology of the network recent experimental results on the broad flow distribution in metabolic networks

approximate reanalysis based on the exact analytic expressions the first author has recently given the exact analytic expressions of the inverse of the stiffness matrix the nodal displacements and the stress resultants in linear elastic structures composed of prismatic elements for structures of constant geometry the expressions are explicit in terms of the unimodal stiffnesses of the components of the structures however the expressions are intractable in their exact form due to their inordinate length it all has to do with the number of statically determinate substructures embedded in common engineering structures this paper describes some preliminary results obtained from approximate analysis models for the internal forces using truncated expressions that are similar in form to the exact analytic ones the approach is illustrated with numerical examples he classical methods of structural analysis methods are numerical techniques that compute the structural response to external loads for a given configuration of the structure and for given stiffnesses of its elements very often engineers are faced with the need to reanalyze structures for the same geo metrical configuration and for the same applied loads but with modified stiffnesses such instances occur in structural design and in particular in optimal structural design where one usually tests many candidate stiffness distributions before an acceptable structure is found 1 structural reanalysis is also present in the analysis of framed structures when the p d effect is accounted for 2 since the axial load in a member modifies its flexural rigidity such structures are solved itera tively at a given iteration the flexural stiffnesses of the members are computed based on the axial loads of the previ ous iteration the procedure is repeated until convergence when the stiffnesses of the individual members change the analysis procedure must in principle be repeated with the modified parameters there is no formula for the structural response in terms of the rigidities of its members to alleviate the computational cost of multiple reanalysis engineers have over the years developed reanalysis techniques that use results of previous analysis to predict the structural response when some or all of the rigidities are modified 3 4 noteworthy is the reciprocal approximation 1 which expresses the nodal dis placements as a linear taylor polynomial in terms of the reciprocals of the member stiffnesses recently the first author has written a series of papers 5 7 describing the analytic solution of the structural analysis prob lem for structures composed of uniform prismatic members of given geometry and assuming linear elastic behavior the analytic expressions are exact equations for the inverse of the stiffness matrix for the nodal displacements and for the internal forces in the members explicit in terms of the cross sectional structural characteristics of the components of the structure however what seems to be eldorado for the struc tural designer an exact formula for the structural response

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

the explicit inverse of the stiffness matrix abstract in two recent papers the author has given the exact analytic expression of the internal forces in linear elastic structures composed of uniform prismatic elements it was shown that the member forces are the ratios of two multilinear homogeneous polynomials in the unimodal stiffnesses of the elements of the structure the order of the polynomials is equal to the number of nodal degrees of freedom of the structure the number of terms of each polynomial is equal to the number of statically determinate stable substructures which can be derived from the original structure the coefficients of the polynomials can be computed by employing the equilibrium equations and by enforcing global compatibility of deformations it was found empirically that the coefficients of the polynomial in the denominator were numerically equal to the square of the determinants of the statics matrices of the respective statically determinate substructures as a consequence the denominator became the sum of the stiffness matrices of the statically determinate substructures this is in fact the binet cauchy form of the determinant of the stiffness matrix of the structure bearing in mind that the inverse of the stiffness matrix can be expressed as the ratio of the adjoint matrix of the stiffness matrix divided by the determinant of the stiffness matrix it became clear that the expressions of the stress resultants stem from an explicit expression of the adjoint the explicit expression of the adjoint of the stiffness matrix lies at the heart of this paper it is shown that the adjoint is a congruent transformation of the n 1 compound of the stiffness matrix where n is the number of degrees of freedom of the structure this cleared the way to use the binct cauchy theorem on the product of compound matrices to obtain an explicit expression for the adjoint and ipso facto for the inverse of the stiffness matrix having now the displacements of the structure the expression of the stress resultants which was obtained independently emerges in a very elegant manner the member forces in a structure can be expressed as the weighted sum of the member forces in all its determinate substructures when subjected to the applied loads the weighting factors are the ratios of the determinants of he stiffness matrices of the substructures to the determinant of the stiffness matrix of the original structure both the explicit inverse of the stiffness matrix and the expression of the internal forces in the structure are at present of a theoretical nature the number of terms involved in the polynomials is simply excessive for common engineering structures however ongoing reseach may yield more applicable expressions to be used for instance in he field of automated design of structures the theory is illustrated with the explicit analysis of a stayed mast

multivalley structure of attractor neural networks we numerically investigate the structure of the basins of attraction and the nature of the spurious attractors of the pseudo inverse and the optimal weights attractor neural networks we show that the number of attractors in the optimal weights model increases as the margin parameter increases and that the basins of attraction of the stored patterns are not significantly enlarged by moreover the number of attractors is smaller than in the pseudo inverse model

two state on off intermittency caused by unstable dimension variability in periodically forced drift waves certain high dimensional dynamical systems present two or more attractors characterized by different energy branches for some parameter values the dynamics oscillates between these two branches in a seemingly random fashion a phenomenon called two state on off intermittency in this work we show that the dynamical mechanism underlying this intermittency involves the severe breakdown of hyperbolicity of the attractors through a mechanism known as unstable dimension variability we characterize the parametric evolution of this variability using statistical properties of the finite time lyapunov exponents as a model system that exhibits this behavior we consider periodically forced and damped drift waves in this spatiotemporal example there is a low dimensional chaotic attractor that is created by an interior crisis already presenting unstable dimension variability

the relations between network operation and topological property in a scale free and small world network with community structure abstract it is always an open demanding and difficult task for generating available model to simulate dynamical functions and reveal inner principles from complex systems and networks in this article due to lots of real life and artificial networks are built from series of simple and small groups components we discuss some interesting and helpful network operation to generate more realistic network models in view of community structure modular topology we present a class of sparse network models n t m at the moment we capture the fact the n t 4 has not only scale free feature which means that the probability that a randomly selected vertex with degree k decays as a power law following p k k where is the degree exponent but also small world property which indicates that the typical distance between two uniform randomly chosen vertices grows proportionally to logarithm of the order of n t 4 namely relatively shorter diameter and lower average path length simultaneously displays higher clustering coefficient next as a new topological parameter correlating to reliability synchronization capability and diffusion properties of networks the number of spanning trees over a network is studied in more detail an exact analytical solution for the number of spanning trees of the n t 4 is obtained based on the network operation part hub vertex linking with each other will be helpful for structuring various network models and investigating the rules related with real life networks

attractors in fully asymmetric neural networks the statistical properties of the length of the cycles and of the weights of the attraction basins in fully asymmetric neural networks i e with completely uncorrelated synapses are computed in the framework of the annealed approximation which we previously introduced for the study of kauffman networks our results show that this model behaves essentially as a random map possessing a reversal symmetry comparison with numerical results suggests that the approximation could become exact in the large size limit

amplitude modulated and chaotic dynamics in resonant motion of strings abstract the weakly non linear resonant motions of a stretched string to a harmonic planar excitation are investigated one mode truncation of the in plane and out of plane equation results in a two degree to freedom system which is examined by using the method of averaging and the method of integral manifolds it is shown that the averaged equations for small enough damping possess non planar constant solutions which become unstable by a hopf bifurcation the resulting limit cycle solutions correspond to amplitude modulated whirling or ballooning motions of the string there are two limit cycle branches for the averaged equations one arising due to the hopf bifurcation and the other due to a global saddle node bifurcation the two branches merge as the damping is further reduced also with variation in detuning the isolated branch exhibits period doubling bifurcations chaotic attractors and merging of attractors giving rise to rossler as well as lorentz type solutions the truncated string equations are also directly integrated it is shown that there are non planar periodic responses that bifurcate into amplitude modulated motions on a 2 torus changes in detuning result in torus doubling merging of tori and then destruction of the torus leading to chaotic amplitude modulations the results of asymptotic analysis via the method of averaging are found to be in qualitative agreement with the actual response

chaos in high dimensional neural and gene networks abstract neural and gene networks are often modeled by differential equations if the continuous threshold functions in the differential equations are replaced by step functions the equations become piecewise linear pl equations the flow through the state space is represented schematically by paths and directed graphs on am n dimensional hypercube closed pathways called cycles may reflect periodic orbits with associated fixed points in a chosen poincare section a return map in the poincare section can be constructed by the composition of fractional linear maps the stable and unstable manifolds of the fixed points can be determined analytically these methods allow us to analyze the dynamics in higher dimensional networks as exemplified by a four dimensional network that displays chaotic behavior the three dimensional poincare map is projected to a two dimensional plane this much simpler piecewise linear two dimensional map conserves the important qualitative features of the flow

the phase space of interactions in neural networks with definite symmetry we calculate the typical fraction of the phase space of interactions which solve the problem of storing a given set of p patterns represented as n spin configurations as a function of the storage ratio a p n of the stability parameter k and of the symmetry 7 of the interaction matrices the calculation is performed for strongly diluted networks where the connectivity of each spin c is of the order of in n for each value of k and 17 there is a maximal value of a above which the volume of solutions vanishes for each value of k and a there is a typical value of 7 at which this volume is maximal the analytical studies are supplemented by numerical simulations on fully connected and diluted networks using specific learning algorithms

multivalley structure in kauffman s model analogy with spin glasses kauffman s model describes a random network of automata the authors calculations indicate that the multivalley structure of the basins of attraction in kauffman s model is very similar to that of infinite range spin glasses the similarity with spin glasses is tested quantitatively by computing the probability that two initial configurations fall into the same valley

flow transitions resembling bifurcations of the logistic map in simulations of the baroclinic rotating annulus we present evidence for a sequence of bifurcations in simulations of the differentially heated baroclinic rotating annulus similar to bifurcations of the logistic map the met office oxford rotating annulus laboratory simulation morals code is used to construct a detailed numerical regime diagram for the annulus and the distribution of regimes in parameter space is discussed the bifurcations are observed in a sequence of runs at high temperature forcing identified by poincare sections of the dominant temperature mode amplitude time series higher order return maps and predictions using quadratic fits to the data are used to verify this result and lyapunov exponents are calculated to identify and quantify the chaotic parts of the sequence

approximating stable and unstable manifolds in experiments we introduce a procedure to reveal invariant stable and unstable manifolds given only experimental data we assume a model is not available and show how coordinate delay embedding coupled with invariant phase space regions can be used to construct stable and unstable manifolds of an embedded saddle we show that the method is able to capture the fine structure of the manifold is independent of dimension and is efficient relative to previous techniques

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

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

statistical properties of randomly broken objects and of multivalley structures in disordered systems the statistical properties of the multivalley structure of disordered systems and of randomly broken objects have many features in common for all these problems if w denotes the weight of the sth piece we show that the probability distributions pi w of the largest piece w p2 w of the second largest piece and n y of y 1 wi always have singularities at w l n w2 l n and y ijn n 1 2 3

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

cycles in a synchronous neural network we extend the standard calculation of the number of metastable states fixed points so as to take into account cyclic attractors as well we calculate analytically the average number of cycles in the little model as a function of the fraction of flipping spins 1 q 2 we find that the cycles of length two are by far the most common type of attractors in particular for small values of the storage parameter the inverse cycles q 1 are the dominant attractors

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

relative stability in the dynamics of a two pattern neural net the authors investigate how patterns of different acquisition strengths influence each other s stability a neural network model with two strictly stable patterns stored on a noisy background is studied by a novel approximation of short time dynamics that singles out coherent contributions systematically and uses a gaussian approximation for incoherent sums the basin of attraction of the weaker pattern is found to shrink depending on the two acquisition strengths for a diluted hopfield type version of the model the weaker pattern may become unrecognisable

subtle dynamic behaviour of finite size sherrington kirkpatrick spin glasses with nonsymmetric couplings the authors have studied numerically the parallel dynamics of nonsymmetric sherrington kirkpatrick spin glasses varying the degree of symmetry eta ji kjk i ji k2 of the coupling coefficients between 0 and 1 for systems of finite size n in the limit t to infinity and at zero temperature t 0 they find subtle behaviour of the function c2 t si t 1 si t 1 which characterizes the appearance of 2 cycles or fixed point attractors quantitatively one has to distinguish the two cases eta 0 5 where c2 infinity to 1 i e the system is eventually trapped with probability 1 in a fixed point or a 2 cycle if after t to infinity the limit n to infinity is taken and eta 0 5 happens only for t 0 and at time scales tau n which increase exponentially with n whereas for t 0 or if for t 0 the limit n to infinity would be taken before t to infinity the quantity c2 t to infinity would decrease smoothly and monotonically with decreasing eta right from eta 1 in quantitative agreement with the mean field simulation of eissfeller and opper for t 0 the transient behaviour of c2 t between the mean field value which is reached already after typically 100 time steps and the trapping event is found to be governed by log normal statistics with size depending parameters

a method to derive approximate explicit solutions for structural mechanics problems the availability of explicit solutions i e analytical relationships between the structural response and the design variables allows a more direct and plain treatment of several structural problems this paper is devoted to derive approximate explicit solutions in the framework of linear static analysis of finite element modeled structures with a given layout fixed node positions the proposed procedure is based on a factorization of the element stiffness matrix following the unimodal components concept which allows a non conventional assembly of the global stiffness matrix the exact inversion of that matrix is a trivial task for the case of statically determinate structures structures with few redundancies or few design variables an approximate inverse of the stiffness matrix is herein derived for more general structural problems by resorting to the sherman morrison woodbury formula 2006 elsevier ltd all rights reserved

bifurcation behavior of the generalized lorenz equations at large rotation numbers the bifurcation structure and periodic orbits of the lorenz stenflo equations at large rotation numbers are given it is shown that rotation can lead to a much richer dynamical behavior than that of the original lorenz system and can be used to control or modify the latter s chaos behavior orbits with new topology arising from the merging and splitting of different periodic windows are observed abrupt changes in the one dimensional map are pointed out and studied in terms of the interaction of the interior and exterior boundaries

analytic representation of member forces in linear elastic redundant trusses abstract the process of designing a truss of given geometry and material properties where the design variables are the cross sectional areas of the bars is hampered by the need to reanalyse the structure many times until an acceptable design is obtained currently approximate explicit analysis models based on truncated linear taylor series expansions are used to evaluate the structural response at the various candidate design points due to the approximate nature of the analysis model the structure is designed iteratively until convergence of both the analysis equations and the design process this paper presents for the first time the exact analytic expressions of the internal loads in a truss which is subjected to static loads the stress resultants are the ratio of two multilinear polynomials in the element stillness the number of terms of the polynomials is equal to the number of combinations of statically determinate stable structures which can be derived from the original structures the coefficients of the polynomial expansions can be obtained from equilibrium considerations and from enforcing global compatibility of deformations the expressions are explicit in both the external loads and the element stiffness the applicability of the analytic equations hinges on the number of combinations of statically determinate stable substructures in the case of small size structures the present explicit equations circumvent the need for approximate reanalysis in common engineering structures the number of stable subsets is prohibitively large which renders the analytic expressions intractable the exact analytic expressions may however constitute a starting point for constructing approximate explicit analysis equations of improved quality