growing scale free networks with tunable clustering we extend the standard scale free network model to include a triad formation step we analyze the geometric properties of networks generated by this algorithm both analytically and by numerical c

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

schwinger and pegg barnett approaches and a relationship between angular and cartesian quantum descriptions ii phase spaces following the discussion in state space language presented in a preceding paper we work on the passage from the phase space description of a degree of freedom described by a finite number of states without classical counterpart to one described by an infinite and continuously labelled number of states with this it is possible to relate an original schwinger idea to the pegg barnett approach to the phase problem in phase space language this discussion shows that one can obtain the weyl wigner formalism for both cartesian and angular coordinates as limiting elements of the discrete phase space formalism

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

continuum quantum systems as limits of discrete quantum systems i state vectors abstract dynamical systems on continuum hilbert spaces may be realized as limits of dynamical systems on discrete possibly finite dimensional hilbert spaces in this first of four papers on the topic the continuum and discrete spaces are interfaced to one another algebraically convergence of vectors is defined in such a way as to preserve inner products and a necessary and sufficient coordinate wise criterion for convergence is proved

optimal systolic design for the transitive closure and the shortest path problems due to vlsi technological progress algorithm oriented array architectures such as systolic arrays appear to be very effective feasible and economic this paper discusses how to design systolic arrays for the transitive closure and the shortest path problems we shall focus on the warshall algorithm for the transitive closure problem and the floyd algorithm for the shortest path problem these two algorithms share exactly the same structural formulation therefore they lead to the same systolic array design in this paper we first present a general method for mapping algorithms to systolic arrays using this methodology two new systolic designs for the warshall floyd algorithm will be derived the first one is a spiral array which is easy to derive and can be further simplified to a hexagonal array the other is an orthogonal systolic array which is optimal in terms of pipelining rate block pipelining rate and the number of input output connections

random walks on site disordered lattices random walk models with disordered waiting times site disorder are analysed using a real space renormalisation group method anomalous diffusion is found in all dimensions whenever the first moment of the distribution of waiting times diverges above two dimensions the fixed points are described by continuous time random walks with non analytic waiting time functions below two dimensions the fixed points are themselves disordered random walks where the disorder is characterised by a stable probability distribution a fractal dimension is defined for site disordered lattices and below two dimensions a hyperscaling law is found which relates the fractal dimension the euclidean dimension and the exponent describing the displacement of the walker

growing network with local rules preferential attachment clustering hierarchy and degree correlations the linear preferential attachment hypothesis has been shown to be quite successful in explaining the existence of networks with power law degree distributions it is then quite important to determine if this mechanism is the consequence of a general principle based on local rules in this work it is claimed that an effective linear preferential attachment is the natural outcome of growing network models based on local rules it is also shown that the local models offer an explanation for other properties like the clustering hierarchy and degree correlations recently observed in complex networks these conclusions are based on both analytical and numerical results for different local rules including some models already proposed in the literature

network formation determined by the diffusion process of random walkers we studied the diffusion process of random walkers in networks formed by their traces this model considers the rise and fall of links determined by the frequency of transports of random walkers in order to examine the relation between the formed network and the diffusion process a situation in which multiple random walkers start from the same vertex is investigated the difference in diffusion rate of random walkers according to the difference in dimension of the initial lattice is very important for determining the time evolution of the networks for example complete subgraphs can be formed on a one dimensional lattice while a graph with a power law vertex degree distribution is formed on a two dimensional lattice we derived some formulae for predicting network changes for the 1d case such as the time evolution of the size of nearly complete subgraphs and conditions for their collapse the networks formed on the 2d lattice are characterized by the existence of clusters of highly connected vertices and their life time as the life time of such clusters tends to be small the exponent of the power law distribution changes from 1 2 to 3

continuum quantum systems as limits of discrete quantum systems iv affine canonical transforms affine canonical transforms complex order fourier transforms and their associated coherent states appear in two scenarios finite discrete and continuum we examine the relationship between the two scenarios making systematic use of inductive limits which were developed in the preceding articles in this series

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

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

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

properties of highly clustered networks we propose and solve exactly a model of a network that has both a tunable degree distribution and a tunable clustering coefficient among other things our results indicate that increased clustering leads to a decrease in the size of the giant component of the network we also study susceptible infective recovered type epidemic processes within the model and find that clustering decreases the size of epidemics but also decreases the epidemic threshold making it easier for diseases to spread in addition clustering causes epidemics to saturate sooner meaning that they infect a near maximal fraction of the network for quite low transmission rates

corrections to scaling for diffusion in disordered media the authors study the diffusion of a particle in a d dimensional lattice where disorder arises from a random distribution of waiting times associated with each site of the lattice using scaling arguments they derive in addition to the leading asymptotic behaviour the correction to scaling terms for the mean square displacement they also perform detailed monte carlo simulations for one two and three dimensions which give results in substantial agreement with the scaling argument predictions

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

structural transitions in scale free networks real growing networks such as the world wide web or personal connection based networks are characterized by a high degree of clustering in addition to the small world property and the absence of a characteristic scale appropriate modifications of the barabasi albert preferential attachment network growth capture all these aspects we present a scaling theory to describe the behavior of the generalized models and the mean field rate equation for clustering this is solved for a specific case with the result c k approximately 1 k for the clustering of a node of degree k this mean field exponent agrees with simulations and reproduces the clustering of many real networks

the architecture of complex weighted networks networked structures arise in a wide array of different contexts such as technological and transportation infrastructures social phenomena and biological systems these highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity along with a complex topological structure real networks display a large heterogeneity in the capacity and intensity of the connections these features however have mainly not been considered in past studies where links are usually represented as binary states i e either present or absent here we study the scientific collaboration network and the world wide air transportation network which are representative examples of social and large infrastructure systems respectively in both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network we define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices this information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network these results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks

a model for social networks social networks are organized into communities with dense internal connections giving rise to high values of the clustering coefficient in addition these networks have been observed to be assortative i e highly connected vertices tend to connect to other highly connected vertices and have broad degree distributions we present a model for an undirected growing network which reproduces these characteristics with the aim of producing efficiently very large networks to be used as platforms for studying sociodynamic phenomena the communities arise from a mixture of random attachment and implicit preferential attachment the structural properties of the model are studied analytically and numerically using the k clique method for quantifying the communities r 2006 elsevier b v all rights reserved

optimal systolic array algorithms for tensor product in this paper we examine the computational complexity of optimal systolic array algorithms for tensor product we provide a time minimal schedule that meets the computed processor lower and upper bounds including one for tensor product our principle contribution is to find lower and upper bounds in the algorithm and its implementation for generating function and find processor time minimal schedule

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

continuum quantum systems as limits of discrete quantum systems iii operators convergence of a discrete operator to a continuum operator is defined as examples the circular rotor the one dimensional box the harmonic oscillator and the fractional fourier transform are realized as limits of finite dimensional quantum systems limits thus defined preserve algebraic structure the results prepare for a sequel in which some affine canonical transforms will be discretized

a phase transition in the dynamics of an exact model for hopping transport the authors analyse an exact model for hopping transport of a random walker in the presence of randomly distributed deep trapping sites for an exponential distribution of the depth of traps they find a phase transition in the diffusive behaviour as a function of temperature they show that above a critical temperature transport is purely diffusive while below it one finds anomalous diffusion characterised by a diffusion exponent that increases with decreasing temperature the analysis combines theory with accurate simulation in one and two dimensions the analytical and numerical results indicate that the upper critical dimension is dc 2 i e for d or dc 2 the mean field theory can be applied