design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

world wide web scaling exponent from simon s 1955 model the statistical properties of the world wide web have attracted considerable attention recently since self similar regimes were first observed in the scaling of its link structure one characteristic quantity is the number of in links k that point to a particular web page its probability distribution p k shows a pronounced power law scaling p k approximately k gamma that is not readily explained by standard random graph theory here we recall a simple and elegant model for scaling phenomena in general copy and growth processes as proposed by simon in 1955 when combined with an experimental measurement of network growth in the world wide web this classical model is able to model the in link dynamics and predicts the scaling exponent gamma 2 1 in accordance with observation

edge overload breakdown in evolving networks we investigate growing networks based on barabasi and albert s algorithm for generating scale free networks but with edges sensitive to overload breakdown the load is defined through edge betweenness centrality we focus on the situation where the average number of connections per vertex is like the number of vertices linearly increasing in time after an initial stage of growth the network undergoes avalanching breakdowns to a fragmented state from which it never recovers this breakdown is much less violent if the growth is by random rather than by preferential attachment as defines the barabasi and albert model we briefly discuss the case where the average number of connections per vertex is constant in this case no breakdown avalanches occur implications to the growth of real world communication networks are discussed

pinning control of complex dynamical networks with general topology abstract recently the researches on pinning control of complex dynamical networks have mainly focused on such networks with very specific coupling schemes e g symmetric coupling uniform coupling and linear coupling however most real networks often consist of local units which interact with each other via asymmetric and heterogeneous connections in this paper pinning control of a continuous time complex dynamical network with general coupling topologies is studied some generic stability criteria based on master stability function msf are derived for such a general controlled network which guarantee that the whole network can be pinned to its equilibrium by placing feedback control only on a small fraction of nodes then these results are extended to discrete time case previous results about symmetric uniform or linear coupled networks in this area are included as special cases of the present work numerical simulations of directed networks with weighted coupling pinned by specifically selective pinning scheme are given for illustration and verification

response time of internauts abstract a new experiment measuring the dynamical response of the internet population to a point like perturbation has been performed the nature of the perturbation was that of an announcement specifically a web interview on stock market crashes which contained the url to the author s articles on the subject it was established that the download rate obeys the relation 1 t in qualitative agreement with previously reported results

hierarchical sequencing of online social graphs abstract in online communications patterns of conduct of individual actors and use of emotions in the process can lead to a complex social graph exhibiting multilayered structure and mesoscopic communities using simplicial complexes representation of graphs we investigate in depth topology of the online social network constructed from myspace dialogs which exhibits original community structure a simulation of emotion spreading in this network leads to the identification of two emotion propagating layers three topological measures are introduced referred to as the structure vectors which quantify graph s architecture at different dimension levels notably structures emerging through shared links triangles and tetrahedral faces frequently occur and range from tree like to maximal 5 cliques and their respective complexes on the other hand the structures which spread only negative or only positive emotion messages appear to have much simpler topology consisting of links and triangles the node s structure vector represents the number of simplices at each topology level in which the node resides and the total number of such simplices determines what we define as the node s topological dimension the presented results suggest that the node s topological dimension provides a suitable measure of the social capital which measures the actor s ability to act as a broker in compact communities the so called simmelian brokerage we also generalize the results to a wider class of computer generated networks investigating components of the node s vector over network layers reveals that same nodes develop different socio emotional relations and that the influential nodes build social capital by combining their connections in different layers

exact solutions for models of evolving networks with addition and deletion of nodes there has been considerable recent interest in the properties of networks such as citation networks and the worldwide web that grow by the addition of vertices and a number of simple solvable models of network growth have been studied in the real world however many networks including the web not only add vertices but also lose them here we formulate models of the time evolution of such networks and give exact solutions for a number of cases of particular interest for the case of net growth and so called preferential attachment in which newly appearing vertices attach to previously existing ones in proportion to vertex degree we show that the resulting networks have power law degree distributions but with an exponent that diverges as the growth rate vanishes we conjecture that the low exponent values observed in real world networks are thus the result of vigorous growth in which the rate of addition of vertices far exceeds the rate of removal were growth to slow in the future for instance in a more mature future version of the web we would expect to see exponents increase potentially without bound

spreading and shortest paths in systems with sparse long range connections spreading according to simple rules e g of fire or diseases and shortest path distances are studied on d dimensional systems with a small density p per site of long range connections small world lattices the volume v t covered by the spreading quantity on an infinite system is exactly calculated in all dimensions as a function of time t from this the average shortest path distance l r can be calculated as a function of euclidean distance r it is found that l r approximately r for r r c the characteristic length r c which governs the behavior of shortest path lengths diverges logarithmically with l for all p 0

epidemic spreading in correlated complex networks we study a dynamical model of epidemic spreading on complex networks in which there are explicit correlations among the node s connectivities for the case of markovian complex networks showing only correlations between pairs of nodes we find an epidemic threshold inversely proportional to the largest eigenvalue of the connectivity matrix that gives the average number of links which from a node with connectivity k go to nodes with connectivity k ensuremath numerical simulations on a correlated growing network model provide support for our conclusions

design of serially concatenated systems depending on the block length based on extrinsic information transfer exit charts the convergence behavior of iterative decoding is studied for a number of serially concatenated systems such as a serially concatenated code coded data transmission over an intersymbol interference channel bit interleaved coded modulation or trellis coded modulation efficient optimization algorithms based on simplified exit chart construction are devised to find irregular codes improving the convergence of iterative decoding one optimization criterion is to find concatenated systems exhibiting thresholds of successful decoding convergence which are close to information theoretic limits however these thresholds are approached only for very long block lengths to overcome this problem the decoding convergence after a fixed finite number of iterations is optimized which yields systems performing very well for short block lengths too as an example optimal system configurations for communication over an additive white gaussian noise channel are presented

involutive automorphisms and iwasawa decomposition of some hyperbolic kac moody algebras the involutive automorphisms of hyperbolic kac moody algebras are computed from satake diagrams of these algebras which are then used to furnish a general procedure of their iwasawa decomposition in particular 1 1 and 2 1 are considered as representative examples to illustrate the efficacy of the underlying treatment

general random pseudofractal networks we introduce a general random pseudofractal network model by assigning fitness to each edge in this model continuous growth and attachment determined by their fitness of already existing edges are the two ingredients we obtain the analytical results that our model exhibits a power law degree distribution with exponent 2 m 1 m 1 where m and are tunable parameters we also show that a general random pseudofractal network has a large clustering coefficient and a small average distance leading to a small world effect these theoretical results agree well with numerical simulations

extrinsic information transfer functions model and erasure channel properties extrinsic information transfer exit charts are a tool for predicting the convergence behavior of iterative processors for a variety of communication problems a model is introduced that applies to decoding problems including the iterative decoding of parallel concatenated turbo codes serially concatenated codes low density parity check ldpc codes and repeat accumulate ra codes exit functions are defined using the model and several properties of such functions are proved for erasure channels one property expresses the area under an exit function in terms of a conditional entropy a useful consequence of this result is that the design of capacity approaching codes reduces to a curve fitting problem for all the aforementioned codes a second property relates the exit function of a code to its helleseth klove levenshtein information functions and thereby to the support weights of its subcodes the relation is via a refinement of information functions called split information functions and via a refinement of support weights called split support weights split information functions are used to prove a third property that relates the exit function of a linear code to the exit function of its dual

epidemic dynamics in finite size scale free networks many real networks present a bounded scale free behavior with a connectivity cutoff due to physical constraints or a finite network size we study epidemic dynamics in bounded scale free networks with soft and hard connectivity cutoffs the finite size effects introduced by the cutoff induce an epidemic threshold that approaches zero at increasing sizes the induced epidemic threshold is very small even at a relatively small cutoff showing that the neglection of connectivity fluctuations in bounded scale free networks leads to a strong overestimation of the epidemic threshold we provide the expression for the infection prevalence and discuss its finite size corrections the present paper shows that the highly heterogeneous nature of scale free networks does not allow the use of homogeneous approximations even for systems of a relatively small number of nodes

self similar disk packings as model spatial scale free networks the network of contacts in space filling disk packings such as the apollonian packing is examined these networks provide an interesting example of spatial scale free networks where the topology reflects the broad distribution of disk areas a wide variety of topological and spatial properties of these systems is characterized their potential as models for networks of connected minima on energy landscapes is discussed

dynamics of directed graphs the world wide web we introduce and simulate a growth model of the world wide web based on the dynamics of outgoing links that is motivated by the conduct of the agents in the real web to update outgoing links re directing them towards constantly changing selected nodes emergent statistical correlation between the distributions of outgoing and incoming links is a key feature of the dynamics of the web the growth phase is characterized by temporal fractal structures which are manifested in the hierarchical organization of links we obtain quantitative agreement with the recent empirical data in the real web for the distributions of in and out links and for the size of the connected component in a fully grown network of n nodes we study the structure of connected clusters of nodes that are accessible along outgoing links from a randomly selected node the distributions of size and depth of the connected clusters with a giant component exhibit supercritical behavior by decreasing the control parameter average fraction of updated and added links per time step towards c n 10 the web can resume a critical structure with no giant component in it we find a different universality class when the updates of links are not allowed i e for 0 corresponding to the network of science citations

a classification of involutive automorphisms of an affine kac moody lie algebra abstract in this thesis we classify the conjugacy classes of involutions in aut g where g is an affine kac moody lie algebra we distinguish between two kinds of involutions those which preserve the conjugacy class of a borel subalgebra and those which don t we give a complete and non redundant list of representatives of involutions of the first kind and we compute their fixed points sets we prove that any involution of the first kind has a conjugate which leaves invariant the components of the gauss decomposition g n h n we also give a complete list of representatives of the conjugacy classes of involutions of the second kind

exact solution of site and bond percolation on small world networks we study percolation on small world networks which has been proposed as a simple model of the propagation of disease the occupation probabilities of sites and bonds correspond to the susceptibility of individuals to the disease and the transmissibility of the disease respectively we give an exact solution of the model for both site and bond percolation including the position of the percolation transition at which epidemic behavior sets in the values of the critical exponents governing this transition the mean and variance of the distribution of cluster sizes disease outbreaks below the transition and the size of the giant component epidemic above the transition

satake superdiagrams real forms and iwasawa decomposition of classical lie superalgebras satake superdiagrams corresponding to the lie superalgebras c n together with the exceptional ones are constructed from their dynkin diagrams with a view to determine the real forms of these superalgebras the involutive automorphisms are computed from a modified formula to account for all pertinent cases in a consistent way this mechanism is also used to furnish a general treatment of the iwasawa and langlands decompositions of these superalgebras in particular we compute these decompositions for for illustrational purposes and obtain the corresponding induced representations using the method of schmidt construction

a comprehensive weighted evolving network model abstract many social technological biological and economical systems are best described by weighted networks whose properties and dynamics depend not only on their structures but also on the connection weights among their nodes however most existing research work on complex network models are concentrated on network structures with connection weights among their nodes being either 1 or 0 in this paper we propose a new weighted evolving network model numerical simulations indicate that this network model yields three power law distributions for the node degrees connection weights and node strengths respectively particularly some other properties of the distributions such as the droop head and heavy tail effects can also be reflected by this model

maximal planar networks with large clustering coefficient and power law degree distribution in this article we propose a simple rule that generates scale free networks with very large clustering coefficient and very small average distance these networks are called random apollonian networks rans as they can be considered as a variation of apollonian networks we obtain the analytic results of power law exponent ensuremath gamma 3 and clustering coefficient c frac 46 3 ensuremath 36 phantom rule 0 2em 0ex mathrm ln phantom rule 0 2em 0ex frac 3 2 ensuremath approx 0 74 which agree with the simulation results very well we prove that the increasing tendency of average distance of rans is a little slower than the logarithm of the number of nodes in rans since most real life networks are both scale free and small world networks rans may perform well in mimicking the reality the rans possess hierarchical structure as c k ensuremath sim k ensuremath 1 that are in accord with the observations of many real life networks in addition we prove that rans are maximal planar networks which are of particular practicability for layout of printed circuits and so on the percolation and epidemic spreading process are also studied and the comparisons between rans and barab asi albert ba as well as newman watts nw networks are shown we find that when the network order n the total number of nodes is relatively small as n ensuremath sim 10 4 the performance of rans under intentional attack is not sensitive to n while that of ba networks is much affected by n and the diseases spread slower in rans than ba networks in the early stage of the suseptible infected process indicating that the large clustering coefficient may slow the spreading velocity especially in the outbreaks

search and topology aspects in transport on scale free networks abstract using a recently proposed model physica a 332 2004 566 of information transport on complex networks we study the role of network substrates on the statistics of queuing times and correlations in traffic streams when navigation with an enlarged information horizon is applied the waiting time distribution on structured networks has a power law tail which can be parametrized by 1 1 q where the respective value of the non extensivity parameter q increases systematically with increasing graph complexity the corresponding distribution on the small world graph appears to be a stretched exponential function that cannot be classified within the same scheme we further demonstrate by looking at the anti persistence in flow time series and the transit time distributions how traffic efficiency increases with an extended information horizon a horizon of two layers approaches the critical horizon on the correlated cyclic scale free graph suggesting that the mechanisms of signaling over two to three layers ensure efficient transport processes on networks with this organizational complexity

temporal fractal structures origin of power laws in the world wide web abstract using numerical simulations and scaling theory we study the dynamics of the world wide web from the growth rules recently proposed in tadic physica a 293 2001 273 with appropriate parameters we demonstrate that the emergence of power law behavior of the out and in degree distributions in the web involves occurrence of temporal fractal structures that are manifested in the scale free growth of the local connectivity and in first return time statistics we also show how the scale free behavior occurs in the statistics of random walks on the web where the walkers use information on the local graph connectivity

robustness of the in degree exponent for the world wide web we consider a stochastic model for directed scale free networks following power laws in the degree distributions in both incoming and outgoing directions in our model the number of vertices grow geometrically with time with a growth rate p at each time step i each newly introduced vertex is connected to a constant number of already existing vertices with the probability linearly proportional to in degree distribution of a selected vertex and ii each existing vertex updates its outgoing edges through a stochastic multiplicative process with mean growth rate of outgoing edges g and its variance ensuremath sigma 2 using both analytic treatment and numerical simulations we show that while the out degree exponent ensuremath gamma mathrm out depends on the parameters the in degree exponent ensuremath gamma mathrm in has two distinct values ensuremath gamma mathrm in 2 for pgg and 1 for plg independent of different parameters values the latter case has logarithmic correction to the power law since the vertex growth rate p is larger than the degree growth rate g for the world wide web www nowadays the in degree exponent appears robust as ensuremath gamma mathrm in 2 for the www

satake superdiagrams and iwasawa decomposition of some hyperbolic kac moody superalgebras the involutive automorphisms of hyperbolic kac moody superalgebras are computed from the satake superdiagrams corresponding to these algebras these are then used to furnish a general treatment of the iwasawa decomposition of these algebras in particular we consider 1 2 1 and 1 0 1 as representative examples for the purpose of illustration

complex scale free networks with tunable power law exponent and clustering abstract we introduce a network evolution process motivated by the network of citations in the scientific literature in each iteration of the process a node is born and directed links are created from the new node to a set of target nodes already in the network this set includes m ambassador nodes and l of each ambassador s descendants where m and l are random variables selected from any choice of distributions p l and q m the process mimics the tendency of authors to cite varying numbers of papers included in the bibliographies of the other papers they cite we show that the degree distributions of the networks generated after a large number of iterations are scale free and derive an expression for the power law exponent in a particular case of the model where the number of ambassadors is always the constant m and the number of selected descendants from each ambassador is the constant l the power law exponent is 2 l 1 l for this example we derive expressions for the degree distribution and clustering coefficient in terms of l and m we conclude that the proposed model can be tuned to have the same power law exponent and clustering coefficient of a broad range of the scale free distributions that have been studied empirically

statistical mechanics of topological phase transitions in networks we provide a phenomenological theory for topological transitions in restructuring networks in this statistical mechanical approach energy is assigned to the different network topologies and temperature is used as a quantity referring to the level of noise during the rewiring of the edges the associated microscopic dynamics satisfies the detailed balance condition and is equivalent to a lattice gas model on the edge dual graph of a fully connected network in our studies based on an exact enumeration method monte carlo simulations and theoretical considerations we find a rich variety of topological phase transitions when the temperature is varied these transitions signal singular changes in the essential features of the global structure of the network depending on the energy function chosen the observed transitions can be best monitored using the order parameters f s5s max m i e the size of the largest connected component divided by the number of edges or f k5k max m the largest degree in the network divided by the number of edges if for example the energy is chosen to be e52s max the observed transition is analogous to the percolation phase transition of random graphs for this choice of the energy the phase diagram in the k t plane is constructed single vertex energies of the form e5 i f k i where k i is the degree of vertex i are also studied depending on the form of f ki first order and continuous phase transitions can be observed in case of f k i 52 k i1a ln ki the transition is continuous and at the critical temperature scale free graphs can be recovered finally by abruptly decreasing the temperature nonequilibrium processes e g nucleation and growth of particular topological phases can also be interpreted by the present approach

dynkin diagrams for hyperbolic kac moody algebras hyperbolic algebras such as e10 are based on minkowskian root spaces that revert to finite or affine root systems upon the removal of any simple root in string theory terms finite lie affine kac moody and hyperbolic kac moody algebras are generated by tachyon vertex operators trachyon plus photon vertex operators and vertex operators for all mass levels respectively the 136 possible hyperbolic dynkin diagrams between the ranks 3 and 10 are classified and exhibited completing an earlier enumeration by kac of the 18 rank 7 10 cases the rank 2 hyperbolic algebras infinite in number have been classified by lepowsky and moody 1979

spectral and dynamical properties in classes of sparse networks with mesoscopic inhomogeneities we study structure eigenvalue spectra and random walk dynamics in a wide class of networks with subgraphs modules at mesoscopic scale the networks are grown within the model with three parameters controlling the number of modules their internal structure as scale free and correlated subgraphs and the topology of connecting network within the exhaustive spectral analysis for both the adjacency matrix and the normalized laplacian matrix we identify the spectral properties which characterize the mesoscopic structure of sparse cyclic graphs and trees the minimally connected nodes the clustering and the average connectivity affect the central part of the spectrum the number of distinct modules leads to an extra peak at the lower part of the laplacian spectrum in cyclic graphs such a peak does not occur in the case of topologically distinct tree subgraphs connected on a tree whereas the associated eigenvectors remain localized on the subgraphs both in trees and cyclic graphs we also find a characteristic pattern of periodic localization along the chains on the tree for the eigenvector components associated with the largest eigenvalue lambda l 2 of the laplacian further differences between the cyclic modular graphs and trees are found by the statistics of random walks return times and hitting patterns at nodes on these graphs the distribution of first return times averaged over all nodes exhibits a stretched exponential tail with the exponent sigma approximately 1 3 for trees and sigma approximately 2 3 for cyclic graphs which is independent of their mesoscopic and global structure

percolation in directed scale free networks many complex networks in nature have directed links a property that affects the network s navigability and large scale topology here we study the percolation properties of such directed scale free networks with correlated in and out degree distributions we derive a phase diagram that indicates the existence of three regimes determined by the values of the degree exponents in the first regime we regain the known directed percolation mean field exponents in contrast the second and third regimes are characterized by anomalous exponents which we calculate analytically in the third regime the network is resilient to random dilution i e the percolation threshold is pc 1

renormalization group for evolving networks we propose a renormalization group treatment of stochastically growing networks as an example we study percolation on growing scale free networks in the framework of a real space renormalization group approach as a result we find that the critical behavior of percolation on the growing networks differs from that in uncorrelated networks

self organized scale free networks growth and preferential attachments have been coined as the two fundamental mechanisms responsible for the scale free feature in complex networks as characterized by an algebraic degree distribution there are situations particularly in biological networks where growth is absent or not important yet some of these networks still exhibit the scale free feature with a small degree exponent here we propose two classes of models to account for this phenomenon we show analytically and numerically that in the first model a spectrum of algebraic degree distributions with a small exponent can be generated the second model incorporates weights for nodes and it is able to generate robust scale free degree distribution with larger algebraic exponents our results imply that it is natural for a complex network to self organize itself into a scale free

relationship between the in degree and out degree of www abstract in this paper the relationship between the in degree and out degree of world wide web is studies at each time step a new node with out degree k out is added where k out obeys the power law distribution and its mean value is m the analytical and simulation results suggest that the exponent of in degree distribution would be i 2 1 m depending on the average out degree this finding is supported by the empirical data which has not been emphasized by the previous studies on directed networks

highly clustered scale free networks we propose a model for growing networks based on a finite memory of the nodes the model shows stylized features of real world networks power law distribution of degree linear preferential attachment of new links and a negative correlation between the age of a node and its link attachment rate notably the degree distribution is conserved even though only the most recently grown part of the network is considered as the network grows the clustering reaches an asymptotic value larger than that for regular lattices of the same average connectivity and similar to the one observed in the networks of movie actors coauthorship in science and word synonyms these highly clustered scale free networks indicate that memory effects are crucial for a correct description of the dynamics of growing networks

topology of technology graphs small world patterns in electronic circuits recent theoretical studies and extensive data analyses have revealed a common feature displayed by biological social and technological networks the presence of small world patterns here we analyze this problem by using several graphs obtained from one of the most common technological systems electronic circuits it is shown that both analogic and digital circuits exhibit small world behavior we conjecture that the small world pattern arises from the compact design in which many elements share a small close physical neighborhood plus the fact that the system must define a single connected component which requires shortcuts connecting different integrated clusters the degree distributions displayed are consistent with a conjecture concerning the sharp cutoffs associated to the presence of costly connections amaral et al proc natl acad sci usa 97 11 149 2000 thus providing a limit case for the classes of universality of small world patterns from real artificial networks the consequences for circuit design are outlined

involutive automorphisms and iwasawa decomposition of affine kac moody superalgebras the involutive automorphisms of affine kac moody superalgebras are computed from satake superdiagrams corresponding to these algebras these are then used to furnish a general treatment of the iwasawa decomposition of these algebras in particular we consider a 1 0 1 and b 1 1 1 as representative examples for the purpose of illustration

small world phenomena and the statistics of linear polymers a regular lattice in which the sites can have long range connections at a distance l with a probabilty p l l in addition to the short range nearest neighbour connections shows small world behaviour for 0 c in the most appropriate physical example of such a system namely the linear polymer network the exponent is related to the exponents of the corresponding n vector model in the n 0 limit and its value is less than c still the polymer networks do not show small world behaviour here we show that this is due to a small value constraint on the number q of long range connections per monomer in the network in the general q space we obtain a phase boundary separating regions with and without small world behaviour and show that the polymer network falls marginally in the regular lattice region

coded sampling bound how much training is needed for iterative semi blind channel estimation coherent detection is commonly facilitated by means of pilot aided channel estimation while this approach offers a high estimation accuracy the spectral and power efficiencies are reduced due to the pilot overhead in the case that redundancy due to channel coding is not used for the purpose of channel estimation the maximum pilot spacing for which a time varying channel can be reconstructed without distortion is given by the nyquist shannon sampling theorem in this paper it is shown that if channel coding is taken into account for channel estimation in an iterative fashion the maximum spacing can be significantly extended which results in an increased spectral efficiency towards this goal a so called coded sampling bound is derived this semi analytical bound is compared with the nyquist shannon sampling theorem the presented results indicate that the maximum pilot spacing can be arbitrarily large given a suitable channel code and code rate

directed accelerated growth application in citation network abstract in many real world networks the number of links increases non linearly with the number of nodes models of such accelerated growth have been considered earlier with deterministic and stochastic number of links here we consider stochastic accelerated growth in a network where links are directed with the number of outgoing links following a power law distribution the results are similar to the undirected case as the accelerated growth is enhanced the degree of distribution becomes independent of the initial attractiveness a parameter which plays a key role in directed networks as an example of a directed model with accelerated growth the citation network is considered in which the distribution of the number of outgoing link has an exponential tail the role of accelerated growth is examined here with two different growth laws

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

dynamical scaling behavior of percolation clusters in scale free networks in this work we investigate the spectra of laplacian matrices that determine many dynamic properties of scale free networks below and at the percolation threshold we use a replica formalism to develop analytically based on an integral equation a systematic way to determine the ensemble averaged eigenvalue spectrum for a general type of treelike networks close to the percolation threshold we find characteristic scaling functions for the density of states rho lambda of scale free networks rho lambda shows characteristic power laws rho lambda approximately lambda alpha 1 or rho lambda approximately lambda d 2 for small lambda where alpha 1 holds below and alpha 2 at the percolation threshold in the range where the spectra are accessible from a numerical diagonalization procedure the two methods lead to very similar results

iterative decoding of binary block and convolutional codes iterative decoding of two dimensional systematic convolutional codes has been termed turbo de coding using log likelihood algebra we show that any decoder can be used which accepts soft inputs including a priori values and delivers soft outputs that can be split into three terms the soft channel and a priori inputs and the extrinsic value the extrinsic value is used as an a priori value for the next iteration decoding algorithms in the log likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code the iteration is controlled by a stop criterion derived from cross entropy which results in a minimal number of iterations optimal and suboptimal decoders with reduced complexity are presented simulation results show that very simple component codes are sufficient block codes are appropriate for high rates and convolutional codes for lower rates less than 2 3 any combination of block and convolutional component codes is possible several interleaving techniques are described at a bit error rate ber of 10 sup 4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block convolutional component codes interleaver sizes less than 1000 and for three to six iterations

analysis of diffusion and trapping efficiency for random walks on non fractal scale free trees abstract in this paper the discrete random walks on the recursive non fractal scale free trees nfsft are studied and a kind of method to calculate the analytic solutions of the mean first passage time mfpt for any pair of nodes the mean trapping time mtt for any target node and mean diffusing time mdt for any starting node are proposed furthermore we compare the trapping efficiency and diffusion efficiency between any two nodes of nfsft by using the mtt and the mdt as the measures of trapping efficiency and diffusion efficiency respectively and find the best or worst trapping sites and the best or worst diffusion sites the results show that the two hubs of nfsft are not only the best trapping site but also the worst diffusion site and that the nodes which are the farthest nodes from the two hubs are not only the worst trapping sites but also the best diffusion sites furthermore we find that the ratio between the maximum and minimum of mtt grows logarithmically with network order but the ratio between the maximum and minimum of mdt is almost equal to 1 the results imply that the trap s position has great effect on the trapping efficiency but the position of starting node has little effect on diffusion efficiency finally the simulation for random walks on nfsft is done and it is consistent with the derived results

dynkin diagrams of hyperbolic kac moody superalgebras hyperbolic kac moody superalgebras are classified in terms of their dynkin diagrams these types of kac moody superalgebras are those whose diagrams revert to either that of simple or affine superalgebras upon deletion of one of the vertices it is found that the maximum rank of this type of algebra is 6

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

infection dynamics on scale free networks we discuss properties of infection processes on scale free networks relating them to the node connectivity distribution that characterizes the network considering the epidemiologically important case of a disease that confers permanent immunity upon recovery we derive analytic expressions for the final size of an epidemic in an infinite closed population and for the dependence of infection probability on an individual s degree of connectivity within the population as in an earlier study r pastor satorras and a vesipignani phys rev lett 86 3200 2001 phys rev e 63 006117 2001 for an infection that did not confer immunity upon recovery the epidemic process in contrast with many traditional epidemiological models does not exhibit threshold behavior and we demonstrate that this is a consequence of the extreme heterogeneity in the connectivity distribution of a scale free network finally we discuss effects that arise from finite population sizes showing that networks of finite size do exhibit threshold effects infections cannot spread for arbitrarily low transmission probabilities

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

iwasawa decomposition of affine kac moody algebras using satake diagrams we carry out the iwasawa decomposition of the affine kac moody algebras within the general framework of involutive automorphisms which are determined with the help of the satake diagrams corresponding to these algebras the parabolic subalgebras essential for the construction of induced representations are computed from the langlands decomposition in particular we consider the untwisted versions viz a1 1 and a2 1 as representative examples for the purpose of illustration

small world networks links with long tailed distributions small world networks swn obtained by randomly adding to a regular structure additional links al are of current interest in this paper we explore based on physical models a new variant of swn in which the probability of realizing an al depends on the chemical distance between the connected sites we assume a power law probability distribution and study random walkers on the network focusing especially on their probability of being at the origin we connect the results to l evy flights which follow from a mean field variant of our model

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 abelian sandpile and related models abstract the abelian sandpile model is the simplest analytically tractable model of self organized criticality this paper presents a brief review of known results about the model the abelian group structure of the algebra of operators allows an exact calculation of many of its properties in particular when there is a preferred direction one can calculate all the critical exponents characterizing the distribution of avalanche sizes in all dimensions for the undirected case the model is related to q 0 state potts model this enables exact calculation of some exponents in two dimensions and there are some conjectures about others we also discuss a generalization of the model to a network of communicating reactive processors this includes sandpile models with stochastic toppling rules as a special case we also consider a nondashabelian stochastic variant which lies in a different universality class related to directed percolation

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

quantum group symmetries and non local currents in 2d qft we construct and study the implications of some new non local conserved currents that exist is a wide variety of massive integrable quantum field theories in 2 dimensions including the sine gordon theory and its generalization to affine toda theory these non local currents provide a non perturbative formulation of the theories the symmetry algebras correspond to the quantum affine kac moody algebras thes matrices are completely characterized by these symmetries formals matrices for the imaginary coupling affine toda theories are thereby derived the application of theses matrices to perturbed coset conformal field theory is studied non local charges generating the finite dimensional quantum group in the liouville theory are briefly presented the formalism based on non local charges we describe provides an algernative to the quantum inverse scattering method for solving integrable quantum field theories in 2d

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

correlated random networks we develop a statistical theory of networks a network is a set of vertices and links given by its adjacency matrix c and the relevant statistical ensembles are defined in terms of a partition function z summation operator exp betah c the simplest cases are uncorrelated random networks such as the well known erdos renyi graphs here we study more general interactions h c which lead to correlations for example between the connectivities of adjacent vertices in particular such correlations occur in optimized networks described by partition functions in the limit beta infinity they are argued to be a crucial signature of evolutionary design in biological networks

scale free networks from a hamiltonian dynamics contrary to many recent models of growing networks we present a model with fixed number of nodes and links where a dynamics favoring the formation of links between nodes with degree of connectivity as different as possible is introduced by applying a local rewiring move the network reaches equilibrium states assuming broad degree distributions which have a power law form in an intermediate range of the parameters used interestingly in the same range we find nontrivial hierarchical clustering

optimal placement of training for frequency selective block fading channels the problem of placing training symbols optimally for orthogonal frequency division multiplexing ofdm and single carrier systems is considered the channel is assumed to be quasi static with a finite impulse response of length l 1 samples under the assumptions that neither the transmitter nor the receiver knows the channel and that the receiver forms a minimum mean square error mmse channel estimate based on training symbols only training is optimized by maximizing a tight lower bound on the ergodic training based independent and identically distributed i i d capacity for ofdm systems it is shown that the lower bound is maximized by placing the known symbols periodically in frequency for single carrier systems under the assumption that the training symbols are placed in clusters of length spl alpha spl ges 2l 1 it is shown that the lower bound is maximized by a family of placement schemes called qpp spl alpha where qpp stands for quasi periodic placement these placement schemes are formed by grouping the known symbols into as many clusters as possible and then placing these clusters periodically in the packet for both ofdm and single carrier systems the optimum energy tradeoff between training and data is also obtained

generalized kac moody algebras on etudie une classe d algebres de lie qui ont une forme bilineaire contravariante qui est presque definie positive ces algebres generalisent celles de kac moody et peuvent etre considerees comme des algebres de kac moody avec des racines simples imaginaires

scale free characteristics of random networks the topology of the world wide web the world wide web forms a large directed graph whose vertices are documents and edges are links pointing from one document to another here we demonstrate that despite its apparent random character the topology of this graph has a number of universal scale free characteristics we introduce a model that leads to a scale free network capturing in a minimal fashion the self organization processes governing the world wide web c 2000 elsevier science b v all rights reserved pacs 84 35 i 64 60 fr 87 23 ge

growing random networks with fitness abstract three models of growing random networks with fitness dependent growth rates are analysed using the rate equations for the distribution of their connectivities in the first model a a network is built by connecting incoming nodes to nodes of connectivity k and random additive fitness with rate k 1 for 0 we find the connectivity distribution is power law with exponent 2 in the second model b the network is built by connecting nodes to nodes of connectivity k random additive fitness and random multiplicative fitness with rate k 1 this model also has a power law connectivity distribution but with an exponent which depends on the multiplicative fitness at each node in the third model c a directed graph is considered and is built by the addition of nodes and the creation of links a node with fitness i incoming links and j outgoing links gains a new incoming link with rate i 1 and a new outgoing link with rate j 1 the distributions of the number of incoming and outgoing links both scale as power laws with inverse logarithmic corrections