a tqft from quantum teichmuller theory by using quantum teichmuller theory we construct a one parameter family of tqfts on the categroid of admissible leveled shaped 3 manifolds

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

scaling of load in communications networks we show that the load at each node in a preferential attachment network scales as a power of the degree of the node for a network whose degree distribution is p k k we show that the load is l k k with 1 implying that the probability distribution for the load is p l 1 l 2 independent of the results are obtained through scaling arguments supported by finite size scaling studies they contradict earlier claims but are in agreement with the exact solution for the special case of tree graphs results are also presented for real communications networks at the ip layer using the latest available data our analysis of the data shows relatively poor power law degree distributions as compared to the scaling of the load versus degree this emphasizes the importance of the load in network analysis

mechanism for linear preferential attachment in growing networks abstract the network properties of a graph ensemble subject to the constraints imposed by the expected degree sequence are studied it is found that the linear preferential attachment is a fundamental rule as it keeps the maximal entropy in sparse growing networks this provides theoretical evidence in support of the linear preferential attachment widely exists in real networks and adopted as a crucial assumption in growing network models besides in the sparse limit we develop a method to calculate the degree correlation and clustering coefficient in our ensemble model which is suitable for all kinds of sparse networks including the ba model proposed by barabasi and albert

wavelet thresholding for non necessarily gaussian noise idealism for various types of noise exponential normal mixture compactly supported wavelet thresholding methods are studied problems linked to the existence of optimal thresholds are tackled and minimaxity properties of the methods also analyzed a coefficient dependent method for choosing thresholds is also briefly presented

addition deletion networks we study structural properties of growing networks where both addition and deletion of nodes are possible our model network evolves via two independent processes with rate r a node is added to the system and this node links to a randomly selected existing node with rate 1 a randomly selected node is deleted and its parent node inherits the links of its immediate descendants we show that the in component size distribution decays algebraically ck k as k the exponent 2 r 1 1 varies continuously with the addition rate r structural properties of the network including the height distribution the diameter of the network the average distance between two nodes and the fraction of dangling nodes are also obtained analytically interestingly the deletion process leads to a giant hub a single node with a macroscopic degree whereas all other nodes have a microscopic degree

accuracy analysis of measurements on a stable power law distributed series of events we investigate how finite measurement time limits the accuracy with which the parameters of a stably distributed random series of events can be determined the model process is generated by timing the emigration of individuals from a population that is subject to deaths and a particular choice of multiple immigration events this leads to a scale free discrete random process where customary measures such as mean value and variance do not exist however converting the number of events occurring in fixed time intervals to a 1 bit clipped process allows the construction of well behaved statistics that still retain vestiges of the original power law and fluctuation properties these statistics include the clipped mean and correlation function from measurements of which both the power law index of the distribution of events and the time constant of its fluctuations can be deduced we report here a theoretical analysis of the accuracy of measurements of the mean of the clipped process this indicates that for a fixed experiment time the error on measurements of the sample mean is minimized by an optimum choice of the number of samples it is shown furthermore that this choice is sensitive to the power law index and that the approach to poisson statistics is dominated by rare events or outliers our results are supported by numerical simulation

universal fir mmse filtering we consider the problem of causal estimation i e filtering of a real valued signal corrupted by zero mean time independent real valued additive noise under the mean squared error mse criterion we build a universal filter whose per symbol squared error for every bounded underlying signal is essentially as small as that of the best finite duration impulse response fir filter of a given order we do not assume a stochastic mechanism generating the underlying signal and assume only that the variance of the noise is known to the filter the regret of the expected mse of our scheme is shown to decay as o logn n where n is the length of the signal moreover we present a stronger concentration result which guarantees the performance of our scheme not only in expectation but also with high probability our result implies a conventional stochastic setting result i e when the underlying signal is a stationary process our filter achieves the performance of the optimal fir filter we back our theoretical findings with several experiments showcasing the potential merits of our universal filter in practice our analysis combines tools from the problems of universal filtering and competitive online regression

fragmentation of random trees we study fragmentation of a random recursive tree into a forest by repeated removal of nodes the initial tree consists of n nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly selected existing node as nodes are removed from the tree one at a time the tree dissolves into an ensemble of separate trees namely a forest we study statistical properties of trees and nodes in this heterogeneous forest and find that the fraction of remaining nodes m characterizes the system in the limit we obtain analytically the size density of trees of size s the size density has power law tail with exponent therefore the tail becomes steeper as further nodes are removed and the fragmentation process is unusual in that exponent increases continuously with time we also extend our analysis to the case where nodes are added as well as removed and obtain the asymptotic size density for growing trees

approximate message passing algorithm with universal denoising and gaussian mixture learning we study compressed sensing cs signal reconstruction problems where an input signal is measured via matrix multiplication under additive white gaussian noise our signals are assumed to be stationary and ergodic but the input statistics are unknown the goal is to provide reconstruction algorithms that are universal to the input statistics we present a novel algorithmic framework that combines 1 the approximate message passing cs reconstruction framework which solves the matrix channel recovery problem by iterative scalar channel denoising 2 a universal denoising scheme based on context quantization which partitions the stationary ergodic signal denoising into independent and identically distributed i i d subsequence denoising and 3 a density estimation approach that approximates the probability distribution of an i i d sequence by fitting a gaussian mixture gm model in addition to the algorithmic framework we provide three contributions 1 numerical results showing that state evolution holds for nonseparable bayesian sliding window denoisers 2 an i i d denoiser based on a modified gm learning algorithm and 3 a universal denoiser that does not need information about the range where the input takes values from or require the input signal to be bounded we provide two implementations of our universal cs recovery algorithm with one being faster and the other being more accurate the two implementations compare favorably with existing universal reconstruction algorithms in terms of both reconstruction quality and runtime

weighted self similar networks under preferential attachment abstract we introduce an abstract evolutionary formalism that generates weighted networks whose growth under stochastic preferential attachment triggers unrestricted weight rearrangements in existing links the class of resulting algorithms for different parameter values includes the barabasi albert and barrat barthelemy vespignani models as special cases we solve the recursions that describe the average growth to derive exact solutions for the expected degree and strength distribution the individual strength and weight development and the joint distribution of neighboring degrees we find that the network exhibits a particular form of self similarity namely every sufficiently interconnected node has on average the same constitution of small degree neighbors as any other node of large degree finally we suggest potential applications in several fields of interest

the asymptotic degree distribution of a kind of evolving random tree in this paper we introduce an evolving random tree the model is a combination of a preferential attachment model and a uniform model we show that the distribution of the number of leaves at time n is approximated by the poisson distribution with parameter the proportional degree sequence obeys a power law with parameter and is a constant number that describes the probability of choosing the connecting form in the evolving random tree model

evolutionary dynamics on networks of selectively neutral genotypes effects of topology and sequence stability networks of selectively neutral genotypes underlie the evolution of populations of replicators in constant environments previous theoretical analysis predicted that such populations will evolve toward highly connected regions of the genome space we first study the evolution of populations of replicators on simple networks and quantify how the transient time to equilibrium depends on the initial distribution of sequences on the neutral network on the topological properties of the latter and on the mutation rate second network neutrality is broken through the introduction of an energy for each sequence this allows to study the competition between two features neutrality and energetic stability relevant for survival and subjected to different selective pressures in cases where the two features are negatively correlated the population experiences sudden migrations in the genome space for values of the relevant parameters that we calculate the numerical study of larger networks indicates that the qualitative behavior to be expected in more realistic cases is already seen in representative examples of small networks

quasistatic scale free networks the nodal degree distribution function of a scale free net work sfn has a power law tail 1 empirical data obtainedfor several social biological and computational networkshave con rmed the existence of such probability distribu tions decaying as power laws 2 4 for example the world wide web 5 which is a network of web pages and the hy perlinks among various pages and the internet network 6 ofrouters or autonomous systems follow power laws as p k k

on dynamic random graphs with degree homogenization via anti preferential attachment probabilities abstract we analyze a dynamic random undirected graph in which newly added vertices are connected to those already present in the graph either using with probability p an anti preferential attachment mechanism or with probability 1 p a preferential attachment mechanism we derive the asymptotic degree distribution in the general case and study the asymptotic behavior of the expected degree process in the general and that of the degree process in the pure anti preferential attachment case degree homogenization mainly affects convergence rates for the former case and also the limiting degree distribution in the latter lastly we perform a simulative study of a variation of the introduced model allowing for anti preferential attachment probabilities given in terms of the current maximum degree of the graph

a universal scheme for wyner ziv coding of discrete sources we consider the wyner ziv wz problem of lossy compression where the decompressor observes a noisy version of the source whose statistics are unknown a new family of wz coding algorithms is proposed and their universal optimality is proven compression consists of sliding window processing followed by lempel ziv lz compression while the decompressor is based on a modification of the discrete universal denoiser dude algorithm to take advantage of side information the new algorithms not only universally attain the fundamental limits but also suggest a paradigm for practical wz coding the effectiveness of our approach is illustrated with experiments on binary images and english text using a low complexity algorithm motivated by our class of universally optimal wz codes

discrete denoising with shifts we introduce s dude a new algorithm for denoising discrete memoryless channel dmc corrupted data the algorithm which generalizes the recently introduced dude discrete universal denoiser aims to compete with a genie that has access in addition to the noisy data also to the underlying clean data and that can choose to switch up to m times between sliding window denoisers in a way that minimizes the overall loss when the underlying data form an individual sequence we show that the s dude performs essentially as well as this genie provided that m is sublinear in the size of the data when the clean data are emitted by a piecewise stationary process we show that the s dude achieves the optimum distribution dependent performance provided that the same sublinearity condition is imposed on the number of switches to further substantiate the universal optimality of the s dude we show that when the number of switches is allowed to grow linearly with the size of the data any sequence of scheme s fails to compete in the above sense using dynamic programming we derive an efficient implementation of the s dude which has complexity time and memory growing linearly with the data size and the number of switches m preliminary experimental results are presented suggesting that s dude has the capacity to improve on the performance attained by the original dude in applications where the nature of the data abruptly changes in time or space as is often the case in practice

weighted scale free networks in euclidean space using local selection rule received 23 july 2005 revised manuscript received 21 july 2006 published 22 september 2006a spatial scale free network is introduced and studied whose motivation originated in the growing internetas well as airport networks we argue that in these real world networks a new node necessarily selects one ofits neighboring local nodes for connection and is not controlled by preferential attachment as in the barabasi albert ba model this observation is mimicked in our model where the nodes pop up at randomly locatedpositions in the euclidean space and are connected to one end of the nearest link in spite of this crucialdifference it is observed that the leading behavior of our network is like that of the ba model de ning the linkweight as an algebraic power of its euclidean length the weight distribution and the nonlinear dependence ofthe nodal strength on the degree are analytically calculated it is claimed that a power law decay of the linkweights with time ensures such nonlinear behavior switching off the euclidean space from the same modelyields a much simpler de nition of the ba model where numerical effort grows linearly with

phase transition in the rich get richer mechanism due to finite size effects the rich get richer mechanism agents increase their wealth randomly at a rate proportional to their holdings is often invoked to explain the pareto power law distribution observed in many physical situations such as the degree distribution of growing scale free nets we use two different analytical approaches as well as numerical simulations to study the case where the number of agents is fixed and finite but large and the rich get richer mechanism is invoked a fraction r of the time the remainder of the time wealth is disbursed by a homogeneous process at short times we recover the pareto law observed for an unbounded number of agents in later times the moving distribution can be scaled to reveal a phase transition with a gaussian asymptotic form for and a pareto like tail on the positive side and a novel stretched exponential decay on the negative side for

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

stationary and nonstationary properties of evolving networks with preferential linkage networks evolving by preferential attachment of both external and internal links are investigated the rate of adding an external link is assumed to depend linearly on the degree of a preexisting node to which a new node is connected the process of creating an internal link between a pair of existing vertices is assumed to be controlled entirely by the vertex that has more links than the other vertex in the pair and the rate of creation of such a link is assumed to be in general nonlinear in the degree of the more strongly connected vertex it is shown that degree distributions of networks evolving only by creating internal links display for large degrees a nonstationary power law decay with a time dependent scaling exponent nonstationary power law behaviors are numerically shown to persist even when the number of nodes is not fixed and both external and internal connections are introduced provided that the rate of preferential attachment of internal connections is nonlinear it is argued that nonstationary effects are not unlikely in real networks although these effects may not be apparent especially in networks with a slowly varying mean degree

preferential attachment with information filtering node degree probability distribution properties abstract a network growth mechanism based on a two step preferential rule is investigated as a model of network growth in which no global knowledge of the network is required in the first filtering step a subset of fixed size m of existing nodes is randomly chosen in the second step the preferential rule of attachment is applied to the chosen subset the characteristics of thus formed networks are explored using two approaches computer simulations of network growth and a theoretical description based on a master equation the results of the two approaches are in excellent agreement special emphasis is put on the investigation of the node degree probability distribution it is found that the tail of the distribution has the exponential form given by exp k m implications of the node degree distribution with such tail characteristics are briefly discussed

image denoising using scale mixtures of gaussians in the wavelet domain we describe a method for removing noise from digital images based on a statistical model of the coefficients of an overcomplete multiscale oriented basis neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables a gaussian vector and a hidden positive scalar multiplier the latter modulates the local variance of the coefficients in the neighborhood and is thus able to account for the empirically observed correlation between the coefficient amplitudes under this model the bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable we demonstrate through simulations with images contaminated by additive white gaussian noise that the performance of this method substantially surpasses that of previously published methods both visually and in terms of mean squared error

percolation on correlated networks we reconsider the problem of percolation on an equilibrium random network with degree degree correlations between nearest neighboring vertices focusing on critical singularities at a percolation threshold we obtain criteria for degree degree correlations to be irrelevant for critical singularities we present examples of networks in which assortative and disassortative mixing leads to unusual percolation properties and new critical exponents

resilience to damage of graphs with degree correlations the existence or nonexistence of a percolation threshold on power law correlated graphs is a fundamental question for which a general criterion is lacking in this work we investigate the problems of site and bond percolation on graphs with degree correlations and their connection with spreading phenomena we obtain some general expressions that allow the computation of the transition thresholds or their bounds using these results we study the effects of assortative and disassortative correlations on the resilience to damage of networks

highly dispersed networks by enhanced redirection we introduce a class of networks that grow by enhanced redirection nodes are introduced sequentially and each either attaches to a randomly chosen target node with probability 1 ensuremath r or to the parent of the target with probability r where r is an increasing function of the degree of the parent this mechanism leads to highly dispersed networks with unusual properties i existence of multiple macrohubs nodes whose degree is a finite fraction of the total number of network nodes n ii lack of self averaging and iii anomalous scaling in which n k the number of nodes of degree k scales as n k ensuremath sim n ensuremath nu ensuremath 1 k ensuremath nu with 1l ensuremath nu l2

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

the nested chinese restaurant process and bayesian nonparametric inference of topic hierarchies we present the nested chinese restaurant process ncrp a stochastic process that assigns probability distributions to ensembles of infinitely deep infinitely branching trees we show how this stochastic process can be used as a prior distribution in a bayesian nonparametric model of document collections specifically we present an application to information retrieval in which documents are modeled as paths down a random tree and the preferential attachment dynamics of the ncrp leads to clustering of documents according to sharing of topics at multiple levels of abstraction given a corpus of documents a posterior inference algorithm finds an approximation to a posterior distribution over trees topics and allocations of words to levels of the tree we demonstrate this algorithm on collections of scientific abstracts from several journals this model exemplifies a recent trend in statistical machine learning the use of bayesian nonparametric methods to infer distributions on flexible data structures

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

general connectivity distribution functions for growing networks with preferential attachment of fractional power we study the general connectivity distribution functions for growing networks with preferential attachment pa of fractional power i k using simon s method we first show that the heart of the previously known methods of the rate equations for the connectivity distribution functions is nothing but simon s method for word problem secondly we show that for the case of fractional the z transformation of the rate equation provides a fractional differential equation of a new type which coincides with that for pa with linear power when 1 we show that to solve such a fractional differential equation we need to define a transcendental function a b c z that we call upsilon function most of all the previously known results are obtained consistently in the framework of a unified theory

nonlinear barabasi albert network abstract in recent years there has been considerable interest in the structure and dynamics of complex networks one of the most studied networks is the linear barabasi albert model here we investigate the nonlinear barabasi albert growing network in this model a new node connects to a vertex of degree k with a probability proportional to k real each vertex adds m new edges to the network we derive an analytic expression for the degree distribution p k which is valid for all values of m and 1 in the limit the network is homogeneous if 1 there is a gel phase with m super connected nodes it is proposed a formula for the clustering coefficient which is in good agreement with numerical simulations the assortativity coefficient r is determined and it is shown that the nonlinear barabasi albert network is assortative disassortative if 1 and no assortative only when 1 in the limit the assortativity coefficient can be exactly calculated we find r 7 13 when m 2 finally the minimum average shortest path length l min is numerically evaluated increasing the network size l min diverges for 1 and it is equal to 1 when 1

universal denoising of discrete time continuous amplitude signals we consider the problem of reconstructing a discrete time signal sequence with continuous valued components corrupted by a known memoryless channel when performance is measured using a per symbol loss function satisfying mild regularity conditions we develop a sequence of denoisers that although independent of the distribution of the underlying ldquocleanrdquo sequence is universally optimal in the limit of large sequence length this sequence of denoisers is universal in the sense of performing as well as any sliding window denoising scheme which may be optimized for the underlying clean signal our results are initially developed in a ldquosemi stochasticrdquo setting where the noiseless signal is an unknown individual sequence and the only source of randomness is due to the channel noise it is subsequently shown that in the fully stochastic setting where the noiseless sequence is a stationary stochastic process our schemes universally attain optimum performance the proposed schemes draw from nonparametric density estimation techniques and are practically implementable we demonstrate efficacy of the proposed schemes in denoising gray scale images in the conventional additive white gaussian noise awgn setting with additional promising results for less conventional noise distributions

scale free properties of weighted networks with connectivity driven topology abstract the rate equations are used to study the scale free behavior of the weight distribution in evolving networks whose topology is determined only by degrees of preexisting vertices an analysis of these equations shows that the degree distribution and thereby the weight distribution remain unchanged when the probability rate of attaching new nodes is replaced with an unnormalized rate determined by the ratio of the degree of a randomly selected old node to the maximal node degree at the current stage of the network evolution such a modification of the attachment rule is argued to accelerate considerable numerical simulations of both unweighted and weighted networks belonging to the class of investigated evolving systems it is also proved that the studied rate equations have a solution corresponding to the total weight concentrated at individual vertices distribution displaying the power law behavior for asymptotically large weights

statistics of changes in lead node in connectivity driven networks we study statistical properties of the highest degree or most popular nodes in growing networks we show that the number of lead changes increases logarithmically with network size n independent of the details of the growth mechanism the probability that the first node retains the lead approaches a finite constant for popularity driven growth and decays as n phi ln n 1 2 with phi 0 08607 em leader for growth with no popularity bias

universal denoising for the finite input general output channel we consider the problem of reconstructing a finite alphabet signal corrupted by a known memoryless channel with a general output alphabet the goodness of the reconstruction is measured by a given loss function we constructively establish the existence of a universal sequence of denoiser s attaining asymptotically the optimum distribution dependent performance for any stationary source that may be generating the noiseless signal we show in fact that there is a whole family of denoiser sequences with this property these schemes are shown to be universal also in a semistochastic setting where the only randomness assumed is that associated with the channel noise the scheme is practical requiring o n sup 1 spl epsiv operations for any spl epsiv 0 and working storage size sublinear in the input data length this extends recent work that presented a discrete universal denoiser for recovering a discrete source corrupted by a discrete memoryless channel dmc

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

the l sub 1 and l sub 2 strong consistency of recursive kernel density estimation from dependent samples the l sub 1 and l sub 2 strong consistency of recursive kernel density estimators is established for mixing and ergodic stationary processes sharp rates of almost sure convergence are obtained in the l sub 2 case for mixing processes in addition the concept and properties of hilbert space valued mixingales are developed and strong laws of large numbers are given

stratification in the preferential attachment network we study structural properties of trees grown by preferential attachment in this mechanism nodes are added sequentially and attached to existing nodes at a rate that is strictly proportional to the degree we classify nodes by their depth n defined as the distance from the root of the tree and find that the network is strongly stratified most notably the distribution f n k of nodes with degree k at depth n has a power law tail f n k k n the exponent grows linearly with depth where the brackets denote an average over all nodes therefore nodes that are closer to the root are better connected and moreover the degree distribution strongly varies with depth similarly the in component size distribution has a power law tail and the characteristic exponent grows linearly with depth qualitatively these behaviors extend to a class of networks that grow by redirection

statistics of weighted treelike networks we study the statistics of growing networks with a tree topology in which each link carries a weight k i k j theta where k i and k j are the node degrees at the end points of link ij network growth is governed by preferential attachment in which a newly added node attaches to a node of degree k with rate a k k lambda for general values of theta and lambda we compute the total weight of a network as a function of the number of nodes n and the distribution of link weights generically the total weight grows as n for lambda theta 1 and superlinearly otherwise the link weight distribution is predicted to have a power law form that is modified by a logarithmic correction for the case lambda 0 we also determine the node strength defined as the sum of the weights of the links that attach to the node as function of k using known results for degree correlations we deduce the scaling of the node strength on k and n

diversity of graphs with highly variable connectivity a popular approach for describing the structure of many complex networks focuses on graph theoretic properties that characterize their large scale connectivity while it is generally recognized that such descriptions based on aggregate statistics do not uniquely characterize a particular graph and also that many such statistical features are interdependent the relationship between competing descriptions is not entirely understood this paper lends perspective on this problem by showing how the degree sequence and other constraints e g connectedness no self loops or parallel edges on a particular graph play a primary role in dictating many features including its correlation structure building on recent work we show how a simple structural metric characterizes key differences between graphs having the same degree sequence more broadly we show how the often implicit choice of a background set against which to measure graph features has serious implications for the interpretation and comparability of graph theoretic descriptions

aggregation kinetics of popularity abstract the tools of aggregation kinetics are applied to the popularity phenomena of single lane traffic clustering and to the growth of a network that mimics citations of scientific publications in the latter the network is built by introducing papers new nodes one at a time with preferential linking to more popular previously existing nodes from the rate equations the distribution of node degree as well as various global properties can be determined easily a simple extension of the model appears to describe the degree distributions of the world wide web

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

stochastic win stay lose shift strategy with dynamic aspirations in evolutionary social dilemmas in times of plenty expectations rise just as in times of crisis they fall this can be mathematically described as a win stay lose shift strategy with dynamic aspiration levels where individuals aspire to be as wealthy as their average neighbor here we investigate this model in the realm of evolutionary social dilemmas on the square lattice and scale free networks by using the master equation and monte carlo simulations we find that cooperators coexist with defectors in the whole phase diagram even at high temptations to defect we study the microscopic mechanism that is responsible for the striking persistence of cooperative behavior and find that cooperation spreads through second order neighbors rather than by means of network reciprocity that dominates in imitation based models for the square lattice the master equation can be solved analytically in the large temperature limit of the fermi function while for other cases the resulting differential equations must be solved numerically either way we find good qualitative agreement with the monte carlo simulation results our analysis also reveals that the evolutionary outcomes are to a large degree independent of the network topology including the number of neighbors that are considered for payoff determination on lattices which further corroborates the local character of the microscopic dynamics unlike large scale spatial patterns that typically emerge due to network reciprocity here local checkerboard like patterns remain virtually unaffected by differences in the macroscopic properties of the interaction network

tetrahedral forms in monoidal categories and 3 manifold invariants we introduce systems of objects and operators in linear monoidal categories called widehat psi systems a widehat psi system satisfying several additional assumptions gives rise to a topological invariant of triples a closed oriented 3 manifold m a principal bundle over m a link in m this construction generalizes the quantum dilogarithmic invariant of links appearing in the original formulation of the volume conjecture we conjecture that all quantum groups at odd roots of unity give rise to widehat psi systems and we verify this conjecture in the case of the borel subalgebra of quantum sl 2

spr based markov chain method for degree distributions of evolving networks abstract in this paper we develop a stochastic process rules spr based markov chain method to calculate the degree distributions of evolving networks this new approach overcomes two shortcomings of shi chen and liu s use of the markov chain method shi et al 2005 21 in addition we show how an spr based markov chain method can be effectively used to calculate degree distributions of random birth and death networks which we believe to be novel first spr are introduced to replace traditional evolving rules tr making it possible to compute degree distributions in one sample space then the spr based markov chain method is introduced and tested by using it to calculate two kinds of evolving network finally and most importantly the spr based method is applied to the problem of calculating the degree distributions of random birth and death networks

human sexual contact network as a bipartite graph abstract a simple model to encapsulate the essential growth properties of the web of human sexual contacts is presented in the model only heterosexual connection is considered and represented by a random growing bipartite graph where both male female contact networks grow simultaneously the time evolution of the model is analysed by a rate equation approach leading to confirm that male and female sexual contact distributions decay as power laws with exponents depending on influx and charisma of the sexes

partition of networks into basins of attraction we study partition of networks into basins of attraction based on a steepest ascent search for the node of highest degree each node is associated with or attracted to its neighbor of maximal degree as long as the degree is increasing a node that has no neighbors of higher degree is a peak attracting all the nodes in its basin maximally random scale free networks exhibit different behavior based on their degree distribution exponent gamma for small gamma broad distribution networks are dominated by a giant basin whereas for large gamma narrow distribution there are numerous basins with peaks attracting mainly their nearest neighbors we derive expressions for the first two moments of the number of basins we also obtain the complete distribution of basin sizes for a class of hierarchical deterministic scale free networks that resemble random nets finally we generalize the problem to regular networks and lattices where all degrees are equal and thus the attractiveness of a node must be determined by an assigned weight rather than the degree we derive the complete distribution of basins of attraction resulting from randomly assigned weights in one dimensional chains

geographical networks evolving with an optimal policy in this article we propose a growing network model based on an optimal policy involving both topological and geographical measures in this model at each time step a new node having randomly assigned coordinates in a 1 1 square is added and connected to a previously existing node i which minimizes the quantity r 2 k where ri is the geographical distance ki the degree and a free parameter the degree distribution obeys a power law form when 1 and an exponential

a context quantization approach to universal denoising we revisit the problem of denoising a discrete time continuous amplitude signal corrupted by a known memoryless channel by modifying our earlier approach to the problem we obtain a scheme that is much more tractable than the original one and at the same time retains the universal optimality properties the universality refers to the fact that the proposed denoiser asymptotically with increasing block length of the data achieves the performance of an optimum denoiser that has full knowledge of the distribution of a source generating the underlying clean sequence the only restriction being that the distribution is stationary the optimality in a sense we will make precise of the denoiser also holds in the case where the underlying clean sequence is unknown and deterministic and the only source of randomness is in the noise the schemes involve a simple preprocessing step of quantizing the noisy symbols to generate quantized contexts the quantized context value corresponding to each sequence component is then used to partition the unquantized symbols into subsequences a universal symbol by symbol denoiser for unquantized sequences is then separately employed on each of the subsequences we identify a rate at which the context length and quantization resolution should be increased so that the resulting scheme is universal the proposed family of schemes is computationally attractive with an upper bound on complexity which is independent of the context length and the quantization resolution initial experimentation seems to indicate that these schemes are not only superior from a computational viewpoint but also achieve better denoising in practice

random networks with tunable degree distribution and clustering we present an algorithm for generating random networks with arbitrary degree distribution and clustering frequency of triadic closure we use this algorithm to generate networks with exponential power law and poisson degree distributions with variable levels of clustering such networks may be used as models of social networks and as a testable null hypothesis about network structure finally we explore the effects of clustering on the point of the phase transition where a giant component forms in a random network and on the size of the giant component some analysis of these effects is presented

effect of aging on network structure in network evolution the effect of aging is universal in scientific collaboration network scientists have a finite time span of being active in movie actors network once popular stars are retiring from stage devices on the internet may become outmoded with techniques developing so rapidly here we find in citation networks that this effect can be represented by an exponential decay factor e ensuremath ensuremath beta ensuremath tau where ensuremath tau is the node age while other evolving networks the internet for instance may have different types of aging for example a power law decay factor which is also studied and compared it has been found that as soon as such a factor is introduced to the barabasi albert scale free model the network will be significantly transformed the network will be clustered even with infinitely large size and the clustering coefficient varies greatly with the intensity of the aging effect i e it increases linearly with ensuremath beta for small values of ensuremath beta and decays exponentially for large values of ensuremath beta at the same time the aging effect may also result in a hierarchical structure and a disassortative degree degree correlation generally the aging effect will increase the average distance between nodes but the result depends on the type of the decay factor the network appears like a one dimensional chain when exponential decay is chosen but with power law decay a transformation process is observed i e from a small world network to a hypercubic lattice and to a one dimensional chain finally the disparities observed for different choices of the decay factor in clustering average node distance and probably other aspects not yet identified are believed to bear significant meaning on empirical data acquisition

rate equation approach for correlations in growing network models we propose a rate equation approach to compute two vertex correlations in scale free growing network models based on the preferential attachment mechanism the formalism based on previous work of szabo et al fphys rev e 67 056102 s2002dg for the clustering spectrum measuring three vertex correlations is based on a rate equation in the continuous degree and time approximation for the average degree of the nearest neighbors of vertices of degree k with an appropriate boundary condition we study the properties of both two and three vertex correlations for linear preferential attachment models and also for a model yielding a large clustering coefficient the expressions obtained are checked by means of extensive numerical simulations the rate equation proposed can be generalized to more sophisticated growing network models and also extended to deal with related correlation measures as an example we consider the case of a recently proposed model of weighted networks for which we are able to compute a weighted two vertex correlation function taking into account the strength of the interactions between connected vertices

the origin of preferential attachment and the generalized preferential attachment for weighted networks abstract in this paper we first discuss the origin of preferential attachment then we establish the generalized preferential attachment gpa which has two new properties first it encapsulates both the topological and weight aspects of a network which makes it is neither entirely degree preferential nor entirely weight preferential second it can tell us not only the chance that each already existing vertex being connected but also how much weight each new edge has the gpa can generate four power law distributions besides the three for vertex degrees vertex strengths and edge weights it yields a new power law distribution for the subgraph degrees

scale free networks from varying vertex intrinsic fitness a new mechanism leading to scale free networks is proposed in this letter it is shown that in many cases of interest the connectivity power law behavior is neither related to dynamical properties nor to preferential attachment assigning a quenched fitness value x i to every vertex and drawing links among vertices with a probability depending on the fitnesses of the two involved sites gives rise to what we call a good get richer mechanism in which sites with larger fitness are more likely to become hubs i e to be highly connected

short ranged memory model with preferential growth in this work we introduce a variant of the yule simon model for preferential growth by incorporating a finite kernel to model the effects of bounded memory we characterize the properties of the model combining analytical arguments with extensive numerical simulations in particular we analyze the lifetime and popularity distributions by mapping the model dynamics to corresponding markov chains and branching processes respectively these distributions follow power laws with well defined exponents that are within the range of the empirical data reported in ecologies interestingly by varying the innovation rate this simple out of equilibrium model exhibits many of the characteristics of a continuous phase transition and around the critical point it generates time series with power law popularity lifetime and interevent time distributions and nontrivial temporal correlations such as a bursty dynamics in analogy with the activity of solar flares our results suggest that an appropriate balance between innovation and oblivion rates could provide an explanatory framework for many of the properties commonly observed in many complex systems

scale free property of directed networks with two intrinsic node weights abstract this paper proposes a network model to understand the scale free property of directed networks the proposed model assigns two intrinsic variables incoming and outgoing weights to every node a directed link is established from node i to node j if the sum of the outgoing weight of node i and the incoming weight of node j exceeds a predetermined threshold the proposed model allows us to know the exact analytical expressions for degree distributions and clustering we analytically find that the in degree and out degree distributions have power law tails and their scaling exponents are controllable within the range 1 the average clustering coefficient of nodes with out degree or in degree n also has a power law tail as a function of n we also find that the scaling exponent of the clustering coefficient depends on the correlation between incoming and outgoing weights

multistable binary decision making on networks we propose a simple model for a binary decision making process on a graph motivated by modeling social decision making with cooperative individuals the model is similar to a random field ising model or fiber bundle model but with key differences on heterogeneous networks for many types of disorder and interactions between the nodes we predict discontinuous phase transitions with mean field theory which are largely independent of network structure we show how these phase transitions can also be understood by studying microscopic avalanches and describe how network structure enhances fluctuations in the distribution of avalanches we suggest theoretically the existence of a glassy spectrum of equilibria associated with a typical phase even on infinite graphs so long as the first moment of the degree distribution is finite this behavior implies that the model is robust against noise below a certain scale and also that phase transitions can switch from discontinuous to continuous on networks with too few edges numerical simulations suggest that our theory is accurate

random networks with sublinear preferential attachment the giant component we study a dynamical random network model in which at every construction step a new vertex is introduced and attached to every existing vertex independently with a probability proportional to a concave function f of its current degree we give a criterion for the existence of a giant component which is both necessary and sufficient and which becomes explicit when f is linear otherwise it allows the derivation of explicit necessary and sufficient conditions which are often fairly close we give an explicit criterion to decide whether the giant component is robust under random removal of edges we also determine asymptotically the size of the giant component and the empirical distribution of component sizes in terms of the survival probability and size distribution of a multitype branching random walk associated with f

the evolutionary game of pressure or interference resistance and collaboration in this paper we extend the framework of the evolutionary inspection game put forward recently by the author and coworkers to a large class of conflict interactions to address the pressure executed by the major player or principal on the large group of small players who can resist this pressure or collaborate with the major player we prove rigorous results on the convergence of various markov decision models of interacting small agents including evolutionary growth i e pairwise in groups and by coalition formation to a deterministic evolution on the distributions of the state spaces of small players paying main attention to situations with an infinite state space of small players we supply precise rates of convergence the theoretical results of the paper are applied to the analysis of the processes of inspection corruption cyber security counter terrorism banks and firms merging strategically enhanced preferential attachment and many other

the colored jones polynomials and the simplicial volume of a knot we show that the set of colored jones polynomials and the set of generalized alexander polynomials defined by akutsu deguchi and ohtsuki intersect non trivially moreover it is shown that the intersection is at least includes the set of kashaev s quantum dilogarithm invariants for links therefore kashaev s conjecture can be restated as follows the colored jones polynomials determine the hyperbolic volume for a hyperbolic knot modifying this we propose a stronger conjecture the colored jones polynomials determine the simplicial volume for any knot if our conjecture is true then we can prove that a knot is trivial if and only if all of its vassiliev invariants are trivial in 13 r m kashaev defined a family of complex valued link invariants indexed by integers n 2 using the quantum dilogarithm later he calculated the asymptotic behavior of his invariant and observed that for the three simplest hyperbolic knots it grows as exp vol k n 2 when n goes to the infinity where vol k is the hyperbolic volume of the complement of a knotk 14 this amazing result and his conjecture that the same also holds for any hyperbolic knot have been almost ignored by mathematicians since his definition of the invariant is too complicated though it uses only elementary tools the aim of this paper is to reveal his mysterious definition and to show that his invariant is nothing but a specialization of the colored jones polynomial the colored jones polynomial is defined for colored links each component is decorated with an irreducible representation of the lie algebra sl 2 c the original jones polynomial corresponds to the case that all the colors are identical to the 2 dimensional fundamental representation we show that kashaev s invariant with parameter n coincides with the colored jones polynomial in a certain normalization with every color the n dimensional representation evaluated at the primitive n th root of unity we have to normalize the colored jones polynomial so that the value for the trivial knot is one for otherwise it always vanishes on the other hand there are other colored polynomial invariants the generalized multivariable alexander polynomial defined by y akutsu t deguchi and t ohtsuki 1 they used the same lie algebra sl 2 c but a different hierarchy of representations their invariants are parameterized by c 1 parameters an integer date february 1 2008 1991 mathematics subject classification 57m25 57m50 17b37 81r50

degree distribution rank size distribution and leadership persistence in mediation driven attachment networks abstract we investigate the growth of a class of networks in which a new node first picks a mediator at random and connects with m randomly chosen neighbors of the mediator at each time step we show that the degree distribution in such a mediation driven attachment mda network exhibits power law p k k m with a spectrum of exponents depending on m to appreciate the contrast between mda and barabasi albert ba networks we then discuss their rank size distribution to quantify how long a leader the node with the maximum degree persists in its leadership as the network evolves we investigate the leadership persistence probability f i e the probability that a leader retains its leadership up to time we find that it exhibits a power law f m with persistence exponent m 1 51 m in mda networks and m 1 53 exponentially with m in ba networks

evolutionary mixed games in structured populations cooperation and the benefits of heterogeneity evolutionary games on networks traditionally involve the same game at each interaction here we depart from this assumption by considering mixed games where the game played at each interaction is drawn uniformly at random from a set of two different games while in well mixed populations the random mixture of the two games is always equivalent to the average single game in structured populations this is not always the case we show that the outcome is in fact strongly dependent on the distance of separation of the two games in the parameter space effectively this distance introduces payoff heterogeneity and the average game is returned only if the heterogeneity is small for higher levels of heterogeneity the distance to the average game grows which often involves the promotion of cooperation the presented results support preceding research that highlights the favorable role of heterogeneity regardless of its origin and they also emphasize the importance of the population structure in amplifying facilitators of cooperation

greedy connectivity of geographically embedded graphs we introduce a measure of greedy connectivity for geographical networks graphs embedded in space and where the search for connecting paths relies only on local information such as a node s location and that of its neighbors constraints of this type are common in everyday life applications greedy connectivity accounts also for imperfect transmission across established links and is larger the higher the proportion of nodes that can be reached from other nodes with a high probability greedy connectivity can be used as a criterion for optimal network design

network growth by copying we introduce a growing network model in which a new node attaches to a randomly selected node as well as to all ancestors of the target node this mechanism produces a sparse ultrasmall network where the average node degree grows logarithmically with network size while the network diameter equals 2 we determine basic geometrical network properties such as the size dependence of the number of links and the in and out degree distributions we also compare our predictions with real networks where the node degree also grows slowly with time the internet and the citation network of all physical review papers

a mathematical model for generating bipartite graphs and its application to protein networks complex systems arise in many different contexts from large communication systems and transportation infrastructures to molecular biology most of these systems can be organized into networks composed of nodes and interacting edges here we present a theoretical model that constructs bipartite networks with the particular feature that the degree distribution can be tuned depending on the probability rate of fundamental processes we then use this model to investigate protein domain networks a protein can be composed of up to hundreds of domains each domain represents a conserved sequence segment with specific functional tasks we analyze the distribution of domains in homo sapiens and arabidopsis thaliana organisms and the statistical analysis shows that while a the number of domain types shared by k proteins exhibits a power law distribution b the number of proteins composed of k types of domains decays as an exponential distribution the proposed mathematical model generates bipartite graphs and predicts the emergence of this mixing of a power law and b exponential distributions our theoretical and computational results show that this model requires 1 growth process and 2 copy mechanism

cluster growth in two growing network models we analyse a simple model for growing tree networks and find that although it never percolates there is an anomalously large cluster at finite size we study the growth of both the maximal cluster and the cluster containing the original vertex and find that they obey power laws this property is also observed through simulations in a nonlinear model with loops and a true percolating phase

zero pearson coefficient for strongly correlated growing trees we obtained pearson s coefficient of strongly correlated recursive networks growing by preferential attachment of every new vertex by m edges we found that the pearson coefficient is exactly zero in the infinite network limit for the recursive trees m 1 if the number of connections of new vertices exceeds one mg1 then the pearson coefficient in the infinite networks equals zero only when the degree distribution exponent ensuremath gamma does not exceed 4 we calculated the pearson coefficient for finite networks and observed a slow power law like approach to an infinite network limit our findings indicate that pearson s coefficient strongly depends on size and details of networks which makes this characteristic virtually useless for quantitative comparison of different networks

filtering random noise from deterministic signals via data compression we present a novel technique for the design of filters for random noise leading to a class of filters called occam filters the essence of the technique is that when a lossy data compression algorithm is applied to a noisy signal with the allowed loss set equal to the noise strength the loss and the noise tend to cancel rather than add we give two illustrative applications of the technique to univariate signals we also prove asymptotic convergence bounds on the effectiveness of occam filters

network evolution by different rewiring schemes many real world networks such as social networks are characterized by rearrangements of the links between nodes rewiring indeed very few natural networks are static in time and it is therefore important to study the properties of networks in which rewiring occurs in this paper two different rewiring schemes are formulated and compared using a general ordinary differential equation ode model the equilibrium distributions are analytically derived it is found that by uniformly choosing a node and a link connected to it rewiring from different ends of the link yields different equilibrium degree distributions rewiring from the neighbor generally produces more high degree nodes the equilibrium distributions of the ode model are compared with simulation results of the corresponding stochastic process for rewiring conditions are discussed under which our ode provides a good approximation for the mean of the corresponding stochastic process

the transition towards immortality non linear autocatalytic growth of citations to scientific papers we discuss microscopic mechanisms of complex network growth with the spe cial emphasis of how these mechanisms can be evaluated from the measurements on real networks as an example we consider the network of citations to scientific papers contrary to common belief that its growth is determined by the linear preferential attachment our mi croscopic measurements show that it is driven by the nonlinear autocatalytic growth t his invalidates the scale free hypothesis for the citation network the nonlinearity is responsible for a dramatic dynamical phase transition while the citation lifetime of majority of papers is 6 10 years the highly cited papers have practically infinite lifetime

complex networks created by aggregation we study aggregation as a mechanism for the creation of complex networks in this evolution process vertices merge together which increases a number of highly connected hubs we study a range of complex network architectures produced by the aggregation fat tailed in particular scale free distributions of connections are obtained for both networks with a finite number of vertices and growing networks we observe a strong variation of a network structure with growing density of connections and find the phase transition of the condensation of edges finally we demonstrate the importance of structural correlations in these networks

hopf algebras with trace and representations we study the restriction of representations of cayley hamilton algebras to subalgebras this theory is applied to determine tensor products and branching rules for representations of quantum groups at roots of 1