on the descending central sequence of absolute galois groups let p be an odd prime number and f a field containing a primitive p th root of unity we prove a new restriction on the group theoretic structure of the absolute galois group g f of f namely the third subgroup g f 3 in the descending p central sequence of g f is the intersection of all open normal subgroups n such that g f n is 1 bbb z p 2 or the extra special group m p 3 of order p 3 and exponent p 2

weighted scale free networks induced by group preferential mechanism abstract based on the bbv model a barrat m barthelemy a vespignani phys rev lett 92 22 2004 we propose a weighted group preferential model which is generated by the group preferential mechanism we derive analytically the various statistical properties such as the distribution of degree strength and weight the degree strength relationship finally we provide a contrast with the bbv model on the synchronization robustness and fragility through numerical simulation

statistical analysis of airport network of china through the study of airport network of china anc composed of 128 airports nodes and 1165 flights edges we show the topological structure of anc conveys two characteristics of small worlds a short average path length 2 067 and a high degree of clustering 0 733 the cumulative degree distributions of both directed and undirected anc obey two regime power laws with different exponents i e the so called double pareto law in degrees and out degrees of each airport have positive correlations whereas the undirected degrees of adjacent airports have significant linear anticorrelations it is demonstrated both weekly and daily cumulative distributions of flight weights frequencies of anc have power law tails besides the weight of any given flight is proportional to the degrees of both airports at the two ends of that flight it is also shown the diameter of each subcluster consisting of an airport and all those airports to which it is linked is inversely proportional to its density of connectivity efficiency of anc and of its subclusters is measured through a simple definition in terms of that the efficiency of anc s subclusters increases as the density of connectivity does anc is found to have an efficiency of 0 484

design of on line algorithms using hitting times random walks are well known for playing a crucial role in the design of randomized off line as well as on line algorithms in this work we prove some basic identities for ergodic markov chains e g an interesting characterization of reversibility in markov chains is obtained in terms of first passage times besides providing new insight into random walks on weighted graphs we show how these identities give us a way of designing competitive randomized on line algorithms for certain well known problems

quotients of absolute galois groups which determine the entire galois cohomology for a prime power q pd and a field f containing a root of unity of order q we show that the galois cohomology ring h g f mathbb z q is determined by a quotient g f 3 of the absolute galois group gf related to its descending q central sequence conversely we show that g f 3 is determined by the lower cohomology of gf this is used to give new examples of pro p groups which do not occur as absolute galois groups of fields

weighted scale free networks with variable power law exponents abstract we present a weighted scale free network model in which the power law exponents can be controlled by the model parameters the network is generated through the weight driven preferential attachment of new nodes to existing nodes and the growth of the weights of existing links the simplicity of the model enables us to derive analytically the various statistical properties such as the distributions of degree strength and weight the degree strength and degree weight relationship and the dependencies of these power law exponents on the model parameters finally we demonstrate that networks of words coauthorship of researchers and collaboration of actor actresses are quantitatively well described by this model

unwinding the hairball graph pruning algorithms for weighted complex networks empirical networks of weighted dyadic relations often contain noisy edges that alter the global characteristics of the network and obfuscate the most important structures therein graph pruning is the process of identifying the most significant edges according to a generative null model and extracting the subgraph consisting of those edges here we focus on integer weighted graphs commonly arising when weights count the occurrences of an event relating the nodes we introduce a simple and intuitive null model related to the configuration model of network generation and derive two significance filters from it the marginal likelihood filter mlf and the global likelihood filter glf the former is a fast algorithm assigning a significance score to each edge based on the marginal distribution of edge weights whereas the latter is an ensemble approach which takes into account the correlations among edges we apply these filters to the network of air traffic volume between us airports and recover a geographically faithful representation of the graph furthermore compared with thresholding based on edge weight we show that our filters extract a larger and significantly sparser giant component

analysis of weighted networks the connections in many networks are not merely binary entities either present or not but have associated weights that record their strengths relative to one another recent studies of networks have by and large steered clear of such weighted networks which are often perceived as being harder to analyze than their unweighted counterparts here we point out that weighted networks can in many cases be analyzed using a simple mapping from a weighted network to an unweighted multigraph allowing us to apply standard techniques for unweighted graphs to weighted ones as well we give a number of examples of the method including an algorithm for detecting community structure in weighted networks and a simple proof of the maximum flow minimum cut theorem

multifractal analysis and topological properties of a new family of weighted koch networks abstract weighted complex networks especially scale free networks which characterize real life systems better than non weighted networks have attracted considerable interest in recent years studies on the multifractality of weighted complex networks are still to be undertaken in this paper inspired by the concepts of koch networks and koch island we propose a new family of weighted koch networks and investigate their multifractal behavior and topological properties we find some key topological properties of the new networks their vertex cumulative strength has a power law distribution there is a power law relationship between their topological degree and weight strength the networks have a high weighted clustering coefficient of 0 41004 which is independent of the scaling factor c in the limit of large generation t the second smallest eigenvalue 2 and the maximum eigenvalue n are approximated by quartic polynomials of the scaling factor c for the general laplacian operator while 2 is approximately a quartic polynomial of c and n 1 5 for the normalized laplacian operator then we find that weighted koch networks are both fractal and multifractal their fractal dimension is influenced by the scaling factor c we also apply these analyses to six real world networks and find that the multifractality in three of them are strong

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

synchronization in weighted uncorrelated complex networks in a noisy environment optimization and connections with transport efficiency motivated by synchronization problems in noisy environments we study the edwards wilkinson process on weighted uncorrelated scale free networks we consider a specific form of the weights where the strength and the associated cost of a link is proportional to kikj beta with ki and kj being the degrees of the nodes connected by the link subject to the constraint that the total edge cost is fixed we find that in the mean field approximation on uncorrelated scale free graphs synchronization is optimal at beta 1 numerical results based on exact numerical diagonalization of the corresponding network laplacian confirm the mean field results with small corrections to the optimal value of beta employing our recent connections between the edwards wilkinson process and resistor networks and some well known connections between random walks and resistor networks we also pursue a naturally related problem of optimizing performance in queue limited communication networks utilizing local weighted routing schemes

generating structured networks based on a weight dependent deactivation mechanism motivated by the degree dependent deactivation model generating networks with high clustering coefficient k klemm et al phys rev e 65 036123 2002 a weight dependent version is studied to model evolving networks the growth dynamics of the network is based on a naive weight driven deactivation mechanism which couples the establishment of new active vertices and the weights dynamical evolution both analytical solutions and numerical simulations show that the generated networks possess a high clustering coefficient larger than that for regular lattices of the same average connectivity weighted structured scale free networks are obtained as the deactivated vertex is target selected at each time step and weighted structured exponential networks are realized for the random selected case

cayley bacharach theorems and conjectures a theorem of pappus of alexandria proved in the fourth century a d began a long development in algebraic geometry in its changing expressions one can see reflected the changing concerns of the field from synthetic geometry to projective plane curves to riemann surfaces to the modern development of schemes and duality we survey this development historically and use it to motivate a brief treatment of a part of duality theory we then explain one of the modern developments arising from it a series of conjectures about the linear conditions imposed by a set of points in projective space on the forms that vanish on them we give a proof of the conjectures in a new

on the cohomology of galois groups determined by witt rings let f denote a field of characteristic different from two in this paper we describe the mod 2 cohomology of a galois group gf called the w group of f which is known to essentially characterize the witt ring wf of anisotropic quadratic modules over f we show that h gf f2 contains the mod 2 galois cohomology of f and that its structure will reflect important properties of the field we construct a space xf endowed with an action of an elementary abelian group e such that the computation of the cohomology of gf reduces to calculating the equivariant cohomology h e xf f2 for the case of a field which is not formally real this amounts to computing the cohomology of an explicit euclidean space form an object which is interesting in its own right we provide a number of examples and a substantial combinatorial computation for the cohomology of the universal w groups

world trade web topological properties dynamics and evolution this paper studies the statistical properties of the web of import export relationships among world countries using a weighted network approach we analyze how the distributions of the most important network statistics measuring connectivity assortativity clustering and centrality have coevolved over time we show that all node statistic distributions and their correlation structure have remained surprisingly stable in the last 20 years and are likely to do so in the future conversely the distribution of positive link weights is slowly moving from a log normal density towards a power law we also characterize the autoregressive properties of network statistics dynamics we find that network statistics growth rates are well proxied by fat tailed densities like the laplace or the asymmetric exponential power finally we find that all our results are reasonably robust to a few alternative economically meaningful weighting schemes

covering problems for brownian motion on spheres bounds are given on the mean time taken by a strong markov process to visit all of a finite collection of subsets of its state space these bounds are specialized to brownian motion on the surface of the unit sphere sigma p in r p this leads to bounds on the mean time taken by this brownian motion to come within a distance varepsilon of every point on the sphere and bounds on the mean time taken to come within varepsilon of every point or its opposite the second case is related to the grand tour a technique of multivariate data analysis that involves a search of low dimensional projections in both cases the bounds are asymptotically tight as varepsilon rightarrow 0 on sigma p for p geq 4

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

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

an empirical study of an agglomeration network recently researchers have reported many models mimicking real network evolution growth among which some are based on network aggregation growth however until now relatively few experiments have been reported accordingly in this paper photomicrographs of real materials the agglomeration in the filtrate of slurry formed by a gap nanoparticle conglomerate dispersed in water are analyzed within the framework of complex network theory by data mapping from photomicrographs we generate undirected networks and as a definition of degree we adopt the number of pixel s nearest neighbors while adjacent pixels define a connection or an edge we study the topological structure of these networks including degree distribution clustering coefficient and average path length in addition we discuss the self similarity and synchronizability of the networks we find that the synchronizability of high concentration agglomeration is better than that of low concentration agglomeration we also find that agglomeration networks possess good self similar features

improved approximation of the minimum cover time feige and rabinovich in feige and rabinovich rand struct algorithms 23 1 2003 1 22 gave a deterministic o log4n approximation for the time it takes a random walk to cover a given graph starting at a given vertex this approximation algorithm was shown to work for arbitrary reversible markov chains we build on the results of feige and rabinovich rand struct algorithms 23 1 2003 1 22 and show that the original algorithm gives a o log2n approximation as it is and that it can be modified to give a o log n log log n 2 approximation moreover we show that given any c n approximation algorithm for the maximum cover time maximized over all initial vertices of a reversible markov chain we can give a corresponding algorithm for the general cover time of a random walk or reversible markov chain with approximation ratio o c n log n

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

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

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

finding modules and hierarchy in weighted financial network using transfer entropy we study the modular structure of financial network based on the transfer entropy te from the comparison with the obtained modular structure using the cross correlation cc we find that te and cc both provide well organized modular structure and the hierarchical relationship between each industrial group when the time scale of the measurement is less than one month however when the time scale of the measurement becomes larger than one month we find that the modular structure from cc cannot correctly reflect the known industrial classification and their hierarchy in addition the measured maximum modularity qmax for te is always larger than that for cc which indicates that te is a better weight measure than cc for the system with asymmetric relationship

modeling the evolution of weighted networks we present a general model for the growth of weighted networks in which the structural growth is coupled with the edges weight dynamical evolution the model is based on a simple weight driven dynamics and a weights reinforcement mechanism coupled to the local network growth that coupling can be generalized in order to include the effect of additional randomness and nonlinearities which can be present in real world networks the model generates weighted graphs exhibiting the statistical properties observed in several real world systems in particular the model yields a nontrivial time evolution of vertices properties and scale free behavior with exponents depending on the microscopic parameters characterizing the coupling rules very interestingly the generated graphs spontaneously achieve a complex hierarchical architecture characterized by clustering and connectivity correlations varying as a function of the vertices degree

a highly clustered scale free network evolved by random walking abstract in present paper we propose a highly clustered weighted network model that incorporates the addition of a new node with some links new links between existing nodes and the edge s weight dynamical evolution based on weight dependent walks at each time step the analytical approach and numerical simulation show that the system grows into a weighted network with the power law distributions of strength weight and degree the weight dependent walk length l will not influence the strength distribution but the clustering coefficient of the network is sensitive to l particularly the clustering coefficient is especially high and almost independent of the network size when l 2

scaling of average receiving time and average weighted shortest path on weighted koch networks abstract in this paper we present weighted koch networks based on classic koch networks a new method is used to determine the average receiving time art whose key step is to write the sum of mean first passage times mfpts for all nodes to absorption at the trap located at a hub node as a recursive relation we show that the art exhibits a sublinear or linear dependence on network order thus the weighted koch networks are more efficient than classic koch networks in receiving information moreover average weighted shortest path awsp is calculated in the infinite network order limit the awsp depends on the scaling factor the weighted koch network grows unbounded but with the logarithm of the network size while the weighted shortest paths stay bounded

influence of weight heterogeneity on random walks in scale free networks many systems are best described by weighted networks in which the weights of the edges are heterogeneous in this paper we focus on random walks in weighted network investigating the impacts of weight heterogeneity on the behavior of random walks we study random walks in a family of weighted scale free tree like networks with power law weight distribution we concentrate on three cases of random walk problems with a trap located at a hub node a leaf adjacent to a hub node and a farthest leaf node from a hub for all these cases we calculate analytically the global mean first passage time gmfpt measuring the efficiency of random walk as well as the leading scaling of gmfpt we find a significant decrease in the dominating scaling of gmfpt compared with the corresponding binary networks in all three random walk problems which implies that weight heterogeneity has a significant influence on random walks in scale free networks

covering problems for markov chains on donne des bornes superieures et inferieures sur la fonction generatrice des moments du temps pris par une chaine de markov pour visiter au moins n des n sous ensembles selectionnes de son espace d etats

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

weight driven growing networks we study growing networks in which each link carries a certain weight randomly assigned at birth and fixed thereafter the weight of a node is defined as the sum of the weights of the links attached to the node and the network grows via the simplest weight driven rule a newly added node is connected to an already existing node with the probability which is proportional to the weight of that node we show that the node weight distribution n w has a universal tail that is it is independent of the link weight distribution n w approximately w 3 as w infinity results are particularly neat for the exponential link weight distribution when n w is algebraic over the entire weight range

random walks on activity driven networks with attractiveness virtually all real world networks are dynamical entities in social networks the propensity of nodes to engage in social interactions activity and their chances to be selected by active nodes attractiveness are heterogeneously distributed here we present a time varying network model where each node and the dynamical formation of ties are characterized by these two features we study how these properties affect random walk processes unfolding on the network when the time scales describing the process and the network evolution are comparable we derive analytical solutions for the stationary state and the mean first passage time of the process and we study cases informed by empirical observations of social networks our work shows that previously disregarded properties of real social systems such as heterogeneous distributions of activity and attractiveness as well as the correlations between them substantially affect the dynamical process unfolding on the network

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

field theory and the cohomology of some galois groups abstract we prove that two arithmetically significant extensions of a field f coincide if and only if the witt ring wf is a group ring z n g furthermore working modulo squares with galois groups which are 2 groups we establish a theorem analogous to hilbert s theorem 90 and show that an identity linking the cohomological dimension of the galois group of the quadratic closure of f the length of a filtration on a certain module over a galois group and the dimension over f 2 of the square class group of the field holds for a number of interesting families of fields finally we discuss the cohomology of a particular galois group in a topological context

laplacian and homology of free two step nilpotent lie algebras abstract in this paper we determine the homology with trivial coefficients of the free two step nilpotent lie algebras over the complex numbers this is done by working out the structure of the homology as a module under the general linear group the main tool is a laplacian for the free two step nilpotent lie algebras which turns out to be closely related to the casimir operator of the general linear group we are able to compute all eigenvalues of the laplacian on the chain complex

entropy of network ensembles in this paper we generalize the concept of random networks to describe network ensembles with nontrivial features by a statistical mechanics approach this framework is able to describe undirected and directed network ensembles as well as weighted network ensembles these networks might have nontrivial community structure or in the case of networks embedded in a given space they might have a link probability with a nontrivial dependence on the distance between the nodes these ensembles are characterized by their entropy which evaluates the cardinality of networks in the ensemble in particular in this paper we define and evaluate the structural entropy i e the entropy of the ensembles of undirected uncorrelated simple networks with given degree sequence we stress the apparent paradox that scale free degree distributions are characterized by having small structural entropy while they are so widely encountered in natural social and technological complex systems we propose a solution to the paradox by proving that scale free degree distributions are the most likely degree distribution with the corresponding value of the structural entropy finally the general framework we present in this paper is able to describe microcanonical ensembles of networks as well as canonical or hidden variable network ensembles with significant implications for the formulation of network constructing algorithms

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

asymmetry between odd and even node weight in complex networks abstract phenomenon of asymmetry between odd and even node weight distribution in some complex networks such as public transportation networks and internet is presented reasons related to this phenomenon are summarized i e 1 edges are solid connections between nodes which are embedded in real geometric background and 2 preferential mechanism during the process of network formation incorporating these two points complex networks are constructed by self avoiding random walk on 2d grid and the simulation results are consistent well with that of empirical researches

random pseudofractal networks with competition abstract in this paper we present a simple rule which assigns fitness to each edge to generate random pseudofractal networks rpns this rpn model is both scale free and small world we obtain the theoretical results that the power law exponent is 2 1 1 for the tunable parameter 1 and that the degree distribution is of an exponential form for others analytical results also show that an rpn has a large clustering coefficient and can process hierarchical structure as c k k 1 that is in accordance with many real networks and we prove that the mean distance l n scales slower logarithmically with network size n in particular we explain the effect of nodes with degree 2 on the clustering coefficient these results agree with numerical simulations very well

self organization versus hierarchy in open source social networks complex networks emerge under different conditions including design i e top down decisions through simple rules of growth and evolution such rules are typically local when dealing with biological systems and most social webs an important deviation from such a scenario is provided by groups collectives of agents engaged in technology development such as open source communities here we analyze their network structure showing that it defines a complex weighted network with scaling laws at different levels as measured by looking at e mail exchanges we also present a simple model of network growth involving nonlocal rules based on betweenness centrality our weighted network analysis suggests that a well defined interplay between the overall goals of the community and the underlying hierarchical organization play a key role in shaping its dynamics

weighted scale free networks with stochastic weight assignments we propose a model of weighted scale free networks incorporating a stochastic scheme for weight assignments to the links taking into account both the popularity and fitness of a node as the network grows the weights of links are driven either by the connectivity with probability p or by the fitness with probability 1 ensuremath p numerical results show that the total weight exhibits a power law distribution with an exponent ensuremath sigma that depends on the probability p the exponent ensuremath sigma decreases continuously as p increases for p 0 the scaling behavior is the same as that of the connectivity distribution an analytical expression for the total weight is derived so as to explain the features observed in the numerical results numerical results are also presented for a generalized model with a fitness dependent link formation mechanism

comparison of different generalizations of clustering coefficient and local efficiency for weighted undirected graphs binary undirected graphs are well established but when these graphs are constructed often a threshold is applied to a parameter describing the connection between two nodes therefore the use of weighted graphs is more appropriate in this work we focus on weighted undirected graphs this implies that we have to incorporate edge weights in the graph measures which require generalizations of common graph metrics after reviewing existing generalizations of the clustering coefficient and the local efficiency we proposed new generalizations for these graph measures to be able to compare different generalizations a number of essential and useful properties were defined that ideally should be satisfied we applied the generalizations to two real world networks of different sizes as a result we found that not all existing generalizations satisfy all essential properties furthermore we determined the best generalization for the clustering coefficient and local efficiency based on their properties and the performance when applied to two networks we found that the best generalization of the clustering coefficient is defined in miyajima and sakuragawa 2014 while the best generalization of the local efficiency is proposed in this letter depending on the application and the relative importance of sensitivity and robustness to noise other generalizations may be selected on the basis of the properties investigated in this letter

mutual selection model for weighted networks for most networks the connection between two nodes is the result of their mutual affinity and attachment in this paper we propose a mutual selection model to characterize the weighted networks by introducing a general mechanism of mutual selection the model can produce power law distributions of degree weight and strength as confirmed in many real networks moreover we also obtained the nontrivial clustering coefficient c degree assortativity coefficient r and degree strength correlation depending on a single parameter m these results are supported by present empirical evidence studying the degree dependent average clustering coefficient c k and the degree dependent average nearest neighbors degree k nn k also provide us with a better description of the hierarchies and organizational architecture of weighted networks

optimal paths in complex networks with correlated weights the worldwide airport network we study complex networks with weights wij associated with each link connecting node i and j the weights are chosen to be correlated with the network topology in the form found in two real world examples a the worldwide airport network and b the e coli metabolic network here wijxijkikj where ki and kj are the degrees of nodes i and j xij is a random number and represents the strength of the correlations the case 0 represents correlation between weights and degree while 0 represents anticorrelation and the case 0 reduces to the case of no correlations we study the scaling of the lengths of the optimal paths opt with the system size n in strong disorder for scale free networks for different we find two different universality classes for opt in strong disorder depending on i if 0 then for 2 the scaling law optn 1 3 where is the power law exponent of the degree distribution of scale free networks and ii if 0 then opt nopt with opt identical to its value for the uncorrelated case 0 we calculate the robustness of correlated scale free networks with different and find the networks with 0 to be the most robust networks when compared to the other values of we propose an analytical method to study percolation phenomena on networks with this kind of correlation and our numerical results suggest that for scale free networks with 0 the percolation threshold pc is finite for 3 which belongs to the same universality class as 0 we compare our simulation results with the real worldwide airport network and we find good agreement

degree based attacks and defense strategies in complex networks we study the stability of random scale free networks to degree dependent attacks we present analytical and numerical results to compute the critical fraction p c of nodes that need to be removed for destroying the network under this attack for different attack parameters we study the effect of different defense strategies based on the addition of a constant number of links on network robustness we test defense strategies based on adding links to either low degree middegree or high degree nodes we find using analytical results and simulations that the middegree nodes defense strategy leads to the largest improvement to the network robustness against degree based attacks we also test these defense strategies on an internet autonomous systems map and obtain similar results

mean field diffusive dynamics on weighted networks diffusion is a key element of a large set of phenomena occurring on natural and social systems modeled in terms of complex weighted networks here we introduce a general formalism that allows to easily write down mean field equations for any diffusive dynamics on weighted networks we also propose the concept of annealed weighted networks in which such equations become exact we show the validity of our approach addressing the problem of the random walk process pointing out a strong departure of the behavior observed in quenched real scale free networks from the mean field predictions additionally we show how to employ our formalism for more complex dynamics our work sheds light on mean field theory on weighted networks and on its range of validity and warns about the reliability of mean field results for complex dynamics

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

primitives and central detection numbers in group cohomology abstract fix a prime p given a finite group g let h g denote its mod p cohomology in the early 1990s henn lannes and schwartz introduced two invariants d 0 g and d 1 g of h g viewed as a module over the mod p steenrod algebra they showed that in a precise sense h g is respectively detected and determined by h d c g v for d d 0 g and d d 1 g with v running through the elementary abelian p subgroups of g the main goal of this paper is to study how to calculate these invariants we find that a critical role is played by the image of the restriction of h g to h c where c is the maximal central elementary abelian p subgroup of g a measure of this is the top degree e g of the finite dimensional hopf algebra h c h g f p a number that tends to be quite easy to calculate our results are complete when g has a p sylow subgroup p in which every element of order p is central using the benson carlson duality we show that in this case d 0 g d 0 p e p and a similar exact formula holds for d 1 as a bonus we learn that h e g p contains nontrivial essential cohomology reproving and sharpening a theorem of adem and karagueuzian in general we are able to show that d 0 g max e c g v v g if certain cases of benson s regularity conjecture hold in particular this inequality holds for all groups such that the difference between the p rank of g and the depth of h g is at most 2 when we look at examples with p 2 we learn that d 0 g 14 for all groups with 2 sylow subgroup of order up to 64 with equality realized when g su 3 4 en route we study two objects of independent interest if c is any central elementary abelian p subgroup of g then h g is an h c comodule and we prove that the subalgebra of h c primitives is always noetherian of krull dimension equal to the p rank of g minus the p rank of c if the depth of h g equals the rank of z g we show that the depth essential cohomology of g is nonzero reproving and extending a theorem of green and cohen macauley in a certain sense and prove related structural results