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

nonlinear growth in weighted networks with neighborhood preferential attachment abstract we propose a nonlinear growing model for weighted networks with two significant characteristics i the new weights triggered by new edges at each time step grow nonlinearly with time and ii a neighborhood local world exists for local preferential attachment which is defined as one selected node and its neighbors global strength driven and local weight driven preferential attachment mechanisms are involved in our model we study the evolution process through both mathematical analysis and numerical simulation and find that the model exhibits a wide range power law distribution for node degree strength and weight in particular a nonlinear degree strength relationship is obtained this nonlinearity implies that accelerating growth of new weights plays a nontrivial role compared with accelerating growth of edges because of the specific local world model a small world property emerges and a significant hierarchical organization independent of the parameters is observed

regular tessellations of surfaces and p q 2 triangle groups the purpose of this paper is a study of the existence and classification of certain regular tessellations of surfaces and their relationship with the schwarz triangle groups of type p q 2 these tessellations are the natural generalizations to arbitrary surfaces of the spherical tessellations corresponding to the five platonic solids a regular tessellation of type p q a p q pattern for short on a surface m is a tessellation of m by p sided faces such that the valence of each vertex is q a more precise definition appears in 1 no global symmetry is assumed a p q pattern on m is said to be geometric if m admits a riemannian metric of constant curvature with respect to which the edges of the pattern are geodesic arcs of equal length and the interior angles at all vertices in all faces are 2 7 q if the curvature happens to he zero then we further insist that the area of each face be 1 it is natural to call two p q patterns on m

the expectation hypothesis of interest rates and network theory the case of brazil abstract this paper investigates the topological properties of the brazilian term structure of interest rates network we build the minimum spanning tree mst which is based on the concept of ultrametricity using the correlation matrix for interest rates of different maturities we show that the short term interest rate is the most important within the interest rates network which is in line with the expectation hypothesis of interest rates furthermore we find that the brazilian interest rates network forms clusters by maturity

accelerated growth in outgoing links in evolving networks deterministic versus stochastic picture in several real world networks such as the internet world wide web etc the number of links grow in time in a nonlinear fashion we consider growing networks in which the number of outgoing links is a nonlinear function of time but new links between older nodes are forbidden the attachments are made using a preferential attachment scheme in the deterministic picture the number of outgoing links m t at any time t is taken as n t ensuremath theta where n t is the number of nodes present at that time the continuum theory predicts a power law decay of the degree distribution p k ensuremath propto k ensuremath 1 ensuremath 2 1 ensuremath ensuremath theta while the degree of the node introduced at time t i is given by k t i t t i ensuremath theta phantom rule 0 2em 0ex t t i 1 ensuremath theta 2 when the network is evolved till time t numerical results show a growth in the degree distribution for small k values at any nonzero ensuremath theta in the stochastic picture m t is a random variable as long as m t is independent of time the network shows a behavior similar to the barab asi albert ba model different results are obtained when m t is time dependent e g when m t follows a distribution p m ensuremath propto m ensuremath ensuremath lambda the behavior of p k changes significantly as ensuremath lambda is varied for ensuremath lambda g3 the network has a scale free distribution belonging to the ba class as predicted by the mean field theory for smaller values of ensuremath lambda it shows different behavior characteristic features of the clustering coefficients in both models have also been discussed

scaling properties of scale free evolving networks continuous approach the scaling behavior of scale free evolving networks arising in areas such as communications scientific citations collaborations etc is studied we derive universal scaling relations describing properties of such networks and indicate the limits of their validity we show that the main properties of scale free evolving networks may be described in the framework of a simple continuous approach the simplest models of networks growing according to a mechanism of preferential attachment of links to nodes are used we consider different forms of this preference and demonstrate that the range of preferential attachments producing scale free networks is wide we also obtain scaling relations for networks with nonlinear accelerating growth and describe the temporal evolution of the arising distributions size effects char22 the cutoffs of these distributions char22 introduce restrictions for the observation of power law dependences mainly we discuss the so called degree distribution i e the distribution of the number of connections of nodes a scaling form of the distribution of links between pairs of individual nodes for a growing network of citations is also studied we describe the effects of differences between nodes the aging of nodes changes the exponents of the distributions the appearance of a single node with high fitness changes the degree distribution of a network dramatically if its fitness exceeds some threshold value this node captures a finite part of all links of the network we show that permanent random damage to a growing scale free network char22 a permanent deletion of some links char22 radically changes the values of the scaling exponents results of other kinds of permanent damage are described

modeling the coevolution of topology and traffic on weighted technological networks for many technological networks the network structures and the traffic taking place on them mutually interact the demands of traffic increment spur the evolution and growth of the networks to maintain their normal and efficient functioning in parallel a change of the network structure leads to redistribution of the traffic in this paper we perform an extensive numerical and analytical study extending results of wang phys rev lett 94 188702 2005 by introducing a general strength coupling interaction driven by the traffic increment between any pair of vertices our model generates networks of scale free distributions of strength weight and degree in particular the obtained nonlinear correlation between vertex strength and degree and the disassortative property demonstrate that the model is capable of characterizing weighted technological networks moreover the generated graphs possess both dense clustering structures and an anticorrelation between vertex clustering and degree which are widely observed in real world networks the corresponding theoretical predictions are well consistent with simulation results

synchronizability in complex ad hoc dynamical networks with accelerated growth recent research works have been pursued in connection with network synchronizability under various constraints for different topological structures and evolving mechanisms however the fundamental question of how the synchronizability of the networks relates to accelerated growth and ad hoc property in the evolving processes remains underexplored here we study the ad hoc dynamical models with accelerated growth where the total number of edges increases faster than linearly with network size by adopting three attachment mechanisms random rewired and preferential attachment we investigate the second largest eigenvalues and the ratios of the extremal eigenvalues of coupled matrices in the accelerated models with different evolving parameters for the new models we demonstrate the robustness and fragility of synchronization against random and specific attacks by numerical simulations we find that accelerated growth represents a convenient tool for improving the synchronizability of an evolving network furthermore we show that not only network synchronization but also topological periodicity has robustness and fragility against random failures and specifical removal of the nodes respectively in particular when ad hoc property is suggested in the evolving networks we find that the deletion of nodes is easier to change network synchronizability and robustness compared to the addition of nodes

master equation analysis of accelerating networks in many real world networks the rates of node and link addition are time dependent this observation motivates the definition of accelerating networks there has been relatively little investigation of accelerating networks and previous efforts at analyzing their degree distributions have employed mean field techniques by contrast we show that it is possible to apply a master equation approach to such network development we provide full time dependent expressions for the evolution of the degree distributions for the canonical situations of random and preferential attachment in networks undergoing constant acceleration these results are in excellent agreement with results obtained from simulations we note that a growing nonequilibrium network undergoing constant acceleration with random attachment is equivalent to a classical random graph bridging the gap between nonequilibrium and classical equilibrium networks

hyperbolic geometry of complex networks we develop a geometric framework to study the structure and function of complex networks we assume that hyperbolic geometry underlies these networks and we show that with this assumption heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry conversely we show that if a network has some metric structure and if the network degree distribution is heterogeneous then the network has an effective hyperbolic geometry underneath we then establish a mapping between our geometric framework and statistical mechanics of complex networks this mapping interprets edges in a network as noninteracting fermions whose energies are hyperbolic distances between nodes while the auxiliary fields coupled to edges are linear functions of these energies or distances the geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures finally we show that targeted transport processes without global topology knowledge made possible by our geometric framework are maximally efficient according to all efficiency measures in networks with strongest heterogeneity and clustering and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure

fuchsian groups and algebraic number fields given the signature of a finitely generated fuchsian group we find the minimal extension of the rationals for which there is a fuchsian group having the required signature whose matrix entries lie in this field

finiteness and fluctuations in growing networks we study the role of finiteness and fluctuations about average quantities for basic structural properties of growing networks we first determine the exact degree distribution of finite networks by generating function approaches the resulting distributions exhibit an unusual finite size scaling behaviour and they are also sensitive to the initial conditions we argue that fluctuations in the number of nodes of degree k become gaussian for fixed degree as the size of the network diverges we also characterize the fluctuations between different realizations of the network in terms of higher moments of the degree distribution

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

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

multiple solutions of h systems and rellich s conjecture soit x y r 2 x 2 y 2 1 on considere une fonction u r 3 satisfaisant le systeme h u 2hu x u y sur avec l une des conditions suivantes soit dirichlet u sur soit plateau u x 2 u y 2 u x u y 0 sur u et u non decroissante sur

local world evolving networks with tunable clustering abstract we propose an extended local world evolving network model including a triad formation tf step in the process of network evolution random fluctuation in the number of new edges is involved we derive analytical expressions for degree distribution clustering coefficient and average path length our model can unify the generic properties of real life networks scale free degree distribution high clustering and small inter node separation moreover in our model the clustering coefficient is tunable simply by changing the expected number of tf steps after a single local preferential attachment step

coauthorship networks and patterns of scientific collaboration by using data from three bibliographic databases in biology physics and mathematics respectively networks are constructed in which the nodes are scientists and two scientists are connected if they have coauthored a paper we use these networks to answer a broad variety of questions about collaboration patterns such as the numbers of papers authors write how many people they write them with what the typical distance between scientists is through the network and how patterns of collaboration vary between subjects and over time we also summarize a number of recent results by other authors on coauthorship patterns

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

exactly solvable random graph ensemble with extensively many short cycles we introduce and analyse ensembles of 2 regular random graphs with a tuneable distribution of short cycles the phenomenology of these graphs depends critically on the scaling of the ensembles control parameters relative to the number of nodes a phase diagram is presented showing a second order phase transition from a connected to a disconnected phase we study both the canonical formulation where the size is large but fixed and the grand canonical formulation where the size is sampled from a discrete distribution and show their equivalence in the thermodynamical limit we also compute analytically the spectral density which consists of a discrete set of isolated eigenvalues representing short cycles and a continuous part representing cycles of diverging size

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

the effect of interdependence on the percolation of interdependent networks abstract two stochastic models are proposed to generate a system composed of two interdependent scale free sf or erd s renyi er networks where interdependent nodes are connected with an exponential or power law relation as well as different dependence strength respectively each subnetwork grows through the addition of new nodes with constant accelerating random attachment in the first model but with preferential attachment in the second model the two subnetworks interact with multi support and undirectional dependence links the effects of dependence relations and strength between subnetworks are analyzed in the percolation behavior of fully interdependent networks against random failure both theoretically and numerically and as a result for both relations interdependent sf networks show a second order percolation phase transition and the increased dependence strength decreases the robustness of the system whereas interdependent er networks show the opposite results in addition the power law relation between networks yields greater robustness than the exponential one at the given dependence strength

effects of accelerating growth on the evolution of weighted complex networks many real systems possess accelerating statistics where the total number of edges grows faster than the network size in this paper we propose a simple weighted network model with accelerating growth we derive analytical expressions for the evolutions and distributions for strength degree and weight which are relevant to accelerating growth we also find that accelerating growth determines the clustering coefficient of the networks interestingly the distributions for strength degree and weight display a transition from scale free to exponential form when the parameter with respect to accelerating growth increases from a small to large value all the theoretical predictions are successfully contrasted with numerical simulations

realizability of branched coverings of surfaces absrract a branched covering m n of degree d between closed surfaces determines a collection 6d of partitions of d its branch data corresponding to the set of branch points the collection of partitions must satisfy certain obvious conditions implied by the riemann hurwitz formula this paper investigates the extent to which any such finite collection 6d of partitions of d can be realized as the branch data of a suitable branched covering if n is not the 2 sphere such data can always be realized if 6d contains sufficiently many elements compared to d then it can be realized and whenever d is nonprime examples are constructed of nonrealizable data

scale free property of complex ad hoc networks with accelerated growth a complex ad hoc network model with accelerated growth is proposed in this paper in the evolving process of the model at each time step with the addition probability c1c1 a new vertex with m t m t edges is added into the model which new adding edges are connected with the old vertices according to the preferential attachment rule and with the deletion probability c2 c1 c2 c1 a random chosen vertex is deleted and its associated edges are disappeared simultaneously the accelerated growth means that at each time step the number of new adding edges is the increasing function of the time tt we consider mainly m t t m t t where 0 10 1 to analyze the degree distribution of the model using the continuous approach we prove theoretically and numerically that the model follows the scale free degree distribution and generates both the barabasi and albert ba model and the model introduced by sarshar and roychowdhury sr in addition we present the appropriate numerical range of c1c1 c2c2 and according to theoretical analyzes and numerical simulation the model takes on a certain universality to complex evolving network and has the guiding significance to the application of the peer to peer networks and complex ad hoc networks

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

h bubbles in a perturbative setting the finite dimensional reduction s method given a regular function h colon mathbb r 3 to mathbb r we look for h bubbles that is regular surfaces in mathbb r 3 parametrized on the sphere mathbb s 2 with mean curvature h at every point here we study the case of h u h 0 varepsilon h 1 u h varepsilon u where h 0 is a nonzero constant varepsilon is the smallness parameter and h 1 is any c 2 function we prove that if bar p in mathbb r 3 is a good stationary point for the melnikov type function gamma p int q p h 0 1 h 1 q dq then for varepsilon small there exists an h varepsilon bubble omega varepsilon that converges to a sphere of radius h 0 1 centered at bar p as varepsilon to 0

acceleratingly growing scale free networks with tunable degree exponents abstract in this paper we analytically study the probabilistic accelerating network m j gagen j s mattick phys rev e 72 2005 016123 in its accelerating regimes by using mean field theory in the growing network the number of links added with each new node is a nonlinearly increasing function a n t where n t is the number of nodes present at time t it is found that the network appears to have a power law degree distribution for large degree with tunable degree exponents ranging from 3 0 to theoretically infinity and the degree exponent depends only on the parameter as 1 2 1 the analytical results are found to be in good agreement with those obtained by extensive numerical simulations

link space formalism for network analysis we introduce the link space formalism for analyzing network models with degree degree correlations the formalism is based on a statistical description of the fraction of links l i j connecting nodes of degrees i and j to demonstrate its use we apply the framework to some pedagogical network models namely random attachment barab asi albert preferential attachment and the classical erd ifmmode mbox h o else h o fi s and r enyi random graph for these three models the link space matrix can be solved analytically we apply the formalism to a simple one parameter growing network model whose numerical solution exemplifies the effect of degree degree correlations for the resulting degree distribution we also employ the formalism to derive the degree distributions of two very simple network decay models more specifically that of random link deletion and random node deletion the formalism allows detailed analysis of the correlations within networks and we also employ it to derive the form of a perfectly nonassortative network for arbitrary degree distribution

power law accelerating growth complex networks with mixed attachment mechanisms abstract in this paper motivated by the thoughts and methods of the mixture of preferential and uniform attachments we extend the barabasi albert ba model and establish a network model with the power law accelerating growth and the mixture of the two attachment mechanisms in our model the number of edges generated by each newly introduced node is proportional to the power of 0 1 of time t i e m t by virtue of the continuum approach we have deduced the degree distribution p k t of our model with the extended power law form p k t a t k b t when the number of edges k generated by each new node is much greater than the value of b t the degree distribution p k t will converge to the power law form p k t a t k when k is much less than the value of b t the degree distribution p k t will converge to the exponential law form p k t a t b t e k b t by virtue of numerical simulations we also discuss the dependence of the degree distribution p k t on the model s parameters where t is considered as a constant in the simulations finally we investigate the possible application of our model in the spreading and evolution of epidemics in some real world systems

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

accelerating hyperaccelerating and decelerating networks many growing networks possess accelerating statistics where the number of links added with each new node is an increasing function of network size so the total number of links increases faster than linearly with network size in particular biological networks can display a quadratic growth in regulator number with genome size even while remaining sparsely connected these features are mutually incompatible in standard treatments of network theory which typically require that every new network node possesses at least one connection to model sparsely connected networks we generalize existing approaches and add each new node with a probabilistic number of links to generate either accelerating hyperaccelerating or even decelerating network statistics in different regimes under preferential attachment for example slowly accelerating networks display stationary scale free statistics relatively independent of network size while more rapidly accelerating networks display a transition from scale free to exponential statistics with network growth such transitions explain for instance the evolutionary record of single celled organisms which display strict size and complexity limits

topology of growing networks accelerated by intermediary process abstract we examined the effect of accelerated growth increase in the mean vertex degree caused by internal links between pre existing nodes on the topology of growing networks and showed that a kind of local rule edge additions stimulated by randomly chosen intermediaries intermediary process not only guarantees a highly clustered structure but also induces the scale free property on growing networks without an explicit preferential rule with the aid of accelerating growth we relate the power law exponent describing the degree distribution to the rate of acceleration and an internal factor which expresses the contribution of internal links to the rate of increase in degrees the behavior of the internal factor generates different forms of time dependence of degrees according to whether 2 or 2 accelerating growth plays a dominant role in the structural formation of networks based on local rules in contrast to cases when non local rules are applied because similar studies showed that a simple preferential linking rule a non local rule easily induces power laws without accelerating growth

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

dynamic weight evolution network with preferential attachment abstract a dynamic weight evolution network with preferential attachment is introduced the network includes two significant characteristics i topological growth triggered by newly added node with m links at each time step each new edge carries an initial weight growing nonlinearly with time ii weight dynamics the weight between two existing nodes experiences increasing or decreasing in a nonlinear way by using continuum theory and mean field method we study the strength the degree the weight and their distributions we find that the distributions exhibit a power law feature in particular the relationship between the degree and the strength is nonlinear and the power law exponents of the three are the same all the theoretical predictions are successfully contrasted with numerical simulations

exact solution for the time evolution of network rewiring models we consider the rewiring of a bipartite graph using a mixture of random and preferential attachment the full mean field equations for the degree distribution and its generating function are given the exact solution of these equations for all finite parameter values at any time is found in terms of standard functions it is demonstrated that these solutions are an excellent fit to numerical simulations of the model we discuss the relationship between our model and several others in the literature including examples of urn backgammon and balls in boxes models the watts and strogatz rewiring problem and some models of zero range processes our model is also equivalent to those used in various applications including cultural transmission family name and gene frequencies glasses and wealth distributions finally some voter models and an example of a minority game also show features described by our model

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

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

generalized local world models for weighted networks based on the weighted scale free network model proposed by barrat barth elemy and vespignani phys rev lett 92 228701 2004 and enlightened by our local world concept li and chen physica a 328 274 2003 we propose two generalized local world glw models for weighted complex networks theoretical analysis and numerical simulations show that the glw models generate weighted networks as a crossover between exponential and scale free weighted networks and exhibit an alteration from assortative networks to disassortative networks

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

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

self organization of collaboration networks we study collaboration networks in terms of evolving self organizing bipartite graph models we propose a model of a growing network which combines preferential edge attachment with the bipartite structure generic for collaboration networks the model depends exclusively on basic properties of the network such as the total number of collaborators and acts of collaboration the mean size of collaborations etc the simplest model defined within this framework already allows us to describe many of the main topological characteristics degree distribution clustering coefficient etc of one mode projections of several real collaboration networks without parameter fitting we explain the observed dependence of the local clustering on degree and the degree degree correlations in terms of the aging of collaborators and their physical impossibility to participate in an unlimited number of collaborations

the topology of an accelerated growth network we present and investigate a general nonlinear growth network model which incorporates accelerated growth of nodes and edges where the growth rates of edges and nodes are all time dependent power law functions the acceleration of edges determines the proportion of the internal edges to the external edges which play a key role influencing the structure of the network on the other hand the effects of the acceleration of nodes on the topology of the network are discussed in the present work this model predicts an observable two regime scale free degree distribution where the scaling exponents are 1 2 and 2 3 respectively the crossover point kcross of the degree distribution is adjusted by the growth rates of nodes and edges the nontrivial clustering coefficient and degree assortativity coefficient are relevant to the acceleration of nodes and edges

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

clustering coefficients of growing networks abstract in this paper we develop a general analytical method to compute clustering coefficients of growing networks this method can be applied to any network as long as we can construct and solve the dynamic equation for the degree of any node we also verify the accuracy of the method through simulation

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

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

duplication divergence model of protein interaction network we investigate a very simple model describing the evolution of protein protein interaction networks via duplication and divergence the model exhibits a remarkably rich behavior depending on a single parameter the probability to retain a duplicated link during divergence when this parameter is large the network growth is not self averaging and an average node degree increases algebraically the lack of self averaging results in a great diversity of networks grown out of the same initial condition when less than a half of links are on average preserved after divergence the growth is self averaging the average degree increases very slowly or tends to a constant and a degree distribution has a power law tail the predicted degree distributions are in a very good agreement with the distributions observed in real protein networks

scale free and stable structures in complex ad hoc networks unlike the well studied models of growing networks where the dominant dynamics consist of insertions of new nodes and connections and rewiring of existing links we study ad hoc networks where one also has to contend with rapid and random deletions of existing nodes and hence the associated links we first show that dynamics based only on the well known preferential attachments of new nodes do not lead to a sufficiently heavy tailed degree distribution in ad hoc networks in particular the magnitude of the power law exponent increases rapidly from 3 with the deletion rate becoming ensuremath infty in the limit of equal insertion and deletion rates we then introduce a local and universal compensatory rewiring dynamic and show that even in the limit of equal insertion and deletion rates true scale free structures emerge where the degree distributions obey a power law with a tunable exponent which can be made arbitrarily close to 2 the dynamics reported in this paper can be used to craft protocols for designing highly dynamic peer to peer networks and also to account for the power law exponents observed in existing popular services

degree correlations in growing networks with deletion of nodes in this paper we study the degree distribution and the two node degree correlations in growing networks generated via a general linear preferential attachment of new nodes together with a uniformly random deletion of nodes by using a continuum approach we show that under some suitable combinations of parameters deletion rate and node attractiveness the degree distribution not only loses its scale free character but can even be supported on a small range of degrees moreover we obtain new results on two vertex degree correlations showing that for degree distributions with finite variance such correlations can change under a nonselective removal of nodes

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

exploring the random genesis of co occurrence networks using the network random generation models from gustedt 2009 we simulate and analyze several characteristics such as the number of components the degree sequence and the clustering coefficient of the generated networks this is done for a variety of distributions fixed value bernoulli poisson binomial that are used to control the parameters of generation process these parameters are in particular the size of newly appearing sets of objects the number of contexts in which new elements appear initially the number of objects that are shared with parent contexts and the time period inside which a context may serve as a parent context aging the results show that these models allow to fine tune the generation process such that the graphs adopt properties as they can be found in real world graphs

weighted assortative and disassortative networks model abstract real world networks process structured connections since they have non trivial vertex degree correlation and clustering here we propose a toy model of structure formation in real world weighted network in our model a network evolves by topological growth as well as by weight change in addition we introduce the weighted assortativity coefficient which generalizes the assortativity coefficient of a topological network to measure the tendency of having a high weighted link between two vertices of similar degrees network generated by our model exhibits scale free behavior with a tunable exponent besides a few non trivial features found in real world networks are reproduced by varying the parameter ruling the speed of weight evolution most importantly by studying the weighted assortativity coefficient we found that both topologically assortative and disassortative networks generated by our model are in fact weighted assortative

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

emergence of network structure in models of collective evolution and evolutionary dynamics we consider an evolving network of a fixed number of nodes the allocation of edges is a dynamical stochastic process inspired by biological reproduction dynamics namely by deleting and duplicating existing nodes and their edges the properties of the degree distribution in the stationary state is analysed by use of the fokker planck equation for a broad range of parameters exponential degree distributions are observed the mechanism responsible for this behaviour is illuminated by use of a simple mean field equation and reproduced by the fokker planck equation the latter is treated exactly except for an approximate treatment of the degree degree correlations in the limit of 0 mutations the degree distribution becomes a power law with exponent 1

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

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