uncorrelated random networks we define a statistical ensemble of nondegenerate graphs i e graphs without multiple connections and self connections between nodes the node degree distribution is arbitrary but the nodes are assumed to be uncorrelated this completes our earlier publication phys rev 64 046118 2001 where trees and degenerate graphs were considered an efficient algorithm generating nondegenerate graphs is constructed the corresponding computer code is available on request finite size effects in scale free graphs i e those where the tail of the degree distribution falls like n beta are carefully studied we find that in the absence of dynamical internode correlations the degree distribution is cut at a degree value scaling like n gamma with gamma min 1 2 1 beta 1 where n is the total number of nodes the consequence is that independently of any specific model the internode correlations seem to be a necessary ingredient of the physics of scale free networks observed in nature

spectra of complex networks we propose a general approach to the description of spectra of complex networks for the spectra of networks with uncorrelated vertices and a local treelike structure exact equations are derived these equations are generalized to the case of networks with correlations between neighboring vertices the tail of the density of eigenvalues ensuremath rho ensuremath lambda at large ensuremath lambda is related to the behavior of the vertex degree distribution p k at large k in particular as p k ensuremath sim k ensuremath ensuremath gamma ensuremath rho ensuremath lambda ensuremath sim ensuremath lambda 1 ensuremath 2 ensuremath gamma we propose a simple approximation which enables us to calculate spectra of various graphs analytically we analyze spectra of various complex networks and discuss the role of vertices of low degree we show that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real world networks e g of the internet

fluctuation dissipation relations in complex networks in this paper we study fluctuations over several ensembles of maximum entropy random networks we derive several fluctuation dissipation relations characterizing the susceptibilities of different networks to changes in external fields in the case of networks with a given degree sequence we argue that the scale free topologies of real world networks may arise as a result of the self organization of real systems into sparse structures with low susceptibility to random external disruptions we also show that the ensembles of networks with a given degree sequence and networks characterized by two point correlations are equivalent to random networks with hidden variables

statistical mechanical load balancer for the web the maximum entropy principle from statistical mechanics states that a closed system attains an equilibrium distribution that maximizes its entropy we first show that for graphs with fixed number of edges one can define a stochastic edge dynamic that can serve as an effective thermalization scheme and hence the underlying graphs are expected to attain their maximum entropy states which turn out to be erdos renyi er random graphs we next show that i a rate equation based analysis of node degree distribution does indeed confirm the maximum entropy principle and ii the edge dynamic can be effectively implemented using short random walks on the underlying graphs leading to a local algorithm for the generation of er random graphs the resulting statistical mechanical system can be adapted to provide a distributed and local i e without any centralized monitoring mechanism for load balancing which can have a significant impact in increasing the efficiency and utilization of both the internet e g efficient web mirroring and large scale computing infrastructure e g cluster and grid computing

comment on subgraphs in random networks we point out biases in the algorithms used by itzkovitz et a phys rev e 68 026127 2003 to assess their approximate formulas for the average number of occurrences of certain subgraphs in random graphs with prescribed degree sequences

statistical mechanics of random graphs we discuss various aspects of the statistical formulation of the theory of random graphs with emphasis on results obtained in a series of our recent publications

resilience of complex networks to random breakdown using monte carlo simulations we calculate f c the fraction of nodes that are randomly removed before global connectivity is lost for networks with scale free and bimodal degree distributions our results differ from the results predicted by an equation for f c proposed by cohen et al we discuss the reasons for this disagreement and clarify the domain for which the proposed equation is valid

percolation theory and fragmentation measures in social networks abstract we study the statistical properties of a recently proposed social networks measure of fragmentation f after removal of a fraction q of nodes or links from the network the measure f is defined as the ratio of the number of pairs of nodes that are not connected in the fragmented network to the total number of pairs in the original fully connected network we compare this measure with the one traditionally used in percolation theory p the fraction of nodes in the largest cluster relative to the total number of nodes using both analytical and numerical methods we study erd s renyi er and scale free sf networks under various node removal strategies we find that for a network obtained after removal of a fraction q of nodes above criticality p 1 f 1 2 for fixed p and close to criticality we show that 1 f better reflects the actual fragmentation for a given p 1 f has a broad distribution and thus one can improve significantly the fragmentation of the network we also study and compare the fragmentation measure f and the percolation measure p for a real national social network of workplaces linked by the households of the employees and find similar results

church rosser theorem for a rewriting system on categorical combinators abstract this paper develops the church rosser theorem for the rewriting system ccl on type free categorial combinators introduced by curien the system ccl is not confluent however we show that there are various sets d of categorical combinator terms such that each d satisfies the following two conditions 1 d is closed under reduction by ccl 2 ccl is confluent on d moreover we examine the relation among these sets

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

clustering of random scale free networks we derive the finite size dependence of the clustering coefficient of scale free random graphs generated by the configuration model with degree distribution exponent 2l ensuremath gamma l3 degree heterogeneity increases the presence of triangles in the network up to levels that compare to those found in many real networks even for extremely large nets we also find that for values of ensuremath gamma ensuremath approx 2 clustering is virtually size independent and at the same time becomes a de facto non self averaging topological property this implies that a single instance network is not representative of the ensemble even for very large network sizes

boolean decision problems with competing interactions on scale free networks critical thermodynamics we study the critical behavior of boolean variables on scale free networks with competing interactions ising spin glasses our analytical results for the disorder network decay exponent phase diagram are verified using monte carlo simulations when the probability of positive ferromagnetic and negative antiferromagnetic interactions is the same the system undergoes a finite temperature spin glass transition if the exponent that describes the decay of the interaction degree in the scale free graph is strictly larger than 3 however when the exponent is equal to or less than 3 a spin glass phase is stable for all temperatures the robustness of both the ferromagnetic and spin glass phases suggests that boolean decision problems on scale free networks are quite stable to local perturbations finally we show that for a given decay exponent spin glasses on scale free networks seem to obey universality furthermore when the decay exponent of the interaction degree is larger than 4 in the spin glass sector the universality class is the same as for the mean field sherrington kirkpatrick ising spin glass

statistical mechanics of multiedge networks statistical properties of binary complex networks are well understood and recently many attempts have been made to extend this knowledge to weighted ones there are however subtle yet important considerations to be made regarding the nature of the weights used in this generalization weights can be either continuous or discrete magnitudes and in the latter case they can additionally have undistinguishable or distinguishable nature this fact has not been addressed in the literature insofar and has deep implications on the network statistics in this work we face this problem introducing multiedge networks as graphs where multiple distinguishable connections between nodes are considered we develop a statistical mechanics framework where it is possible to get information about the most relevant observables given a large spectrum of linear and nonlinear constraints including those depending both on the number of multiedges per link and their binary projection the latter case is particularly interesting as we show that binary projections can be understood from multiedge processes the implications of these results are important as many real agent based problems mapped onto graphs require this treatment for a proper characterization of their collective behavior

random walk with priorities in communicationlike networks we study a model for a random walk of two classes of particles a and b where both species are present in the same site the motion of a s takes precedence over that of b s the model was originally proposed and analyzed in maragakis et al phys rev e 77 020103 r 2008 here we provide additional results we solve analytically the diffusion coefficients of the two species in lattices for a number of protocols in networks we find that the probability of a b particle to be free decreases exponentially with the node degree in scale free networks this leads to localization of the b s at the hubs and arrest of their motion to remedy this we investigate several strategies to avoid trapping of the b s including moving an a instead of the hindered b allowing a trapped b to hop with a small probability biased walk toward non hub nodes and limiting the capacity of nodes we obtain analytic results for lattices and networks and we discuss the advantages and shortcomings of the possible strategies

organization of complex networks without multiple connections we find a new structural feature of equilibrium complex random networks without multiple and self connections we show that if the number of connections is sufficiently high these networks contain a core of highly interconnected vertices the number of vertices in this core varies in the range between const x n1 2 and const x n2 3 where is the number of vertices in a network at the birth point of the core we obtain the size dependent cutoff of the distribution of the number of connections and find that its position differs from earlier estimates

optimization of network robustness to random breakdowns we study network configurations that provide optimal robustness to random breakdowns for networks with a given number of nodes n and a given cost which we take as the average number of connections per node hki we find that the network design that maximizes f c the fraction of nodes that are randomly removed before global connectivity is lost consists of q hk i 1 ffiffi

improved inference for respondent driven sampling data with application to hiv prevalence estimation respondent driven sampling is a form of link tracing network sampling which is widely used to study hard to reach populations often to estimate population proportions previous treatments of this process have used a with replacement approximation which we show induces bias in estimates for large sample fractions and differential network connectedness by characteristic of interest we present a treatment of respondent driven sampling as a successive sampling process unlike existing representations our approach respects the essential without replacement feature of the process while converging to an existing with replacement representation for small sample fractions and to the sample mean for a full population sample we present a successive sampling based estimator for population means based on respondent driven sampling data and demonstrate its superior performance when the size of the hidden population is known we present sensitivity analyses for unknown population sizes in addition we note tha

generation of uncorrelated random scale free networks uncorrelated random scale free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks we propose and analyze a model capable of generating random uncorrelated scale free networks with no multiple and self connections the model is based on the classical configuration model with an additional restriction on the maximum possible degree of the vertices we check numerically that the proposed model indeed generates scale free networks with no two and three vertex correlations as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree k respectively complex networks constitute a general framework for the topological characterization of many natural and technological systems whose complexity prevents a more detailed microscopic description f1 3g within this framework these systems are represented in terms of networks or graphs f4g in which vertices stand for the units composing the system while edges among vertices represent the interactions or relations between pairs of units the focus is thus shift to the topological characterization of the representative network a task which is largely more feasible and yields nevertheless a noticeable amount of information on the structure and properties of the original system the empirical analysis of many real complex networks has unveiled the presence of several typical properties widely found in systems belonging to a large variety of realms one of the most relevant is given by the scale free nature of the degree distribution pskd f1 3 5g defined as the probability that a randomly chosen vertex has degree k si e it is connected to other k verticesd in mathematical terms the scale free property translates into a power law function of the form psk d k g

algorithm for counting large directed loops we derive a belief propagation algorithm for counting large loops in a directed network we evaluate the distribution of the number of small loops in a directed random network with given degree sequence we apply the algorithm to a few characteristic directed networks of various network sizes and loop structures and compare the algorithm with exhaustive counting results when possible the algorithm is adequate in estimating loop counts for large directed networks and can be used to compare the loop structure of directed networks and their randomized counterparts

confluence results for the pure strong categorical logic ccl l calculi as subsystems of ccl abstract the strong categorical combinatory logic ccl ccl sp developed by curien 1986 is when typed and augmented with a rule defining a terminal object a presentation of cartesian closed categories furthermore it is equationally equivalent to the lambda calculus with explicit couples and surjective pairing here we study the confluence propertied of ccl ccl sp and of several of its subsystems and the relationship between untyped lambda calculi and ccl ccl sp as rewriting systems we prove that there exists a subset d of ccl and a subsystem sl of ccl sp confluent on d a very simple isomorphism between the classical lambda calculus and a subset sd of d which is extended between derivations of and a class of derivations of sl substitution which is a one step operation belonging to the metalanguage of is now described by rewritings with sl and calculations between several substitutions launched at the same time may be performed by sl this point is a real increase in the calculation capacities of lambda calculus same results for d the same result holds for the lambda calculus with couples and projection rules without surjective pairing the locally confluent subsystem ccl sp that is sl sp is not confluent this result is obtained by firstly designing a new counter example different from j w klop s one for confluence of the lambda calculus with couples and surjective pairing and then translating it into ccl however ccl sp is shown to be confluent on the set derived from sd these results cannot be obtained with classical methods of confluence and we designed a new method called interpretation method based on this trick a given relation r is confluent on a set x if and only if a relation e r induced by r on a set of regularized terms e x is confluent

confluence properties of weak and strong calculi of explicit substitutions categorical combinators curien 1986 1993 hardin 1989 yokouchi 1989 and more recently s calculus abadi 1991 hardin and le vy 1989 have been introduced to provide an explicit treatment of substitutions in the calculus we reintroduce here the ingredients of these calculi in a self contained and stepwise way with a special emphasis on confluence properties the main new results of the paper with respect to curien 1986 1993 hardin 1989 abadi 1991 and hardin and le vy 1989 are the following 1 we present a confluent weak calculus of substitutions where no variable clashes can be feared 2 we solve a conjecture raised in abadi 1991 s calculus is not confluent it is confluent on ground terms only this unfortunate result is repaired by presenting a confluent version of s calculus named the env caldulus in hardin and le vy 1989 called here the confluent s calculus

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

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

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

topological phase transitions of random networks abstract to provide a phenomenological theory for the various interesting transitions in restructuring networks we employ a statistical mechanical approach with detailed balance satisfied for the transitions between topological states this enables us to establish an equivalence between the equilibrium rewiring problem we consider and the dynamics of a lattice gas on the edge dual graph of a fully connected network by assigning energies to the different network topologies and defining the appropriate order parameters we find a rich variety of topological phase transitions defined as singular changes in the essential feature s of the global connectivity as a function of a parameter playing the role of the temperature in the critical point scale free networks can be recovered

correlations in connected random graphs we study the properties of the giant connected component in random graphs with arbitrary degree distribution we concentrate on the degree degree correlations we show that the adjoining nodes in the giant connected component are correlated and derive analytic formulas for the joint nearest neighbor degree probability distribution using those results we describe correlations in maximal entropy connected random graphs we show that connected graphs are disassortative and that correlations are strongly related to the presence of one degree nodes leaves we propose an efficient algorithm for generating connected random graphs we illustrate our results with several examples

generation of scale free networks using a simple preferential rewiring dynamics abstract we propose a simple dynamical process for non growing networks where steady states in the long time limit exhibit power law degree distributions with the exponent 2 at each time step two nodes i and j are randomly selected and one incoming link to i is redirected to j with the rewiring probability r determined only by degrees of two nodes k i and k j while higher degree nodes are preferred to get another link this is an application of the general model introduced earlier s ree phys rev e 73 2006 026115 to take the structure of networks into account we also consider three possible distinctions for the model i how we choose a rewiring link out of all incoming links to i three cases ii whether links are directed or not two cases iii types of networks considering the existence of self loops and multiple links two cases as a result we specify the total of 12 different cases of the model we then observe numerically that networks will evolve to steady states with power law degree distributions when parameters of the model satisfy certain conditions this work is from an effort to find a simple model of the network dynamics generating scale free networks and has a potential to become an underlying mechanism for wide range of scale free non growing networks

k core architecture and k core percolation on complex networks abstract we analytically describe the architecture of uncorrelated complex networks with arbitrary degree distributions as a set of successively enclosed substructures k cores we find the structure of k cores their sizes and their birth points the bootstrap percolation thresholds we show that in networks with a finite mean number z 2 of the second nearest neighbours the emergence of a k core is a unique hybrid phase transition with a jump emergence of the k core as at a first order phase transition but also with a critical singularity as at a continuous phase transition in contrast if z 2 diverges the networks contain an infinite sequence of k cores which are ultra robust against random damage we explain the meaning of the order parameter for k core percolation and reveal the origin of the specific critical phenomena we demonstrate that a so called corona of the k core plays a crucial role it turns out that the k core percolation threshold is at the same time the percolation threshold of finite corona clusters the mean separation of vertices in corona clusters plays the role of the correlation length and diverges at the critical point we demonstrate that a random removal of even one vertex from the k core may result in the collapse of a vast region of the k core around the removed vertex the mean size of this region diverges at the critical point we find an exact mapping of k core percolation to a model of cooperative relaxation this model undergoes critical relaxation with a divergent rate at some critical moment

optimization of the robustness of multimodal networks dirac s delta function x we refer to this type of network as a scale free multimodal network for m 2 the network is a bimodal network in the limit m approaches infinity the network models a scale free network we calculate and optimize the robustness for given values of the number of modes m the total number of nodes n and the average degree k using analytical formulas for the random and targeted node removal thresholds for network collapse we find when n 1 that i the robustness against random and targeted node removal for this multimodal network is controlled by a single combination of variables n 1 m 1 ii the robustness of the multimodal network against targeted node removal decreases rapidly when the number of modes becomes larger than a critical value that is of the order of ln n and iii the values of exponent opt that characterizes the scale free degree distribution of the multimodal network that maximize the robustness against both random and targeted node removal fall between 2 5 and 3

proof of termination of the rewriting system subst on ccl abstract curien 1985 defines a translation of the c calculus in the pure combinatory categorical logic and establishes an equivalence theorem between these two theories the rewriting system subst simulates in particular the substitution of the c calculus this system is locally confluent we shall show here that it is also noetherian

homogeneous complex networks abstract we discuss various ensembles of homogeneous complex networks and a monte carlo method of generating graphs from these ensembles the method is quite general and can be applied to simulate micro canonical canonical or grand canonical ensembles for systems with various statistical weights it can be used to construct homogeneous networks with desired properties or to construct a non trivial scoring function for problems of advanced motif searching

tuning clustering in random networks with arbitrary degree distributions we present a generator of random networks where both the degree dependent clustering coefficient and the degree distribution are tunable following the same philosophy as in the configuration model the degree distribution and the clustering coefficient for each class of nodes of degree k are fixed ad hoc and a priori the algorithm generates corresponding topologies by applying first a closure of triangles and second the classical closure of remaining free stubs the procedure unveils an universal relation among clustering and degree degree correlations for all networks where the level of assortativity establishes an upper limit to the level of clustering maximum assortativity ensures no restriction on the decay of the clustering coefficient whereas disassortativity sets a stronger constraint on its behavior correlation measures in real networks are seen to observe this structural bound

scale free networks with exponent one a majority of studied models for scale free networks have degree distributions with exponents greater than two real networks however can demonstrate essentially more heavy tailed degree distributions we explore two models of scale free equilibrium networks that have the degree distribution exponent ensuremath gamma 1 p q ensuremath sim q ensuremath ensuremath gamma such degree distributions can be identified in empirical data only if the mean degree of a network is sufficiently high our models exploit a rewiring mechanism they are local in the sense that no knowledge of the network structure apart from the immediate neighborhood of the vertices is required these models generate uncorrelated networks in the infinite size limit where they are solved explicitly we investigate finite size effects by the use of simulations we find that both models exhibit disassortative degree degree correlations for finite network sizes in addition we observe a markedly degree dependent clustering in the finite networks we indicate a real world network with a similar degree distribution

statistical ensemble of scale free random graphs a thorough discussion of the statistical ensemble of scale free connected random tree graphs is presented methods borrowed from field theory are used to define the ensemble and to study analytically its properties the ensemble is characterized by two global parameters the fractal and the spectral dimensions which are explicitly calculated it is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified the stability of the scale free regime is also considered when it breaks down either a scale is spontaneously generated or else a singular node appears and the graphs become crumpled a new computer algorithm to generate these random graphs is proposed possible generalizations are also discussed in particular more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary degenerate scale free random graphs

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

complex monge ampere equations on quasi projective varieties we introduce generalized monge ampere capacities and use these to study complex monge ampere equations whose right hand side is smooth outside a divisor we prove in many cases that there exists a unique normalized solution which is smooth outside the divisor our results still hold if the divisor is replaced by any closed subset

scale free networks as entropy competition complex networks describe several and different real world systems consisting of a number of interacting elements a very important characteristic of such networks is the degree distribution that strongly controls their behavior based on statistical mechanics three classes of uncorrelated complex networks are identified here depending on the role played by the connectivities amongst elements in particular by identifying the connectivities of a node with the number of its nearest neighbors we show that the power law is the most probable degree distribution that both nodes and neighbors in a reciprocal competition assume when the respective entropy functions reach their maxima under mutual constraint as a result we obtain scaling exponent values as a function of the structural characteristics of the whole network moreover our approach sheds light on the exponential and poissonian degree distributions derived respectively when connectivities are thought of as degenerated connections or as half edges thus all three classes of degree distributions are derived starting from a common principle and leading to a general and unified framework for investigating the network structure

statistical mechanics of scale free networks at a critical point complexity without irreversibility based on a rigorous extension of classical statistical mechanics to networks we study a specific microscopic network hamiltonian the form of this hamiltonian is derived from the assumption that individual nodes increase or decrease their utility by linking to nodes with a higher or lower degree than their own we interpret utility as an equivalent to energy in physical systems and discuss the temperature dependence of the emerging networks we observe the existence of a critical temperature tc where total energy utility and network architecture undergo radical changes along this topological transition we obtain ensemble averages of scale free networks with complex hierarchical topology the scale free nature emerges strictly within equilibrium with a clearly defined microcanonical ensemble and the principle of detailed balance fulfilled this provides evidence that complex networks may arise without irreversibility the utility approach establishes a link between classical statistical physics and a wide variety of applications in socioeconomic statistical systems

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