absence of the nonpercolating phase for percolation on the nonplanar hanoi network we investigate bond percolation on the nonplanar hanoi network hn np which was studied previously boettcher et al phys rev e 80 041115 2009 we calculate the fractal exponent of a subgraph of the hn np which gives a lower bound for the fractal exponent of the original graph this lower bound leads to the conclusion that the original system does not have a nonpercolating phase where only finite size clusters exist for pg0 or equivalently that the system exhibits either the critical phase where infinitely many infinite clusters exist or the percolating phase where a unique giant component exists monte carlo simulations support our conjecture

phase transitions in dynamical random graphs we study a large time limit of a markov process whose states are finite graphs the number of the vertices is described by a supercritical branching process and the dynamics of edges is determined by the rates of appending and deleting we find a phase transition in our model similar to the one in the random graph model g n p we derive a formula for the line of critical parameters which separates two different phases one is where the size of the largest component is proportional to the size of the entire graph and another one where the size of the largest component is at most logarithmic with respect to the size of the entire graph in the supercritical phase we find the asymptotics for the size of the largest component

absolute value equations abstract we investigate existence and nonexistence of solutions for np hard equations involving absolute values of variables ax x b where a is an arbitrary n n real matrix by utilizing an equivalence relation to the linear complementarity problem lcp we give existence results for this class of absolute value equations aves as well as a method of solution for special cases we also give nonexistence results for our ave using theorems of the alternative and other arguments

revisiting node based sir models in complex networks with degree correlations abstract in this paper we consider two growing networks which will lead to the degree degree correlations between two nearest neighbors in the network when the network grows to some certain size we introduce an sir like disease such as pandemic influenza h1n1 09 to the population due to its rapid spread the population size changes slowly and thus the disease spreads on correlated networks with approximately fixed size to predict the disease evolution on correlated networks we first review two node based sir models incorporating degree correlations and an edge based sir model without considering degree correlation and then compare the predictions of these models with stochastic sir simulations respectively we find that the edge based model even without considering degree correlations agrees much better than the node based models incorporating degree correlations with stochastic sir simulations in many respects moreover simulation results show that for networks with positive correlation the edge based model provides a better upper bound of the cumulative incidence than the node based sir models whereas for networks with negative correlation it provides a lower bound of the cumulative incidence

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

degree product rule tempers explosive percolation in the absence of global information we introduce a guided network growth model which we call the degree product rule process that uses solely local information when adding new edges for small numbers of candidate edges our process gives rise to a second order phase transition but becomes first order in the limit of global choice we provide the set of critical exponents required to characterize the nature of this percolation transition such a process permits interventions which can delay the onset of percolation while tempering the explosiveness caused by cluster product rule processes

community detection in complex networks using density based clustering algorithm and manifold learning abstract like clustering analysis community detection aims at assigning nodes in a network into different communities fdp is a recently proposed density based clustering algorithm which does not need the number of clusters as prior input and the result is insensitive to its parameter however fdp cannot be directly applied to community detection due to its inability to recognize the community centers in the network to solve the problem a new community detection method named isofdp is proposed in this paper first we use isomap technique to map the network data into a low dimensional manifold which can reveal diverse pair wised similarity then fdp is applied to detect the communities in the network an improved partition density function is proposed to select the proper number of communities automatically we test our method on both synthetic and real world networks and the results demonstrate the effectiveness of our algorithm over the state of the art methods

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

weak convergence for random weighting estimation of smoothed quantile processes this paper presents a new random weighting method for smoothed quantile processes a theory is established for random weighting estimation of smoothed quantile processes it proves the weak convergence of the random weighting estimation error experiments and comparison analysis demonstrate that the proposed random weighting method can effectively estimate statistics and the achieved accuracy and convergence speed are much higher than those of the bootstrap method

optimal symmetric networks in terms of minimizing average shortest path length and their sub optimal growth model abstract homogeneous entangled networks characterized by small world large girths and no community structure have attracted much attention due to some of their favorable performances however the optimization algorithm proposed by donetti et al is very time consuming and will lose its efficiency when the size of the target network becomes large in this paper an alternative optimization algorithm is provided to get optimal symmetric networks by minimizing the average shortest path length it is shown that the synchronizability of a symmetric network is enhanced when the average shortest path length of the network is shortened as the optimization proceeds which suggests that the optimal symmetric networks in terms of minimizing average shortest path length will be very close to those entangled networks in order to overcome the time consuming obstacle of the optimization algorithms proposed by us and donetti et al a growth model is proposed to get large scale sub optimal symmetric networks numerical simulations show that the symmetric networks derived by our growth model will have small world property and besides these networks will have many other similar favorable performances as entangled networks e g robustness against errors and attacks very good load balancing ability and strong synchronizability

rigorous results on the threshold network model we analyse the threshold network model in which a pair of vertices with random weights are connected by an edge when the summation of the weights exceeds a threshold we prove some convergence theorems and central limit theorems on the vertex degree degree correlation and the number of prescribed subgraphs we also generalize some results in the spatially extended cases

random graphs with hidden color we propose and investigate a unifying class of sparse random graph models based on a hidden coloring of edge vertex incidences extending an existing approach random graphs with a given degree distribution in a way that admits a nontrivial correlation structure in the resulting graphs the approach unifies a number of existing random graph ensembles within a common general formalism and allows for the analytic calculation of observable graph characteristics in particular generating function techniques are used to derive the size distribution of connected components clusters as well as the location of the percolation threshold where a giant component appears

random geometric graphs we analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present the critical connectivity is found numerically by examining the size of the largest cluster we derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs even for infinite dimensionality insights relevant for graph bipartitioning are included

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

multiple power law structures in heterogeneous complex networks this paper develops a framework for analyzing and designing dynamic networks comprising different classes of nodes that coexist and interact in one shared environment we consider ad hoc i e nodes can leave the network unannounced and no node has any global knowledge about the class identities of other nodes preferentially grown networks where different classes of nodes are characterized by different sets of local parameters used in the stochastic dynamics that all nodes in the network execute we show that multiple scale free structures one within each class of nodes and with tunable power law exponents as determined by the sets of parameters characterizing each class emerge naturally in our model moreover the coexistence of the scale free structures of the different classes of nodes can be captured by succinct phase diagrams which show a rich set of structures including stable regions where different classes coexist in heavy tailed i e the exponent is between 2 and 3 and light tailed i e the exponent is greater than 3 states and sharp phase transitions the topology of the emergent networks is also shown to display a complex structure akin to the distribution of different components of an alloyed material e g nodes with a light tailed scale free structure get embedded to the outside of the network and have most of their edges connected to nodes belonging to the class with a heavy tailed distribution finally we show how the dynamics formulated in this paper will serve as an essential part of ad hoc networking protocols which can lead to the formation of robust and efficiently searchable networks including the well known peer to peer networks even under very dynamic conditions

turing instability in reaction diffusion models on complex networks in this paper the turing instability in reaction diffusion models defined on complex networks is studied here we focus on three types of models which generate complex networks i e the erd os renyi the watts strogatz and the threshold network models from analysis of the laplacian matrices of graphs generated by these models we reveal that the stable unstable regions of a spa tially homogeneous solution completely differ depending on network structures in particular we approximately argue the existence of the stable unstable re gions in the cases of regular enhanced ring lattices which include regular circles and networks generated by the threshold network model when the number of vertices is large enough

connectivity degrees in the threshold preferential attachment model abstract in this paper we present a study of the connectivity degrees of the threshold preferential attachment model a generalization of the barabasi albert model to heterogeneous complex networks the threshold model incorporates the states of the nodes in its preferential linking rule and assumes that the affinity between network nodes follows an inverse relationship with the distance between their states we numerically analyze the connectivity degrees of the model studying the influence of the main parameters on the distribution of connectivity degrees and its statistics the average degree and highest degree of the network we show that such statistics exhibit markedly different behaviors in the dependence on the model parameters particularly as regards the interaction threshold nevertheless we show that the two statistics converge in the limit of null threshold and often exhibit scaling that can be described by power laws of the model parameters

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

general formalism for inhomogeneous random graphs we present and investigate an extension of the classical random graph to a general class of inhomogeneous random graph models where vertices come in different types and the probability of realizing an edge depends on the types of its terminal vertices this approach provides a general framework for the analysis of a large class of models the generic phase structure is derived using generating function techniques and relations to other classes of models are pointed out

generating simple random graphs with prescribed degree distribution let f be a probability distribution with support on the non negative integers four methods for generating a simple undirected graph with approximate degree distribution f are described and compared two methods are based on the so called configuration model with modifications ensuring a simple graph one method is an extension of the classical erdos renyi graph where the edge probabilities are random variables and the last method starts with a directed random graph which is then modified to a simple undirected graph all methods are shown to give the correct distribution in the limit of large graph size but under different assumptions on the degree distribution f and also using different order of operations

assortative model for social networks in this brief report we present a version of a network growth model generalized in order to describe the behavior of social networks the case of study considered is the preprint archive at cul arxiv org each node corresponds to a scientist and a link is present whenever two authors wrote a paper together this graph is a nice example of degree assortative network that is to say a network where sites with similar degree are connected to each other the model presented is one of the few able to reproduce such behavior giving some insight on the microscopic dynamics at the basis of the graph structure

neighbor network in a polydisperse hard disk fluid degree distribution and assortativity the neighbor network in a two dimensional polydisperse hard disk fluid with diameter distribution p 4 is examined using constant pressure monte carlo simulations graphs are constructed from vertices disks with edges links connecting each vertex to k neighboring vertices defined by a radical tessellation at packing fractions in the range 0 24 0 36 the decay of the network degree distribution is observed to be consistent with the power law k where the exponent lies in the range 5 6 6 0 comparisons with the predictions of a maximum entropy theory suggest that this apparent power law behavior is not the asymptotic one and that p k k 4 in the limit k this is consistent with the simple idea that for large disks the number of neighbors is proportional to the disk diameter a power law decay of the network degree distribution is one of the characteristics of a scale free network the assortativity of the network is measured and is found to be positive meaning that vertices of equal degree are connected more often than in a random network finally the equation of state is determined and compared with the prediction from a scaled particle theory very good agreement between simulation and theory is demonstrated

aging and percolation dynamics in a non poissonian temporal network model we present an exhaustive mathematical analysis of the recently proposed non poissonian activity driven nopad model moinet et al phys rev lett 114 108701 2015 a temporal network model incorporating the empirically observed bursty nature of social interactions we focus on the aging effects emerging from the non poissonian dynamics of link activation and on their effects on the topological properties of time integrated networks such as the degree distribution analytic expressions for the degree distribution of integrated networks as a function of time are derived exploring both limits of vanishing and strong aging we also address the percolation process occurring on these temporal networks by computing the threshold for the emergence of a giant connected component highlighting the aging dependence our analytic predictions are checked by means of extensive numerical simulations of the nopad model

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

dynamical random graphs with memory we study the large time dynamics of a markov process whose states are finite but unbounded graphs the number of vertices is described by a supercritical branching process and the edges follow a certain mean field dynamics determined by the rates of appending and deleting the older an edge is the lesser is the probability that it is still in the graph the lifetime of any edge is distributed exponentially we call its mean value common for all edges a parameter of memory since it shows for how long the system keeps a particular connection between the vertices in the graph we show that our model provides a bridge between two well known models when the parameter of memory goes to infinity this is a generalized model of random growth and when this parameter is zero i e no memory our model behaves as a random graph thus by introducing a general class of dynamical graphs we have a unified overview on rather different models and the relations between them we find all the critical values of the parameters at which our model exhibits phase transitions and describe the properties of the phase diagram finally we compare and discuss the efficiency of the corresponding networks

networks based on collisions among mobile agents we investigate in detail a recent model of colliding mobile agents phys rev lett 96 088702 used as an alternative approach to construct evolving networks of interactions formed by the collisions governed by suitable dynamical rules the system of mobile agents evolves towards a quasi stationary state which is apart small fluctuations well characterized by the de nsity of the system and the residence time of the agents the residence time defines a collision rate and by varying the collision ra te the system percolates at a critical value with the emergence of a giant cluster whose critical exponents are the ones of t wodimensional percolation further the degree and clusteri ng coefficient distributions and the average path length sho w that the network associated with such a system presents non trivial features which depending on the collision rule enables one not only to recover the main properties of standard networks such as exponential random and scale free networks but also to obtain other topological structures namely we show a specific example where the obtained structure has topological fe atures which characterize accurately the structure and evolution of social networks in different contexts ranging from networks of acquaintances to networks of sexual contacts

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

strong approximation of density estimators from weakly dependent observations by density estimators from independent observations we derive an approximation of a density estimator based on weakly dependent random vectors by a density estimator built from independent random vectors we construct on a sufficiently rich probability space such a pairing of the random variables of both experiments that the set of observations x 1 x n from the time series model is nearly the same as the set of observations y 1 y n from the i i d model with a high probability all sets of the form x 1 x n delta y 1 y n boolean and a 1 b 1 x x a d b d contain no more than o n 1 2 pi b i a i 1 log n elements respectively although this does not imply very much for parametric problems it has important implications in nonparametric statistics it yields a strong approximation of a kernel estimator of the stationary density by a kernel density estimator in the i i d model moreover it is shown that such a strong approximation is also valid for the standard bootstrap and the smoothed bootstrap using these results we derive simultaneous confidence bands as well as supremum type nonparametric tests based on reasoning for the i i d model

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

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

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

bootstrapping the distance between smooth bootstrap and sample quantile distribution the normalized kolmogorov smirnov and variational distance between the distribution of the sampleq quantile and the pertaining smooth bootstrap distribution are asymptotically distributed like the absolute value of a normal random variable the distribution functions of these random distances may serve as measures of the accuracy of the bootstrap procedure

topological properties of a time integrated activity driven network here we consider the topological properties of the integrated networks emerging from the activity driven model n perra et al sci rep 2 469 2012 a temporal network model recently proposed to explain the power law degree distribution empirically observed in many real social networks by means of a mapping to a hidden variable network model we provide analytical expressions for the main topological properties of the integrated network depending on the integration time and the distribution of activity potential characterizing the model the expressions obtained exacts in some cases the results of controlled asymptotic expansions in others are confirmed by means of extensive numerical simulations our analytical approach which highlights the differences of the model with respect to the empirical observations made in real social networks can be easily extended to deal with improved more realistic modifications of the activity driven network paradigm

a strict inequality for the random triangle model the random triangle model on a graph g is a random graph model where the usual i i d measure is perturbed by a factor qt where q 1 is a constant and t is the number of triangles in the random subgraph here we consider the case where g is the usual two dimensional triangular lattice for which there exists a percolation threshold pc q such that the probability of getting an infinite connected component of retained edges is 0 for p pc q it has previously been shown that pc q is a decreasing function of q here we strengthen this by showing that pc q is strictly decreasing this confirms a conjecture by haggstrom and jonasson

patterns in randomly evolving networks idiotypic networks we present a model for the evolution of networks of occupied sites on undirected regular graphs at every iteration step in a parallel update i randomly chosen empty sites are occupied and occupied sites having occupied neighbor degree outside of a given interval t l t u are set empty depending on the influx i and the values of both lower threshold and upper threshold of the occupied neighbor degree different kinds of behavior can be observed in certain regimes stable long living patterns appear we distinguish two types of patterns static patterns arising on graphs with low connectivity and dynamic patterns found on high connectivity graphs increasing i patterns become unstable and transitions between almost stable patterns interrupted by disordered phases occur for still larger i the lifetime of occupied sites becomes very small and network structures are dominated by randomness we develop methods to analyze the nature and dynamics of these network patterns give a statistical description of defects and fluctuations around them and elucidate the transitions between different patterns results and methods presented can be applied to a variety of problems in different fields and a broad class of graphs aiming chiefly at the modeling of functional networks of interacting antibodies and b cells of the immune system idiotypic networks we focus on a class of graphs constructed by bit chains the biological relevance of the patterns and possible operational modes of idiotypic networks are discussed

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

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

spatially self organized resilient networks by a distributed cooperative mechanism abstract the robustness of connectivity and the efficiency of paths are incompatible in many real networks we propose a self organization mechanism for incrementally generating onion like networks with positive degree degree correlations whose robustness is nearly optimal as a spatial extension of the generation model based on cooperative copying and adding shortcut we show that the growing networks become more robust and efficient through enhancing the onion like topological structure on a space the reasonable constraint for locating nodes on the perimeter in typical surface growth as a self propagation does not affect these properties of the tolerance and the path length moreover the robustness can be recovered in the random growth damaged by insistent sequential attacks even without any remedial measures

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

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

long paths and cycles in dynamical graphs we study the large time dynamics of a markov process whose states are finite directed graphs the number of the vertices is described by a supercritical branching process and the edges follow a certain mean field dynamics determined by the rates of appending and deleting we find sufficient conditions under which asymptotically a s the order of the largest component is proportional to the order of the graph a lower bound for the length of the longest directed path in the graph is provided as well we derive an explicit formula for the limit as time goes to infinity of the expected number of cycles of a given finite length finally we study the phase diagram

criterion for the emergence of explosive synchronization transitions in networks of phase oscillators the emergence of explosive synchronization transitions in networks of phase oscillators recently has become one of the most interesting topics it is widely believed that the large frequency mismatch of a pair of oscillators also known as disassortativity in frequency is a direct cause of an explosive synchronization it is found that besides the disassortativity in frequency the disassortativity in node degree also shows up in connection with the first order synchronization transition in this paper we simulate the kuramoto model on top of a family of networks with different degree degree and frequency frequency correlation patterns results show that only when the degrees and natural frequencies of the network s nodes are both disassortative can an explosive synchronization occur

an efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side our goal in this work is to give an optimum correction of the infeasible absolute value equations ave in order to make the mentioned system feasible we apply the minimal correction using the l 2 norm by changing just the right hand vector we will show that this problem can be formulated as an unconstrained optimization problem with a quadratic objective function we propose an extension of newton s method for solving unconstrained objective optimization some examples are provided to illustrate the efficiency and validity of our proposed method

analysis of scale free networks based on a threshold graph with intrinsic vertex weights many real networks are complex and have power law vertex degree distribution short diameter and high clustering we analyze the network model based on thresholding of the summed vertex weights which belongs to the class of networks proposed by caldarelli et al phys rev lett 89 258702 2002 power law degree distributions particularly with the dynamically stable scaling exponent 2 realistic clustering and short path lengths are produced for many types of weight distributions thresholding mechanisms can underlie a family of real complex networks that is characterized by cooperativeness and the baseline scaling exponent 2 it contrasts with the class of growth models with preferential attachment which is marked by competitiveness and baseline scaling exponent 3

evolution of a modified binomial random graph by agglomeration in the classical erd s renyi random graph g n p there are n vertices and each of the possible edges is independently present with probability p the random graph g n p is homogeneous in the sense that all vertices have the same characteristics on the other hand numerous real world networks are inhomogeneous in this respect such an inhomogeneity of vertices may influence the connection probability between pairs of vertices the purpose of this paper is to propose a new inhomogeneous random graph model which is obtained in a constructive way from the erd s renyi random graph g n p given a configuration of n vertices arranged in n subsets of vertices we call each subset a super vertex we define a random graph with n super vertices by letting two super vertices be connected if and only if there is at least one edge between them in g n p our main result concerns the threshold for connectedness we also analyze the phase transition for the emergence of the giant component and the degree distribution even though our model begins with g n p it assumes the existence of some community structure encoded in the configuration furthermore under certain conditions it exhibits a power law degree distribution both properties are important for real world applications

oscillatory epidemic prevalence in growing scale free networks we study the persistent epidemic prevalence with oscillatory behavior and the extinction of computer viruses via e mails on a contact relational network growing with new users for which scale free structure is estimated from real data typical oscillatory phenomenon is simulated in a stochastic model for the execution and detection of viruses the conditions of extinction by random and targeted immunizations for hubs are derived through bifurcation analysis for simpler deterministic models by using a mean field approximation without the connectivity correlations we can qualitatively understand the mechanisms of the spread in linearly growing scale free networks

a high order modified levenberg marquardt method for systems of nonlinear equations with fourth order convergence fan 2014 presented an accelerated modified levenberg marquardt method for nonlinear equations at every iteration the accelerated modified lm method computed not only a lm trial step but also an additional approximate lm step which employed a line search in this paper based on the accelerated modified lm method we compute the approximate lm step one more time at every iteration and obtain a high order accelerating modified levenberg marquardt method under the local error bound condition which is weaker than nonsingularity the convergence order of this new method is shown to be fourth a globally convergence is also given by the trust region technique numerical results show that the new method is efficient and could save many calculations of the jacobian

emergence of symmetry in complex networks many real networks have been found to have a rich degree of symmetry which is a universal structural property of complex networks yet has been rarely studied so far one of the fascinating problems related to symmetry is exploration of the origin of symmetry in real networks for this purpose we summarized the statistics of local symmetric motifs that contribute to local symmetry of networks analysis of these statistics shows that the symmetry of complex networks is a consequence of similar linkage pattern which means that vertices with similar degrees tend to share common neighbors an improved version of the baraba i albert model integrating similar linkage pattern successfully reproduces the symmetry of real networks indicating that similar linkage pattern is the underlying ingredient that is responsible for the emergence of symmetry in complex networks

levenberg marquardt method for solving systems of absolute value equations in this paper we suggest and analyze the levenberg marquardt method for solving system of absolute value equations a x x b where a r n i n b r n and x r n are unknown we present different line search methods to convey the main idea and the significant modifications we discuss the convergence of the proposed method we consider numerical examples to illustrate the implementation and efficiency of the method results are very encouraging and may stimulate further research in this direction

dynamic random networks in dynamic populations we consider a random network evolving in continuous time in which new nodes are born and old may die and where undirected edges between nodes are created randomly and may also disappear the node population is markovian and so is the creation and deletion of edges given the node population each node is equipped with a random social index and the intensity at which a node creates new edges is proportional to the social index and the neighbour is either chosen uniformly or proportional to its social index in a modification of the model we derive properties of the network as time and the node population tends to infinity in particular the degree distribution is shown to be a mixed poisson distribution which may exhibit a heavy tail e g power law if the social index distribution has a heavy tail the limiting results are verified by means of simulations and the model is fitted to a network of sexual contacts

spin models on random graphs with controlled topologies beyond degree constraints we study ising spin models on finitely connected random interaction graphs which are drawn from an ensemble in which not only the degree distribution p k can be chosen arbitrarily but which allows for further fine tuning of the topology via preferential attachment of edges on the basis of an arbitrary function q k k of the degrees of the vertices involved we solve these models using finite connectivity equilibrium replica theory within the replica symmetric ansatz in our ensemble of graphs phase diagrams of the spin system are found to depend no longer only on the chosen degree distribution but also on the choice made for q k k the increased ability to control interaction topology in solvable models beyond prescribing only the degree distribution of the interaction graph enables a more accurate modeling of real world interacting particle systems by spin systems on suitably defined random graphs

enhancing neural network performance via assortativity the performance of attractor neural networks has been shown to depend crucially on the heterogeneity of the underlying topology we take this analysis a step further by examining the effect of degree degree correlations assortativity on neural network behavior we make use of a method recently put forward for studying correlated networks and dynamics thereon both analytically and computationally which is independent of how the topology may have evolved we show how the robustness to noise is greatly enhanced in assortative positively correlated neural networks especially if it is the hub neurons that store the information

infinite order percolation and giant fluctuations in a protein interaction network we investigate a model protein interaction network whose links represent interactions between individual proteins this network evolves by the functional duplication of proteins supplemented by random link addition to account for mutations when link addition is dominant an infinite order percolation transition arises as a function of the addition rate in the opposite limit of high duplication rate the network exhibits giant structural fluctuations in different realizations for biologically relevant growth rates the node degree distribution has an algebraic tail with a peculiar rate dependence for the associated exponent

a simple asymmetric evolving random network we introduce a new oriented evolving graph model inspired by biological networks a node is added at each time step and is connected to the rest of the graph by random oriented edges emerging from older nodes this leads to a statistical asymmetry between incoming and outgoing edges we show that the model exhibits a percolation transition and discuss its universality below the threshold the distribution of component sizes decreases algebraically with a continuously varying exponent depending on the average connectivity we prove that the transition is of infinite order by deriving the exact asymptotic formula for the size of the giant component close to the threshold we also present a thorough analysis of aging properties we compute local in time profiles for the components of finite size and for the giant component showing in particular that the giant component is always dense among the oldest nodes but invades only an exponentially small fraction of the young nodes close to the threshold

percolation in media with columnar disorder we study a generalization of site percolation on a simple cubic lattice where not only single sites are removed randomly but also entire parallel columns of sites we show that typical clusters near the percolation transition are very anisotropic with different scaling exponents for the sizes parallel and perpendicular to the columns below the critical point there is a griffiths phase where cluster size distributions and spanning probabilities in the direction parallel to the columns have power law tails with continuously varying non universal powers this region is very similar to the griffiths phase in subcritical directed percolation with frozen disorder in the preferred direction and the proof follows essentially the same arguments as in that case but in contrast to directed percolation in disordered media the number of active growth sites in a growing cluster at criticality shows a power law while the probability of a cluster to continue to grow shows logarithmic behavior

diffusion on hierarchical systems of weakly coupled networks abstract we analyzed diffusion dynamics on weakly coupled networks interconnected networks by means of separation of time scales using an adiabatic approximation we reduced the system dynamics to a markov chain with aggregated variables and derived a transport equation that is analogous to fick s first law and includes a driving force entropy production is a sum of microscopic entropy transport which results from the particle s migration between networks of different topologies and macroscopic entropy production of the markov chain equilibrium particles partition between different sub networks depends only on internal sub network parameters by changing structure of networks one can not only modify diffusion constants but can also induce or reverse the direction of the particles flow between different networks our framework confirmed by numerical simulations is also useful for considering diffusion in nested systems corresponding to hierarchical networks with several different time scales thus it can serve to uncover hidden hierarchy levels from observations of diffusion processes

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

properties of a random attachment growing network in this study we introduce and analyze the statistical structural properties of a model of growing networks which may be relevant to social networks at each step a new node is added which selects k possible partners from the existing network and joins them with probability delta by undirected edges the activity of the node ends here it will get new partners only if it is selected by a newcomer the model produces an infinite order phase transition when a giant component appears at a specific value of delta which depends on k the average component size is discontinuous at the transition in contrast the network behaves significantly different for k 1 there is no giant component formed for any delta and thus in this sense there is no phase transition however the average component size diverges for delta or 1 2

growing networks with mixed attachment mechanisms networks grow and evolve when new nodes and links are added in there are two methods to add the links uniform attachment and preferential attachment we take account of the addition of links with mixed attachment between uniform attachment and preferential attachment in proportion by using numerical simulations and analysis based on a continuum theory we obtain that the degree distribution p k has an extended power law form p k k k0 when the number of edges k of a node is much larger than a certain value k0 the degree distribution reduces to the power law form p k k and when k is much smaller than k0 the degree distribution degenerates into the exponential form it has been found that degree distribution possesses this extended power law form for many real networks such as the movie actor network the citation network of scientific papers and diverse protein interaction networks

on m estimators and normal quantiles this paper explores a class of robust estimators of normal quantiles filling the gap between maximum likelihood estimators and empirical quantiles our estimators are linear combinations of m estimators their asymptotic variances can be arbitrarily close to variances of the maximum likelihood estimators compared with empirical quantiles the new estimators offer considerable reduction of variance at near normal probability distributions

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

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

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

preferential attachment in randomly grown networks abstract we reintroduce the model of callaway et al 2001 as a special case of a more general model for random network growth vertices are added to the graph at a rate of 1 while edges are introduced at rate rather than edges being introduced at random we allow for a degree of preferential attachment with a linear attachment kernel parametrised by m the original model is recovered in the limit of no preferential attachment m as expected even weak preferential attachment introduces a power law tail to the degree distribution additionally this generalisation retains a great deal of the tractability of the original along with a surprising range of behaviour although key mathematical features are modified for finite m in particular the critical edge density c which marks the onset of a giant network component is reduced with increasing tendency for preferential attachment the positive degree degree correlation introduced by the unbiased growth process is offset by the skewed degree distribution reducing the network assortativity

class of correlated random networks with hidden variables we study a class of models of correlated random networks in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices we find analytical expressions for the main topological properties of these models as a function of the distribution of hidden variables and the probability of connecting vertices the expressions obtained are checked by means of numerical simulations in a particular example the general model is extended to describe a practical algorithm to generate random networks with an a priori specified correlation structure we also present an extension of the class to map nonequilibrium growing networks to networks with hidden variables that represent the time at which each vertex was introduced in the system

are randomly grown graphs really random we analyze a minimal model of a growing network at each time step a new vertex is added then with probability ensuremath delta two vertices are chosen uniformly at random and joined by an undirected edge this process is repeated for t time steps in the limit of large t the resulting graph displays surprisingly rich characteristics in particular a giant component emerges in an infinite order phase transition at ensuremath delta 1 8 at the transition the average component size jumps discontinuously but remains finite in contrast a static random graph with the same degree distribution exhibits a second order phase transition at ensuremath delta 1 4 and the average component size diverges there these dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph older vertices tend to have higher degree and to link with other high degree vertices merely by virtue of their age we conclude that grown graphs however randomly they are constructed are fundamentally different from their static random graph counterparts

properties of random graphs with hidden color we investigate in some detail a recently suggested general class of ensembles of sparse undirected random graphs based on a hidden stub coloring with or without the restriction to nondegenerate graphs the calculability of local and global structural properties of graphs from the resulting ensembles is demonstrated cluster size statistics are derived with generating function techniques yielding a well defined percolation threshold explicit rules are derived for the enumeration of small subgraphs duality and redundancy is discussed and subclasses corresponding to commonly studied models are identified

coverage probabilities of bootstrap confidence intervals for quantiles an asymptotic expansion of length 2 is established for the coverage probabilities of confidence intervals for the underlying q quantile which are derived by bootstrapping the sample q quantile the corresponding level error turns out to be of order o n 1 2 which is unexpectedly low a confidence interval of even more practical use is derived by using backward critical points the resulting confidence interval is of the same length as the one derived by ordinary bootstrap but it is distribution free and has higher coverage probability

percolation of partially interdependent scale free networks we study the percolation behavior of two interdependent scale free sf networks under random failure of 1 p fraction of nodes our results are based on numerical solutions of analytical expressions and simulations we find that as the coupling strength between the two networks q reduces from 1 fully coupled to 0 no coupling there exist two critical coupling strengths q 1 and q 2 which separate three different regions with different behavior of the giant component as a function of p i for q q 1 an abrupt collapse transition occurs at p p c ii for q 2 3 q 1 decreases with increasing here is the scaling exponent of the degree distribution p k proportionality k b in the hybrid transition at the q 2 0 for 3 thus the known theoretical p c 0 for a single network with 3 is expected to be valid also for strictly partial interdependent networks

clustering determines the dynamics of complex contagions in multiplex networks we present the mathematical analysis of generalized complex contagions in a class of clustered multiplex networks the model is intended to understand spread of influence or any other spreading process implying a threshold dynamics in setups of interconnected networks with significant clustering the contagion is assumed to be general enough to account for a content dependent linear threshold model where each link type has a different weight for spreading influence that may depend on the content e g product rumor political view that is being spread using the generating functions formalism we determine the conditions probability and expected size of the emergent global cascades this analysis provides a generalization of previous approaches and is especially useful in problems related to spreading and percolation the results present nontrivial dependencies between the clustering coefficient of the networks and its average degree in particular several phase transitions are shown to occur depending on these descriptors generally speaking our findings reveal that increasing clustering decreases the probability of having global cascades and their size however this tendency changes with the average degree there exists a certain average degree from which on clustering favors the probability and size of the contagion by comparing the dynamics of complex contagions over multiplex networks and their monoplex projections we demonstrate that ignoring link types and aggregating network layers may lead to inaccurate conclusions about contagion dynamics particularly when the correlation of degrees between layers is high

percolation transitions with nonlocal constraint we investigate percolation transitions in a nonlocal network model numerically in this model each node has an exclusive partner and a link is forbidden between two nodes whose r neighbors share any exclusive pair the r neighbor of a node x is defined as a set of at most n r neighbors of x where n is the total number of nodes the parameter r controls the strength of a nonlocal effect the system is found to undergo a percolation transition belonging to the mean field universality class for r 1 2 the system undergoes a peculiar phase transition from a nonpercolating phase to a quasicritical phase where the largest cluster size g scales as g n with 0 74 1 in the marginal case with r 1 2 the model displays a percolation transition that does not belong to the mean field universality class

analysis of complex contagions in random multiplex networks we study the diffusion of influence in random multiplex networks where links can be of r different types and for a given content e g rumor product or political view each link type is associated with a content dependent parameter ci in 0 that measures the relative bias type i links have in spreading this content in this setting we propose a linear threshold model of contagion where nodes switch state if their perceived proportion of active neighbors exceeds a threshold namely a node connected to mi active neighbors and ki mi inactive neighbors via type i links will turn active if cimi ciki exceeds its threshold under this model we obtain the condition probability and expected size of global spreading events our results extend the existing work on complex contagions in several directions by i providing solutions for coupled random networks whose vertices are neither identical nor disjoint ii highlighting the effect of content on the dynamics of complex contagions and iii showing that content dependent propagation over a multiplex network leads to a subtle relation between the giant vulnerable component of the graph and the global cascade condition that is not seen in the existing models in the literature

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

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