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

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

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

percolation critical exponents in scale free networks we study the behavior of scale free networks having connectivity distribution p k k 2l close to the percolation threshold we show that for networks with 3 l 4 known to undergo a transition at a finite threshold of dilution the critical exponents are different than the expected mean field values of regular percolation in infinite dimensions networks with 2 l 3 possess only a percolative phase nevertheless we show that in this case percolation critical exponents are well defined near the limit of extreme dilution where all sites are removed and that also then the exponents bear a strong l dependence the regular mean field values are recovered only for l 4

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

a note on the variance of a stopping time let sn e xi n 0 be a random walk with positive drift t ex1 0 and finite variance a2 varxl let t b inf n 2 1 sn b rb st b b m minn osn t t o and h s lai and siegmund show that var t b ba2 1t3 k m2 o 1 as b oo but give an unpleasant expression for the constant k using the identity jeh r y dp m y et h h et the expression for k can be simplified to a form that depends only on moments of ladder variables

belief propagation algorithm and the ising model on networks with arbitrary distributions of motifs we generalize the belief propagation algorithm to sparse random networks with arbitrary distributions of motifs triangles loops etc each vertex in these networks belongs to a given set of motifs generalization of the configuration model these networks can be treated as sparse uncorrelated hypergraphs in which hyperedges represent motifs here a hypergraph is a generalization of a graph where a hyperedge can connect any number of vertices these uncorrelated hypergraphs are treelike hypertrees which crucially simplifies the problem and allows us to apply the belief propagation algorithm to these loopy networks with arbitrary motifs as natural examples we consider motifs in the form of finite loops and cliques we apply the belief propagation algorithm to the ferromagnetic ising model with pairwise interactions on the resulting random networks and obtain an exact solution of this model we find an exact critical temperature of the ferromagnetic phase transition and demonstrate that with increasing the clustering coefficient and the loop size the critical temperature increases compared to ordinary treelike complex networks however weak clustering does not change the critical behavior qualitatively our solution also gives the birth point of the giant connected component in these loopy networks

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

symmetry and classes of transport equations abstract a method for classification and determination of local transport equations is presented based on the use of lie symmetries the starting point is the choice of a specific symmetry group and the definition of a class of partial differential equations results associated with the fokker planck equation in 1 1 and 2 1 dimensions are given

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

detection of topological patterns in complex networks correlation profile of the internet abstract a general scheme for detecting and analyzing topological patterns in large complex networks is presented in this scheme the network in question is compared with its properly randomized version that preserves some of its low level topological properties statistically significant deviation of any topological property of a network from this null model likely reflects its design principles and or evolutionary history we illustrate this basic scheme using the example of the correlation profile of the internet quantifying correlations between degrees of its neighboring nodes this profile distinguishes the internet from previously studied molecular networks with a similar scale free degree distribution we finally demonstrate that the clustering in a network is very sensitive to both the degree distribution and its correlation profile and compare the clustering in the internet to the appropriate null model

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

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

on harmonic renewal measures let be a probability and nu h sum limits n 1 infty frac 1 n mu n the corresponding harmonic renewal measure complementing earlier results where is concentrated on a halfline we investigate the behaviour ofvh x x 1 and the harmonic renewal functiong x vh x asx ifm1 x dx 0 we also consider the casem1 0

clustering of correlated networks we obtain the clustering coefficient the degree dependent local clustering and the mean clustering of networks with arbitrary correlations between the degrees of the nearest neighbor vertices the resulting formulas allow one to determine the nature of the clustering of a network

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

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

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

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

gauge symmetry in fokker planck dynamics abstract using a galilean metric approach based on an embedding of the euclidean space into a 4 1 minkowski space we analyze a gauge invariant lagrangian associated with a riemannian manifold r with metric g with a specific choice of the gauge condition the euler lagrange equations are written covariantly in r and then the fokker planck equation is derived such that the drift and the diffusion terms are obtained from g the analysis is carried out for both abelian and non abelian symmetries and an example with the su 2 symmetry is presented

symmetry groups density matrix equations and covariant wigner functions abstract a representation theory for lie groups is developed taking the hilbert space say h w of the w algebra standard representation as the representation space in this context the states describing physical systems are amplitude wave functions but closely connected with the notion of the density matrix then based on symmetry properties a general physical interpretation for the dual variables of thermal theories in particular the thermofield dynamics tfd formalism is introduced the kinematic symmetries galilei and poincare are studied and density amplitude matrix equations are derived for both of these cases in the same context of group theory the notion of phase space in quantum theory is analysed thus in the non relativistic situation the concept of density amplitude is introduced and as an example a spin half system is algebraically studied wigner function representations for the amplitude density matrices are derived and the connection of tfd and the usual wigner function methods are analysed for the poincare symmetries the relativistic density matrix equations are studied for the scalar and spinorial fields the relativistic phase space is built following the lines of the non relativistic case so for the scalar field the kinetic theory is introduced via the klein gordon density matrix equation and a derivation of the juttiner distribution is presented as an example thus making it possible to compare with the standard approaches the analysis of the phase space for the dirac field is carried out in connection with the dual spinor structure induced by the dirac field density matrix equation with the physical content relying on the symmetry groups gauge invariance is considered and as a basic result it is shown that the heinz density operator which has been used to develope a gauge covariant kinetic theory is a particular solution for the klein gordon and dirac density matrix equation

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

search for the symmetry of the fokker planck equation abstract it is shown that the fokker planck equations with linear drift x and nonlinear drift x x i e the rayleigh process have exact symmetries on their own solution manifolds the sl 2 r weyl algebra and the sl 2 r i respectively the symmetries of these two types of processes for 1 2 dimensional cases are explicitly found out too

scaling limits for cumulative input processes we study different scaling behavior of very general telecommunications cumulative input processes the activities of a telecommunication system are described by a marked point process tn zn n z where tn is the arrival time of a packet brought to the system or the starting time of the activity of an individual source and the mark zn is the amount of work brought to the system at time tn this model includes the popular on off process and the infinite source poisson model in addition to the latter models one can flexibly model dependence of the interarrival times tn tn 1 clustering behavior due to the arrival of an impulse generating a flow of activities but also dependence between the arrival process tn and the marks zn similarly to the on off and infinite source poisson model we can derive a multitude of scaling limits for the input process of one source or for the superposition of an increasing number of such sources the memory in the input process depends on a variety of factors such as the tails of the interarrival times or the tails of the distribution of activities initiated at an arrival tn or the number of activities starting at tn it turns out that as in standard results on the scaling behavior of cumulative input processes in telecommunications fractional brownian motion or infinite variance levy stable motion can occur in the scaling limit however the fractional brownian motion is a much more robust limit than the stable motion and many other limits may occur as well

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

edge union of networks on the same vertex set random network generators such as erd s r nyi watts strogatz and barab si albert models are used as models to study real world networks let g1 v e1 and g2 v e2 be two such networks on the same vertex set v this paper studies the degree distribution and clustering coefficient of the resultant networks g v e1 e2

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

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

on subordinated distributions and generalized renewal measures where an n eno is some sequence of nonnegative numbers sn neno is the sequence of partial sums s0 0 sn xflxk of another sequence xk ken of i i d random variables and a c r is a fixed borel set such as 0 1 or 0 oo examples of such convolution series are subordinated distributions f 0oan 1 which arise as distributions of random sums and harmonic and ordinary renewal measures a0 0 an 1 n for all n c n in the first an 1 for all n c no in the second case these examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes they are also of interest in connection with representation theorems such as the levy representation of infinitely divisible distributions a traditional approach to such problems is via regular variation if the underlying random variables are nonnegative we can use laplace transforms and the related abelian and tauberian theorems see e g stam 1973 in the context of subordination and feller 1971 xiv 3 in connection with renewal theory embrechts maejima and omey 1984 is a recent treatment of generalized renewal measures along these lines the approach of the present paper is based on the wiener levy gel fand theorem and has occasionally been called the banach algebra method in gruibel 1983 we gave a new variant of this method for the special case of lattice distributions showing that by using the appropriate banach algebras of sequences arbitrarily fine expansions are possible under certain assumptions on the higher order differences of p x1 n fnen here we give a corresponding treatment of nonlattice distributions we restrict ourselves to an analogue of first order differences and obtain a number of theorems which perhaps are described best as next term results to explain this let us consider a special case in more detail

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

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

galilei invariant gauge symmetries in fokker planck dynamics with logarithmic diffusion and drift terms a tensorial approach to galilean invariance is utilized together with lie symmetries of differential equations in order to derive equations of fokker planck type containing a logarithmic diffusion tensor and drift term the formalism is based on the projection from an extended by one space like dimension minkowski manifold to the usual newtonian spacetime so that non relativistic models are described by manifestly covariant lagrangians in this paper we obtain the fokker planck equations from the euler lagrange equations with the extended manifold by using a specific choice of the gauge condition we work in 1 1 spacetime and carry out the analysis for both abelian and non abelian symmetries

lie group classification of second order ordinary difference equations a group classification of invariant difference models i e difference equations and meshes is presented in the continuous limit the results go over into lie s classification of second order ordinary differential equations the discrete model is a three point one and we show that it can be invariant under lie groups of dimension 0 n 6

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

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

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

anomalous percolation properties of growing networks we describe the anomalous phase transition of the emergence of the giant connected component in scale free networks growing under mechanism of preferential linking we obtain exact results for the size of the giant connected component and the distribution of vertices among connected components we show that all the derivatives of the giant connected component size s over the rate b of the emergence of new edges are zero at the percolation threshold b c and s ensuremath propto mathrm exp ensuremath d ensuremath gamma b ensuremath b c ensuremath 1 2 where the coefficient d is a function of the degree distribution exponent ensuremath gamma in the entire phase without the giant component these networks are in a critical state the probability mathcal p k that a vertex belongs to a connected component of a size k is of a power law form at the phase transition point mathcal p k ensuremath sim 1 k mathrm ln k 2 in the phase with the giant component mathcal p k has an exponential cutoff at k c ensuremath propto 1 s in the simplest particular case we present exact results for growing exponential networks

percolation transition in networks with degree degree correlation we introduce an exponential random graph model for networks with a fixed degree distribution and a tunable degree degree correlation we then investigate the nature of the percolation transition in a correlated network with a poisson degree distribution it is found that negative correlation is irrelevant in that the percolation transition in the disassortative network belongs to the same universality class as in the uncorrelated network positive correlation turns out to be relevant the percolation transition in the assortative network is characterized by the nondiverging mean size of finite clusters and power law scalings of the density of the largest cluster and the cluster size distribution in the nonpercolating phase as well as at the critical point our results suggest that the unusual type of percolation transition in the growing network models reported recently may be inherited from the assortative degree degree correlation

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

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

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

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

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

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

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

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

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

interfaces and the edge percolation map of random directed networks the traditional node percolation map of directed networks is reanalyzed in terms of edges in the percolated phase edges can mainly organize into five distinct giant connected components interfaces bridging the communication of nodes in the strongly connected component and those in the in and out components formal equations for the relative sizes in the number of edges of these giant structures are derived for arbitrary joint degree distributions in the presence of local and two point correlations the uncorrelated null model is fully solved analytically and compared against simulations finding an excellent agreement interfaces and their particular conformations giving place from hairy ball percolation landscapes to bottleneck straits could bring new light to the discussion of how a structure is interwoven with functionality in flow networks

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

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

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

on generalized renewal measures and certain first passage times by relating it to the family x t infln 1 sn t t 0 one of the major results is that up t oo for all t e r iff 0 t u0 t 1 a 1 ua i1as t a oo iff e xj 2 x1 oo provided 4 is ultimately increasing a 2 0 a related result is proved for u t h u t and up t e n 2 o4 n p mn t where mn max0 i n s our results form extensions of earlier ones by heyde kalma gut and others who either considered more specific functions 4 or used stronger moment assumptions the proofs are based on a regeneration technique from renewal theory and two martingale inequalities by burkholder davis and gundy

on lie backlund vector fields of the evolution equations 2u x t f u and u t 2u x2 f u for the evolution equations 2u x t f u and u t 2u x2 f u we derive the analytic functions f where lie backlund vector fields are admitted

fractal dimensions of percolating networks we use the generating function formalism to calculate the fractal dimensions for the percolating cluster at criticality in erdos renyi er and random scale free sf networks with degree distribution p k ck we show that the chemical dimension is dl 2 for er and sf networks with 4 as in percolation in d dc 6 dimensions for 3 4 and df 2 2 3 3 4 and dc 2 1 3 3 4 we discuss the meaning of these dimensions for networks

generalized percolation in random directed networks we develop a general theory for percolation in directed random networks with arbitrary two point correlations and bidirectional edges that is edges pointing in both directions simultaneously these two ingredients alter the previously known scenario and open new views and perspectives on percolation phenomena equations for the percolation threshold and the sizes of the giant components are derived in the most general case we also present simulation results for a particular example of uncorrelated network with bidirectional edges confirming the theoretical predictions