pagerank of integers we build up a directed network tracing links from a given integer to its divisors and analyze the properties of the google matrix of this network the pagerank vector of this matrix is computed numerically and it is shown that its probability is inversely proportional to the pagerank index thus being similar to the zipf law and the dependence established for the world wide web the spectrum of the google matrix of integers is characterized by a large gap and a relatively small number of nonzero eigenvalues a simple semi analytical expression for the pagerank of integers is derived that allows to find this vector for matrices of billion size this network provides a new pagerank order of integers

a sufficient condition for global regularity of the neumann operator abstract a theory of global regularity of the neumann operator is developed which unifies the two principal approaches to date namely the one via compactness due to kohn nirenberg j j kohn l nirenberg non coercive boundary value problems comm pure appl math 18 1965 443 492 and catlin david catlin global regularity of the neumann problem in y t siu ed complex analysis of several variables in proc sympos pure math vol 41 amer math soc providence ri 1984 pp 39 49 and the one via plurisubharmonic defining functions and or vector fields that commute approximately with due to boas and the author harold p boas emil j straube sobolev estimates for the neumann operator on domains in c n admitting a defining function that is plurisubharmonic on the boundary math z 206 1 1991 81 88 harold p boas emil j straube de rham cohomology of manifolds containing the points of infinite type and sobolev estimates for the neumann problem j geom anal 3 3 1993 225 235

random growth urban model with geographical fitness abstract this paper formulates a random growth urban model with a notion of geographical fitness using techniques of complex network theory we study our system as a type of preferential attachment model with fitness and we analyze its macro behavior to clarify the properties of the city size distributions it predicts first restricting the geographical fitness to take positive values and using a continuum approach we show that the city size distributions predicted by our model asymptotically approach pareto distributions with coefficients greater than unity then allowing the geographical fitness to take negative values we perform local coefficient analysis to show that the predicted city size distributions can deviate from pareto distributions as is often observed in actual city size distributions as a result the model we propose can generate a generic class of city size distributions including but not limited to pareto distributions for applications to city population projections our simple model requires randomness only when new cities are created not during their subsequent growth this property leads to smooth trajectories of city population growth in contrast to other models using gibrat s law in addition a discrete form of our dynamical equations can be used to estimate past city populations based on present day data this fact allows quantitative assessment of the performance of our model further study is needed to determine appropriate formulas for the geographical fitness

growth model for fractal scale free networks generated by a random walk abstract the diversity of networked systems with fractal structures suggests that knowing the underlying mechanism that generates the fractality is necessary for building a model of the development of complex networks in the present paper we propose a growth model of a network generated by a random walk and show that the evolving graph forms a fractal structure with various properties including the scale free property if the graph which provides a space where a random walk occurs by itself is formed by the random walk the proposed model is regulated by two parameters p v and p e which define the probability of either a roundabout path via a new vertex or a shortcut being formed by the random walk respectively the power law exponent describing the vertex degree distribution is determined by the ratio p e p v and is related to an internal factor f i via the relation 1 f i 1 where f i is a parameter that describes the local structure generated by the random walk a sufficiently small p v provides the small world property to the model network the small world property is usually considered to be incompatible with the fractal scaling property open m c l c d c where open m c is the average number of vertices which can be reached from a randomly chosen vertex in at most l c steps however we demonstrate that fractality can be reconciled with the small world property by introducing a size dependent fractal cluster dimension d c

local preferential attachment model for hierarchical networks abstract in real life networks incomers may only connect to a few others in a local area for their limited information and individuals in a local area are likely to have close relations accordingly we propose a local preferential attachment model here a local area network stands for a node and all its neighbors and the new nodes perform nonlinear preferential attachment k i k i in local areas the stable degree distribution and clustering degree correlations are analytically obtained with the increasing of the clustering coefficient increases while assortativity decreases from positive to negative in addition by adjusting the parameter the model can generate different kinds of degree distribution from exponential to power law the hierarchical organization independent of is the most significant character of this model

topological properties of integer networks abstract inspired by pythagoras s belief that numbers represent the reality we study the topological properties of networks of composite numbers in which the vertices represent the numbers and two vertices are connected if and only if there exists a divisibility relation between them the network has a fairly large clustering coefficient c 0 34 which is insensitive to the size of the network the average distance between two nodes is shown to have an upper bound that is independent of the size of the network in contrast to the behavior in small world and ultra small world networks the out degree distribution is shown to follow a power law behavior of the form k 2 in addition these networks possess hierarchical structure as c k k 1 in accord with the observations of many real life networks

analyzing percolation of networks inspired by the 3x 1 problem abstract in this paper we investigate percolation in a sort of networks inspired by the observation of a collatz graph cg which is the network version of the famous 3 x 1 problem in mathematics the cg consists of positive integers that are connected based on the iteration relations actually we never mean to solve the 3 x 1 problem exactly but we observe it from the viewpoint of statistical physics we focus on the so called reduced collatz graph rcg that is a subgraph with all odd numbers since even numbers can be iterated into odd ones through 3 x 1 rules considering boundary conditions we obtain a special degree distribution of rcg for finite size set of odd integers and treat the infinite case as its limit with the percolation criterion through the approach of a generating function we determine the critical condition for the network ensemble of rcg furthermore we generalize the graph model with rcg type degree distributions beyond the 3 x 1 problem get the generic criterion of percolation and phase diagram for an ensemble of positive integer networks with rcg as its extreme case

a new small world network created by cellular automata in this paper we generate small world networks by the cellular automaton based on starting with one dimensional regular networks besides the common properties of small world networks with small average shortest path length and large clustering coefficient the small world networks generated in this way have other properties i the edges which are cut in the regular network can be controlled that whether the edges are reconnected or not and ii the number of the edges of the small world network model equals the number of the edges of the original regular network in other words the average degree of the small world network model equals to the average degree of the original regular network

families and clustering in a natural numbers network we develop a network in which the natural numbers are the vertices the decomposition of natural numbers by prime numbers is used to establish the connections we perform data collapse and show that the degree distribution of these networks scales linearly with the number of vertices we explore the families of vertices in connection with prime numbers decomposition we compare the average distance of the network and the clustering coefficient with the distance and clustering coefficient of the corresponding random graph in case we set connections among vertices each time the numbers share a common prime number the network has properties similar to a random graph if the criterion for establishing links becomes more selective only prime numbers greater than p l are used to establish links where the network has high clustering coefficient

a multiplier condition for hypoellipticity of complex vector fields with optimal loss of derivatives abstract in c z r t we consider the function g g z set g 1 z g g 1 1 z z g and define the operator l g z i g 1 t we discuss estimates with loss of derivatives in the sense of kohn for the system l g f k l g where l g l g is 1 2 m subelliptic at 0 and f 0 0 d f 0 0 we prove estimates with a loss l k 1 2 m if the multiplier condition f g 1 1 1 2 m 1 is fulfilled for estimates without cut off subellipticity can be weakened to compactness and this results in a loss of l k 2 m 1 for the choice g f k z 2 m z k this result was obtained by kohn and bove derridj kohn tartakoff for m 1 and m 1 respectively also the loss l k 1 2 m was proven to be optimal we show that it remains optimal for the model g f k x 2 m x k instead for the model g f k z 2 m x k in which the multiplier condition is violated the loss is not lowered by the type and must be k 1 2

weighted self similar networks under preferential attachment abstract we introduce an abstract evolutionary formalism that generates weighted networks whose growth under stochastic preferential attachment triggers unrestricted weight rearrangements in existing links the class of resulting algorithms for different parameter values includes the barabasi albert and barrat barthelemy vespignani models as special cases we solve the recursions that describe the average growth to derive exact solutions for the expected degree and strength distribution the individual strength and weight development and the joint distribution of neighboring degrees we find that the network exhibits a particular form of self similarity namely every sufficiently interconnected node has on average the same constitution of small degree neighbors as any other node of large degree finally we suggest potential applications in several fields of interest

high dimensional random apollonian networks abstract we propose a simple algorithm which produces a new category of networks high dimensional random apollonian networks with small world and scale free characteristics we derive analytical expressions for their degree distributions and clustering coefficients which are determined by the dimension of the network the values obtained for these parameters are in good agreement with simulation results and comparable to those coming from real networks we estimate also analytically that the average path length of the networks increases at most logarithmically with the number of vertices

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

pinning synchronization of delayed complex dynamical networks with nonlinear coupling abstract in this paper we find that complex networks with the watts strogatz or scale free ba random topological architecture can be synchronized more easily by pin controlling fewer nodes than regular systems theoretical analysis is included by means of lyapunov functions and linear matrix inequalities lmi to make all nodes reach complete synchronization numerical examples are also provided to illustrate the importance of our theoretical analysis which implies that there exists a gap between the theoretical prediction and numerical results about the minimum number of pinning controlled nodes

flexible construction of hierarchical scale free networks with general exponent extensive studies have been done to understand the principles behind architectures of real networks recently evidence for hierarchical organization in many real networks has also been reported here we present a hierarchical model that reproduces the main experimental properties observed in real networks scale free of degree distribution p k frequency of the nodes that are connected to k other nodes decays as a power law p k ensuremath sim k ensuremath ensuremath gamma and power law scaling of the clustering coefficient c k ensuremath sim k ensuremath 1 the major points of our model can be summarized as follows a the model generates networks with scale free distribution for the degree of nodes with general exponent ensuremath gamma g2 and arbitrarily close to any specified value being able to reproduce most of the observed hierarchical scale free topologies in contrast previous models cannot obtain values of ensuremath gamma g2 58 b our model has structural flexibility because i it can incorporate various types of basic building blocks e g triangles tetrahedrons and in general fully connected clusters of n nodes and ii it allows a large variety of configurations i e the model can use more than n ensuremath 1 copies of basic blocks of n nodes the structural features of our proposed model might lead to a better understanding of architectures of biological and nonbiological networks

the diederich fornaess index and the global regularity of the bar partial neumann problem we describe along the guidelines of kohn quantitative estimates for global regularity analysis and geometry in several complex variables pp 97 128 trend math birkhauser boston 1999 the constant mathcal e s which is needed to control the commutator of a totally real vector field t mathcal e with bar partial in order to have h s a priori estimates for the bergman projection b k k ge q 1 on a smooth q convex domain d subset subset mathbb c n this statement not explicit in kohn quantitative estimates for global regularity analysis and geometry in several complex variables pp 97 128 trend math birkhauser boston 1999 yields regularity of b k in specific sobolev degree s next we refine the pseudodifferential calculus at the boundary in order to relate for a defining function r of d the operators t frac delta 2 and r frac delta 2 we are thus able to extend to general degree k ge 0 of b k the conclusion of quantitative estimates for global regularity analysis and geometry in several complex variables pp 97 128 trend math birkhauser boston 1999 which only holds for q 1 and k 0 if for the diederich fornaess index delta of d we have 1 delta frac 1 2 le mathcal e s then b k is h s regular

dynamics of directed graphs the world wide web we introduce and simulate a growth model of the world wide web based on the dynamics of outgoing links that is motivated by the conduct of the agents in the real web to update outgoing links re directing them towards constantly changing selected nodes emergent statistical correlation between the distributions of outgoing and incoming links is a key feature of the dynamics of the web the growth phase is characterized by temporal fractal structures which are manifested in the hierarchical organization of links we obtain quantitative agreement with the recent empirical data in the real web for the distributions of in and out links and for the size of the connected component in a fully grown network of n nodes we study the structure of connected clusters of nodes that are accessible along outgoing links from a randomly selected node the distributions of size and depth of the connected clusters with a giant component exhibit supercritical behavior by decreasing the control parameter average fraction of updated and added links per time step towards c n 10 the web can resume a critical structure with no giant component in it we find a different universality class when the updates of links are not allowed i e for 0 corresponding to the network of science citations

phase transition in a stochastic prime number generator we introduce a stochastic algorithm that acts as a prime number generator the dynamics of this algorithm gives rise to a continuous phase transition which separates a phase where the algorithm is able to reduce a whole set of integers into primes and a phase where the system reaches a frozen state with low prime density we present both numerical simulations and an analytical approach in terms of an annealed approximation by means of which the data are collapsed a critical slowing down phenomenon is also outlined

maximal planar networks with large clustering coefficient and power law degree distribution in this article we propose a simple rule that generates scale free networks with very large clustering coefficient and very small average distance these networks are called random apollonian networks rans as they can be considered as a variation of apollonian networks we obtain the analytic results of power law exponent ensuremath gamma 3 and clustering coefficient c frac 46 3 ensuremath 36 phantom rule 0 2em 0ex mathrm ln phantom rule 0 2em 0ex frac 3 2 ensuremath approx 0 74 which agree with the simulation results very well we prove that the increasing tendency of average distance of rans is a little slower than the logarithm of the number of nodes in rans since most real life networks are both scale free and small world networks rans may perform well in mimicking the reality the rans possess hierarchical structure as c k ensuremath sim k ensuremath 1 that are in accord with the observations of many real life networks in addition we prove that rans are maximal planar networks which are of particular practicability for layout of printed circuits and so on the percolation and epidemic spreading process are also studied and the comparisons between rans and barab asi albert ba as well as newman watts nw networks are shown we find that when the network order n the total number of nodes is relatively small as n ensuremath sim 10 4 the performance of rans under intentional attack is not sensitive to n while that of ba networks is much affected by n and the diseases spread slower in rans than ba networks in the early stage of the suseptible infected process indicating that the large clustering coefficient may slow the spreading velocity especially in the outbreaks

propagation of holomorphic extendibility and non hypoellipticity of the neumann problem in an exponentially degenerate boundary abstract we prove that cr lines in an exponentially degenerate boundary are propagators of holomorphic extendibility this explains in the context of the cr geometry why in this situation the induced kohn laplacian b is not hypoelliptic christ 2000 2

uniform synchronous criticality of diversely random complex networks abstract we investigate collective synchronous behaviors in random complex networks of limit cycle oscillators with the non identical asymmetric coupling scheme and find a uniform coupling criticality of collective synchronization which is independent of complexity of network topologies numerical simulations on categories of random complex networks have verified this conclusion

a small world network of prime numbers abstract according to goldbach conjecture any even number can be broken up as the sum of two prime numbers n p q we construct a network where each node is a prime number and corresponding to every even number n we put a link between the component primes p and q in most cases an even number can be broken up in many ways and then we chose one decomposition with a probability p q through computation of average shortest distance and clustering coefficient we conclude that for 1 8 the network is of small world type and for 1 8 it is of regular type we also present a theoretical justification for such behaviour

projective cluster synchronization in drive response dynamical networks abstract in this paper we study the projective cluster synchronization in a drive response dynamical network with 1 n coupled partially linear chaotic systems because the scaling factors characterizing the dynamics of projective synchronization remain unpredictable pinning control ideas are adopted to direct the different scaling factors onto the desired values it is also shown that the projection cluster synchronization can be realized by controlling only one node in each cluster numerical simulations on the chaotic lorenz system are illustrated to verify the theoretical results

necessary geometric and analytic conditions for general estimates in the bar partial neumann problem we show the geometric and analytic consequences of a general estimate in the bar partial neumann problem a gain in the estimate yields a bound in the type of the boundary that is in its order of contact with an analytic curve as well as in the rate of the bergman metric we also discuss the potential theoretical consequence a gain implies a lower bound for the levi form of a bounded weight

maximal hypoellipticity for the neumann problem abstract we establish maximal hypoellipticity for the neumann problem on smooth bounded pseudoconvex domains in c n under the weakest possible condition on the levi form in particular maximal hypoellipticity holds on the level of n 1 forms for pseudoconvex domains of finite commutator type

regularity of the neumann problem at point of infinite type abstract we introduce general estimates for gain of regularity of solutions of the neumann problem and relate it to the existence of weights with large levi form at the boundary this enables us to discuss in a unified framework the classical results on fractional ellipticity subellipticity superlogarithmic ellipticity and compactness for each case we exhibit a corresponding class of domains

effect of node deleting on network structure abstract the ever increasing knowledge of the structure of various real world networks has uncovered their complex multi mechanism governed evolution processes therefore a better understanding of the structure and evolution of these networked complex systems requires us to describe such processes in a more detailed and realistic manner in this paper we introduce a new type of network growth rule which comprises addition and deletion of nodes and propose an evolving network model to investigate the effect of node deleting on network structure it is found that with the introduction of node deleting network structure is significantly transformed in particular degree distribution of the network undergoes a transition from scale free to exponential forms as the intensity of node deleting increases at the same time nontrivial disassortative degree correlation develops spontaneously as a natural result of network evolution in the model we also demonstrate that node deleting introduced in the model does not destroy the connectedness of a growing network so long as the increasing rate of edges is not excessively small in addition it is found that node deleting will weaken but not eliminate the small world effect of a growing network and generally it will decrease the clustering coefficient in a network

regularity of a class of sublaplacians on the 3 dimensional torus abstract in this paper we present a necessary and sufficient condition for a family of sums of squares operators to be globally hypoelliptic on a torus this condition says that either a diophantine condition is satisfied or there exists a point of finite type also we describe the analytic and gevrey versions of this result the proof is based on l 2 estimates and microlocal analysis

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

the complex green operator on cr submanifolds of ℂⁿ of hypersurface type compactness we establish compactness estimates for b on a compact pseudoconvex cr submanifold of c of hypersurface type that satisfies property p when the submanifold is orientable these estimates were proved by a raich in 32 using microlocal methods our proof deduces the estimates from a slight extension when q 1 of those known on hypersurfaces via the fact that locally cr submanifolds of hypersurface type are cr equivalent to a hypersurface the relationship between two potential theoretic conditions is also clarified

how much information is needed to be the majority during the binary state opinion formation abstract in this paper we try to propose a toy model which follows the majority rule with the fermi function to uncover the role of the heterogeneous interaction between individuals in opinion formation in order to do this we define the impact factor i f i says individual i as the exponential function of its connectivity k i with the tunable parameter also shows the public information that can be collected by individuals in the system we realize our model in scale free networks with mean connectivity k we find that much more public information 2 and less public information 1 cannot let either of the two opinions be the majority during the opinion formation furthermore 1 is a constant and equal to 0 76 0 04 and 2 decreases as a power law function of the mean connectivity k of the network our work can provide some perspectives and tools to understand the diversity of opinion in social networks

hypoellipticity of the kohn laplacian box _b and of the bar partial neumann problem by means of subelliptic multipliers we prove local hypoellipticity of the complex laplacian box and of the kohn laplacian box b in a pseudoconvex boundary when for a system of cut off eta the gradient partial b eta and the levi form frac 1 2 partial b bar partial b bar partial b partial b eta are subelliptic multipliers in the sense of kohn acta math 142 79 122 1979

complex networks and economics abstract as the first step toward a study of complex networks in economics we consider business networks categorized into bipartite networks through empirical study we show the possibility that business networks will fall into the scale free category to have an idea about the clustering coefficient and the averaged path length for bipartite networks we perform a one mode reduction we calculate the quantities for networks constructed by banks and companies before after a merger among banks and quantitatively show that the merger among banks increases the cliquishness of companies and decreases the typical separation between two companies

the hardy littlewood lemma and the estimate of the neumann problem in a general norm abstract we prove a generalized hardy littlewood lemma on a non smooth domain in the f norm and give an application to a corresponding estimate for the neumann problem by means of suitable weights

epidemic dynamics behavior in some bus transport networks abstract we abstract bus transport networks btns to complex networks using the space p approach first we select three actual btns in three major cities in china namely beijing shanghai and hangzhou using the sis model we simulate and study the epidemic spreading in the three btns we obtain the density of infected vertices varying with time and the stationary density of infected vertices varying with infection rate second we simulate and study the epidemic spreading in a recently introduced btn evolution model the network properties of which correspond well with those of actual btns third we use mean field theory to analyze the epidemic dynamics behavior of the btn evolution model and obtain the theoretical epidemic threshold of this model the theoretical value agrees well with the simulation results based on the work in this paper we provide the following possible forecasts for epidemic dynamics in actual btns an actual btn should have a finite positive epidemic threshold if the effective infection rate is above this threshold the epidemic spread in the network and the density of infected vertices finally stabilizes in a balanced state below this threshold the number of infected vertices decays exponentially fast and the epidemic cannot spread on a large scale

exploring complex networks via topological embedding on surfaces we demonstrate that graphs embedded on surfaces are a powerful and practical tool to generate to characterize and to simulate networks with a broad range of properties any network can be embedded on a surface with sufficiently high genus and therefore the study of topologically embedded graphs is non restrictive we show that the local properties of the network are affected by the surface genus which determines the average degree which influences the degree distribution and which controls the clustering coefficient the global properties of the graph are also strongly affected by the surface genus which is constraining the degree of interwovenness changing the scaling properties of the network from large world kind small genus to small and ultrasmall world kind large genus two elementary moves allow the exploration of all networks embeddable on a given surface and naturally introduce a tool to develop a statistical mechanics description for these networks within such a framework we study the properties of topologically embedded graphs which dynamically tend to lower their energy towards a ground state with a given reference degree distribution we show that the cooling dynamics between high and low temperatures is strongly affected by the surface genus with the manifestation of a glass like transition occurring when the distance from the reference distribution is low we prove with examples that topologically embedded graphs can be built in a way to contain arbitrary complex networks as subgraphs this method opens a new avenue to build geometrically embedded networks on hyperbolic manifolds

equilibrium distributions in entropy driven balanced processes abstract for entropy driven balanced processes we obtain final states with poisson bernoulli negative binomial and polya distributions we apply this both for complex networks and particle production for random networks we follow the evolution of the degree distribution p n in a system where a node can activate k fixed connections from k possible partnerships among all nodes the total number of connections n is also fixed for particle physics problems p n is the probability of having n particles or other quanta distributed among k states phase space cells while altogether a fixed number of n particles reside on k states

gradient mechanism in a communication network we study the efficiency of the gradient mechanism of message transfer in a two dimensional communication network of regular nodes and randomly distributed hubs each hub on the network is assigned some randomly chosen capacity and hubs with lower capacities are connected to the hubs with maximum capacity the average travel times of single messages traveling on the lattice decrease rapidly as the number of hubs increase the functional dependence of the average travel times on the hub density shows q exponential behavior with a power law tail we also study the relaxation behavior of the network when a large number of messages are created simultaneously at random locations and travel on the network toward their designated destinations for this situation in the absence of the gradient mechanism the network can show congestion effects due to the formation of transport traps we show that if hubs of high betweenness centrality are connected by the gradient mechanism efficient decongestion can be achieved the gradient mechanism is less prone to the formation of traps than other decongestion schemes we also study the spatial configurations of transport traps and propose minimal strategies for their elimination

compactness of the complex green operator on cr manifolds of hypersurface type the purpose of this article is to study compactness of the complex green operator on cr manifolds of hypersurface type we introduce cr pq a potential theoretic condition on 0 q forms that generalizes catlin s property pq to cr manifolds of arbitrary codimension we prove that if an embedded cr manifold of hypersurface type of real dimension at least five satisfies cr pq and cr pn 1 q then the complex green operator is a compact operator on the sobolev spaces h s 0 q m and h s 0 n 1 q m if 1 q n 2 and s 0 we use cr plurisubharmonic functions to build a microlocal norm that controls the totally real direction of the tangent bundle 1 introduction and results in this article we introduce property cr pq a potential theoretic condition on 0 q forms we show that if an embedded cr manifold of hypersurface type satisfies cr pq and cr pn 1 q then the complex green operator is a compact operator on the sobolev spaces h s 0 q m and h s 0 n 1 q m if 1 q n 2 we use cr plurisubharmonic functions to build a microlocal norm that controls the bad direction of the tangent bundle we first prove the closed range and compactness results on l 2 q m and use an elliptic regularization

structure of n clique networks embedded in a complex network we propose the n clique network as a powerful tool for understanding global structures of combined highly interconnected subgraphs and provide theoretical predictions for statistical properties of the n clique networks embedded in a complex network using the degree distribution and the clustering spectrum furthermore using our theoretical predictions we find that the statistical properties are invariant between 3 clique networks and original networks for several observable real world networks with the scale free connectivity and the hierarchical modularity the result implies that structural properties are identical between the 3 clique networks and the original networks

a general geometric growth model for pseudofractal scale free web abstract we propose a general geometric growth model for pseudofractal scale free web psw which is controlled by two tunable parameters we derive exactly the main characteristics of the networks degree distribution second moment of degree distribution degree correlations distribution of clustering coefficient as well as the diameter which are partially determined by the parameters analytical results show that the resulting networks are disassortative and follow power law degree distributions with a more general degree exponent tuned from 2 to 1 ln 3 ln 2 the clustering coefficient of each individual node is inversely proportional to its degree and the average clustering coefficient of all nodes approaches to a large nonzero value in the infinite network order the diameter grows logarithmically with the number of network nodes all these reveal that the networks described by our model have small world effect and scale free topology

tests for interaction in epidemiologic studies a review and a study of power tests for statistical interaction have come into increasing use in epidemiologic analysis with most based on either an additive or multiplicative model for joint effects further procedures have been proposed for testing the goodness of fit and comparing the fit of the latter models this paper reviews the relationships between the various tests and model comparison methods and for the special case of two dichotomous risk factors presents asymptotic power functions for tests of additivity and multiplicativity for a range of sample sizes and factor effects the powers of the tests are computed using both the asymptotic power function and simulation studies the powers of the tests are very low in several commonly encountered situations in addition convergence to the asymptotic distribution appears slow for some of the statistics the results also indicate that likelihood comparison procedures can provide a useful adjunct to the classical hypothesis testing approach

a small world network derived from the deterministic uniform recursive tree by line graph operation the deterministic uniform recursive tree is one of the deterministic versions of the uniform recursive tree zhang et al 2008 eur phys j b 63 507 13 studied the properties of durt including its topological characteristics and spectral properties although durt shows a logarithmic scaling with the size of the network durt is not a small world network since its clustering coefficient is zero lu et al 2012 physica a 391 87 92 proposed a deterministic small world network by adding some edges with a simple rule in each durt iteration in this paper we intoduce a method for constructing a new deterministic small world network by the line graph operation in each durt iteration the line graph operation brings about cliques at each node of the previous given graph and the resulting line graph possesses larger clustering coefficients on the other hand this operation can decrease the diameter at almost one then giving the analytic solutions to several topological characteristics of the model proposed

random boolean network model exhibiting deterministic chaos this paper considers a simple boolean network with n nodes each node s state at time t being determined by a certain number of parent nodes which may vary from one node to another this is an extension of a model studied by andrecut and ali int j mod phys b 15 17 2001 who consider the same number of parents for all nodes we make use of the same boolean rule as andrecut and ali provide a generalization of the formula for the probability of finding a node in state 1 at a time t and use simulation methods to generate consecutive states of the network for both the real system and the model the results match well we study the dynamics of the model through sensitivity of the orbits to initial values bifurcation diagrams and fixed point analysis we show that the route to chaos is due to a cascade of period doubling bifurcations which turn into reversed period halving bifurcations for certain combinations of parameter values

the application of the gsk method to the determination of minimum sample sizes grizzle starmer and koch 1969 biometrics 25 489 503 presented a unified approach for data analysis when the outcome variable is measured on a nominal or ordinal scale the technique uses a weighted least squares methodology and hypotheses are tested using asymptotic chi square statistics in this paper we adapt these procedures to the problem of determining the minimum sample size required for an applied research effort and use the noncentral versions of these chi square statistics the results are compared against several procedures widely used in the literature and are found to concur well with these techniques as well some new situations are considered

universal construction mechanism for networks from one dimensional symbol sequences abstract in this paper we introduce construction mechanisms to generate directed networks from one dimensional symbol sequences we prove that any of these construction mechanism leads to the same undirected network further we prove that the introduced construction mechanisms are universal in the sense that any undirected network can be generated by such a mechanism in addition we provide various numerical examples to demonstrate the applicability of the introduced mechanism

supnorm and f holder estimates for on convex domains of general type in c2 abstract in this paper we study supnorm and modified holder estimates for the integral solution of the equation on a class of convex domains of general type in c 2 that includes many infinite type examples

synchronization diversity and topology of networks of integrate and fire oscillators we study synchronization dynamics of a population of pulse coupled oscillators in particular we focus our attention on the interplay between topological disorder and synchronization features of networks first we analyze synchronization time t in random networks and find a scaling law which relates t to network connectivity then we compare synchronization time for several other topological configurations characterized by a different degree of randomness the analysis shows that regular lattices perform better than a disordered network this fact can be understood by considering the variability in the number of links between two adjacent neighbors this phenomenon is equivalent to having a nonrandom topology with a distribution of interactions and it can be removed by an adequate local normalization of the couplings

boolean game on scale free networks abstract inspired by the local minority game we propose a network boolean game and investigate its dynamical properties on scale free networks the system can self organize to a stable state with better performance than the random choice game although only the local information is available to each agent by introducing the heterogeneity of local interactions we find that the system will achieve the best performance when each agent s interaction frequency is linearly correlated with its information capacity generally the agents with more information gain more than those with less information while in the optimal case each agent almost has the same average profit in addition we investigate the role of irrational factor and find an interesting symmetrical behavior

a small world network derived from the deterministic uniform recursive tree abstract as the deterministic version of the uniform recursive tree urt the deterministic uniform recursive tree durt has been intensively studied by zhang et al 2008 21 they gave several important properties of durt including its topological characteristics and spectral properties although durt shows a logarithmic scaling with the size of the network durt is not a small world network since its clustering coefficient is zero in this paper we propose a new deterministic small world network by adding some edges with a simple rule in each durt iteration and then give the analytic solutions to several topological characteristics of the model proposed

comparison of the bergman and szego kernels abstract the quotient of the szego and bergman kernels for a smooth bounded pseudoconvex domains in c n is bounded from above by a constant multiple of log p for any p n where is the distance to the boundary for a class of domains that includes those of d angelo finite type and those with plurisubharmonic defining functions the quotient is also bounded from below by a constant multiple of log p for any p 1 moreover for convex domains the quotient is bounded from above and below by constant multiples of

real hypersurfaces orders of contact and applications a fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the cauchy riemann equations suppose that d is an open domain in cn and that its boundary m is a smooth real submanifold of cn how does the geometry of m influence the function theory on d one approach to this question is the a neumann problem 6 there one studies the inhomogeneous cauchy riemann equations au a where a is a 0 q form with distribution coefficients on the closed domain d u m satisfying the necessary compatibility condition a a 0 the a neumann problem constructs a particular solution u which is orthogonal to the null space of the complex laplacian kohn 10a has solved this problem and has shown that good local regularity properties for u in terms of a follow from so called subelliptic estimates it is natural to seek geometric conditions on m that are necessary and sufficient for these estimates in this paper we will be concerned with a certain geometric condition on m that we feel is the right one for subellipticity although the motivation for the questions discussed here comes from partial differential equations the techniques come from algebraic geometry to see why we state theorems of kohn greiner and catlin concerning subellipticity kohn 1oa has discovered a sufficient condition for a subelliptic estimate near a fixed point p in m in case d is pseudoconvex and is contained in c2 this condition is equivalent to the finiteness of order of tangency of every complex analytic one dimensional manifold with m at p greiner 13 established the necessity of this condition in

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

embedding binary trees into crossed cubes the recently introduced interconnection network crossed cube has attracted much attention in the parallel processing area due to its many attractive features like the ordinary hypercube the n dimensional crossed cube is a regular graph with 2 sup n vertices and n2 sup n 1 edges the diameter of the crossed cube is approximately half that of the ordinary hypercube these advantages of the crossed cube motivated the study of how well it can simulate other networks such as the complete binary tree we show that the 2 sup n 1 node complete binary tree can be embedded into the n dimensional crossed cube with dilation 1

superlogarithmic estimates on pseudoconvex domains and cr manifolds this paper is concerned with proving superlogarithmic estimates for the operator r b on pseudoconvex cr manifolds and using them to establish hypoellipticity of lb and of the neumann problem these estimates are established under the assumption that subellipticity degenerates in certain specified ways

lobby index in networks abstract we propose a new node centrality measure in networks the lobby index which is inspired by hirsch s h index it is shown that in scale free networks with exponent the distribution of the l index has power tail with exponent 1 properties of the l index and extensions are discussed

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

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

stochastic resonance in coupled threshold elements on a barabasi albert network abstract stochastic resonance is investigated in a system of threshold elements located at nodes and coupled along edges of a barabasi albert network driven by a common subthreshold periodic signal and independent noises array enhanced stochastic resonance is observed i e increase of the spectral power amplification evaluated from the mean output of the network due to proper coupling this enhancement occurs though the response of individual threshold elements to the periodic signal is very diverse due to a power law distribution of their connectivity numerical results are qualitatively explained using simple linear response theory in the mean field approximation

power calculations for likelihood ratio tests in generalized linear models summary recently self and mauritsen 1988 biometrics 44 79 86 described an approach for sample size power calculations within the framework of generalized linear models that is based on a noncentral chi square approximation to the distribution of score statistics in this work we present another approach to this same problem that is based on a noncentral chi square approximation to the distribution of the likelihood ratio statistic this approach is easier to implement than the previous one and simulation studies indicate that it is accurate over a much wider range of parameter values and data configurations than the earlier method the new method is also applied to the problem of power calculations for 1 k matched case control studies which is related to but formally outside of the generalized linear model framework

deterministic weighted scale free small world networks abstract we propose a deterministic weighted scale free small world model for considering pseudofractal web with the co evolution of topology and weight considering the fluctuations in traffic flow constitute a main reason for congestion of packet delivery and poor performance of communication networks we suggest a recursive algorithm to generate the network which restricts the traffic fluctuations on it effectively during the evolutionary process we provide a relatively complete view of topological structure and weight dynamics characteristics of the networks such as weight and strength distribution degree correlations average clustering coefficient and degree cluster correlations as well as the diameter

power sample size and smallest detectable effect determination for multivariate studies this paper discusses some general methods for determining approximate power sample size and smallest detectable effect for studies of multiple risk factors these methods are based on standard large sample formulae for determining the power of chi square tests and emphasis is given to determinations for pearson chi 2 tests in multiway contingency tables the methods are illustrated in application to the design of a clinical trial of the preventive effect of alpha tocopherol ascorbic acid and beta carotene on colon polyp recurrence and a case control study of the joint effect of smoking and asbestos exposure on lung cancer incidence

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

sample size for logistic regression with small response probability abstract the fisher information matrix for the estimated parameters in a multiple logistic regression can be approximated by the augmented hessian matrix of the moment generating function for the covariates the approximation is valid when the probability of response is small with its use one can obtain a simple closed form estimate of the asymptotic covariance matrix of the maximum likelihood parameter estimates and thus approximate sample sizes needed to test hypotheses about the parameters the method is developed for selected distributions of a single covariate and for a class of exponential type distributions of several covariates it is illustrated with an example concerning risk factors for coronary heart disease