a small world of weak ties provides optimal global integration of self similar modules in functional brain networks the human brain is organized in functional modules such an organization presents a basic conundrum modules ought to be sufficiently independent to guarantee functional specialization and sufficiently connected to bind multiple processors for efficient information transfer it is commonly accepted that small world architecture of short paths and large local clustering may solve this problem however there is intrinsic tension between shortcuts generating small worlds and the persistence of modularity a global property unrelated to local clustering here we present a possible solution to this puzzle we first show that a modified percolation theory can define a set of hierarchically organized modules made of strong links in functional brain networks these modules are large world self similar structures and therefore are far from being small world however incorporating weaker ties to the network converts it into a small world preserving an underlying backbone of well defined modules remarkably weak ties are precisely organized as predicted by theory maximizing information transfer with minimal wiring cost this trade off architecture is reminiscent of the strength of weak ties crucial concept of social networks such a design suggests a natural solution to the paradox of efficient information flow in the highly modular structure of the brain

growing scale free networks with tunable clustering we extend the standard scale free network model to include a triad formation step we analyze the geometric properties of networks generated by this algorithm both analytically and by numerical c

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

inferring a possibility distribution from empirical data several transformations from probabilities to possibilities have been proposed in particular dubois and prade s procedure produces the most specific possibility distribution among the ones dominating a given probability distribution in this paper this method is generalized to the case where the probabilities are unknown the only information being a data sample represented by a histogram it is proposed to characterize the probabilities of the different classes by simultaneous confidence intervals with a given confidence level email protected from this imprecise specification a procedure for constructing a possibility distribution is described insuring that the resulting possibility distribution will dominate the true probability distribution in at least 100 email protected of the cases finally a simple efficient algorithm is given which makes the computations tractable even if the number of classes is high

the fractal dimension of percolation cluster hulls large simulations of 2d site percolation on a square lattice provide several estimates of the fractal dimension dh of the hull or outer boundary of percolation clusters close to criticality statistical estimates based on an ensemble of more than 106 clusters with varying sizes yield dh 1 74 or 0 02 geometrical measurements of the hull two point correlation for individual clusters with more than 105 sites give dh 1 76 or 0 01 the observed scaling behaviour is the same both slightly above and below pc

real space renormalization from a geometric fractal perspective abstract an introduction to real space renormalization transformation rt is presented with geometric fractal aspects as focus critical exponents are equivalent to fractal dimensions details of the construction rule of deterministic fractals and the definition of their fractal dimensions illustrate differences in approach between field theory and condensed matter physics the appearance of fractal structures is typically less obvious within the context of partition functions equilibrium statistical mechanics and field theory and master equations dynamic processes an rt amounts to a reformulation of those construction rules into a form similar to the conventional ones for deterministic fractals but on an algebraic level without ever explicitly referring to the geometric structure this is illustrated explicitly in the context of the sierpinsky gasket its construction rule can be reformulated as a growth process and also as a partition function

fractality and the small world effect in sierpinski graphs although some real networks exhibit self similarity there is no standard definition of fractality in graphs on the other hand the small world phenomenon is one of the most important common properties of real interconnection networks in this paper we relate these two properties in order to do so we focus on the family of sierpinski networks for the sierpinski gasket the sierpinski carpet and the sierpinski tetra we give the basic properties and we calculate the box counting dimension as a measure of their fractality we also define a deterministic family of graphs which we call small world sierpinski graphs we show that our construction preserves the structure of sierpinski graphs including its box counting dimension while the small world phenomenon arises thus in this family of graphs fractality and small world effect are simultaneously present

multiple comparisons for analyzing dichotomous response dichotomous response models are common in many experimental settings statistical parameters of interest are typically the probabilities pi that an experimental unit will respond at the various treatment levels herein simultaneous procedures are considered for multiple comparisons among these probabilities with attention directed at construction of simultaneous confidence intervals for various functions of the pi the inferences are based on the asymptotic normality of the maximum likelihood estimator of pi specific applications include all pairwise comparisons and comparisons with a fixed control treatment monte carlo evaluations are undertaken to examine the small sample properties of the various procedures it is seen that use of the usual estimates of variance consistently leads to less than nominal empirical coverage for most sample sizes examined for very large samples total size greater than about 300 nominal coverage is achieved a reformulation of the pairwise comparisons using a construction noted by beal 1987 biometrics 43 941 950 is shown to exhibit generally nominal empirical coverage characteristics and is recommended for use with small to moderate sample sizes

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

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

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

network formation determined by the diffusion process of random walkers we studied the diffusion process of random walkers in networks formed by their traces this model considers the rise and fall of links determined by the frequency of transports of random walkers in order to examine the relation between the formed network and the diffusion process a situation in which multiple random walkers start from the same vertex is investigated the difference in diffusion rate of random walkers according to the difference in dimension of the initial lattice is very important for determining the time evolution of the networks for example complete subgraphs can be formed on a one dimensional lattice while a graph with a power law vertex degree distribution is formed on a two dimensional lattice we derived some formulae for predicting network changes for the 1d case such as the time evolution of the size of nearly complete subgraphs and conditions for their collapse the networks formed on the 2d lattice are characterized by the existence of clusters of highly connected vertices and their life time as the life time of such clusters tends to be small the exponent of the power law distribution changes from 1 2 to 3

highly clustered scale free networks we propose a model for growing networks based on a finite memory of the nodes the model shows stylized features of real world networks power law distribution of degree linear preferential attachment of new links and a negative correlation between the age of a node and its link attachment rate notably the degree distribution is conserved even though only the most recently grown part of the network is considered as the network grows the clustering reaches an asymptotic value larger than that for regular lattices of the same average connectivity and similar to the one observed in the networks of movie actors coauthorship in science and word synonyms these highly clustered scale free networks indicate that memory effects are crucial for a correct description of the dynamics of growing networks

network formed by traces of random walks abstract we propose a model of time evolving networks in which a kind of transport between vertices generates new edges in the graph we call the model network formed by traces of random walks because the transports are represented abstractly by random walks our numerical calculations yield several important properties observed commonly in complex networks although the graph at initial time is only a one dimensional lattice for example the distribution of vertex degree exhibits various behaviors such as exponential power law like and bi modal distribution according to change of probability of extinction of edges another property such as strong clustering structure and small mean vertex vertex distance can also be found the transports represented by random walks in a framework of strong links between regular lattice is a new mechanisms which yields biased acquisition of links for vertices

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

fractal small world dichotomy in real world networks we draw attention to a clear dichotomy between small world networks exhibiting exponential neighborhood growth and fractal like networks where neighborhoods grow according to a power law this distinction is observed in a number of real world networks and is related to the degree correlations and geographical constraints we conclude by pointing out that the status of human social networks in this dichotomy is far from clear

self similar scale free networks and disassortativity self similar networks with scale free degree distribution have recently attracted much attention since these apparently incompatible properties were reconciled in c song s havlin and h a makse nature 433 392 2005 by an appropriate box counting method that enters the measurement of the fractal dimension we study two genetic regulatory networks saccharomyces cerevisiae n m luscombe m m babu h yu m snyder s teichmann and m gerstein nature 431 308 2004 and escherichia coli http www ccg unam mx computationalgenomics regulondb datasets regulonnetdatasets html and http www gbf de systemsbiology and show their self similar and scale free features in extension to the datasets studied by c song s havlin and h a makse nature 433 392 2005 moreover by a number of numerical results we support the conjecture that self similar scale free networks are not assortative from our simulations so far these networks seem to be disassortative instead we also find that the qualitative feature of disassortativity is scale invariant under renormalization but it appears as an intrinsic feature of the renormalization prescription as even assortative networks become disassortative after a sufficient number of renormalization steps

fractal networks induced by movements of random walkers on a tree graph abstract we show that a fractal scale free and small world network can grow on a non fractal cayley tree through the simple random process of shortcut creation by movements of random walkers the cayley tree provides a stage for the walkers to diffuse into a locally one dimensional structure with the small world property self organization of fractal graph by adding shortcut edges to the locally one dimensional structure generated by walkers flow from the root vertex to the outermost shell of the cayley tree is a novel mechanism for generating fractal graphs also we show numerically that the fractal box counting dimension and a power law exponent describing the degree distribution of the resulting graph are mainly determined by the bifurcation number of the original cayley tree in our model the fractal cluster dimension must change with the size of the evolving graph this is a general result that resolves an apparent contradiction between the fractality in the context of the cluster growing method and the small world property of the graph

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

reciprocal relation between the fractal and the small world properties of complex networks the fractal and the small world properties of complex networks are systematically studied both in the box covering bc and the cluster growing cg measurements we elucidate that complex networks possessing the fractal small world nature in the bc measurement are always fractal small world even in the cg measurement and vice versa while the fractal dimensions d text b by the bc measurement and d text c by the cg measurement are generally different this implies that two structural properties of networks fractality and small worldness cannot coexist in the same length scale these properties can however crossover from one to the other by varying the length scale we show that the crossover behavior in a network near the percolation transition appears both in the bc and cg measurements and is scaled by a unique characteristic length ensuremath xi

generating a set of association and decision rules with statistically representative support and anti support abstract the problem of rule evaluation is central to association and decision rule mining in the literature many attractiveness measures assessing the utility of discovered rules have been proposed in particular two dimensional evaluation planes like support confidence or support anti support are a frequently employed scheme the inherent drawback of attractiveness measures is the necessity to provide a cutoff threshold which defines the minimal or maximal acceptable value of a given measure such threshold is usually unintuitive meaningless and difficult to impose with objectivity in this paper we focus on the support anti support evaluation plane we propose a methodology of simultaneously assessing statistical representativeness of both measures by performing multinomial tests on association or decision rules the proposed approach combines assessing statistical soundness of the rules with an implicit eliciting of a threshold in both measures from the user the latter is accomplished by employing the notion of a relative error in both support and anti support we evaluate the proposed method on a number of data sets and provide general conclusions

simultaneous confidence intervals and sample size determination for multinomial proportions abstract simultaneous confidence interval procedures for multinomial proportions are used in many areas of science in this article two new simultaneous confidence interval procedures are introduced numerical results are presented to evaluate these procedures and compare their performance with established methods that have been used in statistical literature from the results presented in this article it is evident that the new procedures are more accurate than the established ones where the accuracy of the procedure is measured by the volume of the confidence region corresponding to the nominal coverage probability and the probability of coverage it achieves in the sample size determination problem the new procedures provide a sizable amount of savings as compared to the procedures that have been used in many applications because both procedures performed equally well the procedure that requires the least amount of computing time is recommended

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

nonequilibrium transitions in complex networks a model of social interaction we analyze the nonequilibrium order disorder transition of axelrod s model of social interaction in several complex networks in a small world network we find a transition between an ordered homogeneous state and a disordered state the transition point is shifted by the degree of spatial disorder of the underlying network the network disorder favoring ordered configurations in random scale free networks the transition is only observed for finite size systems showing system size scaling while in the thermodynamic limit only ordered configurations are always obtained thus in the thermodynamic limit the transition disappears however in structured scale free networks the phase transition between an ordered and a disordered phase is restored

boundary invariants of pseudoconvex domains the purpose of this paper is to describe an invariant ax z which is associated with each point z in the boundary of a smoothly bounded pseudoconvex domain in cn the motivation for defining the invariant ax z referred to as the multitype of z comes from the study of the boundary regularity properties of solutions of the d neumann problem on domains of finite type by a domain of finite type we mean one that satisfies the definition given by d angelo 5 namely a domain such that at any boundary point the maximum order of contact of one dimensional complex analytic varieties with the boundary is bounded in 4 the author showed how one can give a proof of global regularity of the a neumann problem for pseudoconvex domains of finite type if there is an invariant such as the one described in this paper in a forthcoming article this invariant will be used to obtain a proof of subelliptic estimates for the same class of domains before giving the definition of the multitype ax z we must introduce some

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

a model for social networks social networks are organized into communities with dense internal connections giving rise to high values of the clustering coefficient in addition these networks have been observed to be assortative i e highly connected vertices tend to connect to other highly connected vertices and have broad degree distributions we present a model for an undirected growing network which reproduces these characteristics with the aim of producing efficiently very large networks to be used as platforms for studying sociodynamic phenomena the communities arise from a mixture of random attachment and implicit preferential attachment the structural properties of the model are studied analytically and numerically using the k clique method for quantifying the communities r 2006 elsevier b v all rights reserved

fractality in complex networks critical and supercritical skeletons fractal scaling char22 a power law behavior of the number of boxes needed to tile a given network with respect to the lateral size of the box char22 is studied we introduce a box covering algorithm that is a modified version of the original algorithm introduced by song et al nature london 433 392 2005 this algorithm enables easy implementation fractal networks are viewed as comprising a skeleton and shortcuts the skeleton embedded underneath the original network is a special type of spanning tree based on the edge betweenness centrality it provides a scaffold for the fractality of the network when the skeleton is regarded as a branching tree it exhibits a plateau in the mean branching number as a function of the distance from a root for nonfractal networks on the other hand the mean branching number decays to zero without forming a plateau based on these observations we construct a fractal network model by combining a random branching tree and local shortcuts the scaffold branching tree can be either critical or supercritical depending on the small worldness of a given network for the network constructed from the critical supercritical branching tree the average number of vertices within a given box grows with the lateral size of the box according to a power law an exponential form in the cluster growing method the critical and supercritical skeletons are observed in protein interaction networks and the world wide web respectively the distribution of box masses i e the number of vertices within each box follows a power law p m m ensuremath sim m ensuremath ensuremath eta the exponent ensuremath eta depends on the box lateral size ensuremath ell b for small values of ensuremath ell b ensuremath eta is equal to the degree exponent ensuremath gamma of a given scale free network whereas ensuremath eta approaches the exponent ensuremath tau ensuremath gamma ensuremath gamma ensuremath 1 as ensuremath ell b increases which is the exponent of the cluster size distribution of the random branching tree finally we study the perimeter h ensuremath alpha of a given box ensuremath alpha i e the number of edges connected to different boxes from a given box ensuremath alpha as a function of the box mass m b ensuremath alpha it is obtained that the average perimeter over the boxes with box mass m b is likely to scale as h m b ensuremath sim m b irrespective of the box size ensuremath ell b

dynamics of majority rule in two state interacting spin systems we introduce a two state opinion dynamics model where agents evolve by majority rule in each update a group of agents is specified whose members then all adopt the local majority state in the mean field limit where a group consists of randomly selected agents consensus is reached in a time that scales ln n where n is the number of agents on finite dimensional lattices where a group is a contiguous cluster the consensus time fluctuates strongly between realizations and grows as a dimension dependent power of n the upper critical dimension appears to be larger than 4 the final opinion always equals that of the initial majority except in one dimension

impact of initial lattice structures on networks generated by traces of random walks abstract we show that the platform stage of network evolution plays a principal role in the topology of resulting networks generated by short cuts stimulated by the movements of a random walker the mechanism of which tends to produce power law degree distributions to examine the numerical results we have developed a statistical method which relates the power law exponent to random properties of the subgraph developed in the platform stage as a result we find that an important exponent in the network evolution is which characterizes the size of the subgraph in the form v t where v and t denote the number of vertices in the subgraph and the time variable respectively 2d lattices can impose specific limitations on the walker s diffusion which keeps the value of within a moderate range and provides typical properties of complex networks 1d and 3d cases correspond to different ends of the spectrum for with 2d cases in between especially for 2d square lattices a discontinuous change of the network structure is observed which varies according to whether is greater or less than 2 for 1d cases we show that emergence of nearly complete subgraphs is guaranteed by 1 2 although the transient power law is permitted at low increase rates of edges additionally the model exhibits a spontaneous emergence of highly clustered structures regardless of its initial structure

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

quick simultaneous confidence intervals for multinomial proportions abstract opinion polls often give an indication of sampling error based on the standard conservative confidence intervals for a single binomial proportion we give a lower bound for the simultaneous coverage probability when such intervals are applied to all of the proportions in a poll this bound is equal to 1 2 for small values of the same result is shown to apply to the standard confidence intervals for changes in proportions between surveys

tree networks with causal structure a geometry of networks endowed with a causal structure is discussed using the conventional framework of the equilibrium statistical mechanics the popular growing network models appear as particular causal models we focus on a class of tree graphs an analytically solvable case general formulas are derived describing the degree distribution the ancestor descendant correlation and the probability that a randomly chosen node lives at a given geodesic distance from the root it is shown that the hausdorff dimension d h of the causal networks is generically infinite in contrast to the maximally random trees where it is generically finite

a review of fractality and self similarity in complex networks we review recent findings of self similarity in complex networks using the box covering technique it was shown that many networks present a fractal behavior which is seemingly in contrast to their small world property moreover even non fractal networks have been shown to present a self similar picture under renormalization of the length scale these results have an important effect in our understanding of the evolution and behavior of such systems a large number of network properties can now be described through a set of simple scaling exponents in analogy with traditional fractal theory

subellipticity of the neumann problem on a weakly q pseudoconvex concave domain abstract for a domain d of c n which is weakly q pseudoconvex or q pseudoconcave we give a sufficient condition for subelliptic estimates for the neumann problem this extends to domains which are not necessarily pseudoconvex the results and the techniques of catlin 1987 3