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

flat g bundles with canonical metrics on considere la relation entre la theorie invariante des actions de groupes algebriques semi simples sur des varietes algebriques complexes et le comportement de l application de moment et des donnees quantiques fournies par une structure symplectique associee on etudie un exemple de dimension infinie comprenant des fibres plats sur des varietes de riemann compactes

models of social networks based on social distance attachment we propose a class of models of social network formation based on a mathematical abstraction of the concept of social distance social distance attachment is represented by the tendency of peers to establish acquaintances via a decreasing function of the relative distance in a representative social space we derive analytical results corroborated by extensive numerical simulations showing that the model reproduces the main statistical characteristics of real social networks large clustering coefficient positive degree correlations and the emergence of a hierarchy of communities the model is confronted with the social network formed by people that shares confidential information using the pretty good privacy pgp encryption algorithm the so called web of trust of pgp

uniformization of mathcal g bundles we show some of the conjectures of pappas and rapoport concerning the moduli stack bung of g torsors on a curve c where g is a semisimple bruhat tits groupschemeon c inparticularweprovetheanalogoftheuniformizationtheoremof drinfeld simpson in this setting furthermore we apply this to compute the connected components of these moduli stacks and to calculate the picard group of bung in case g is simply connected

evolution of a social network the role of cultural diversity we present a simple deterministic and based on local rules model of evolving social network which leads to a network with the properties of a real social system e g small world topology and assortative mixing the state of an individual si is characterized by the values of q cultural features drawn from gaussian distribution with variance sigma the other control parameter is sociability ti which describes the maximal number of connections of an individual the state of individuals and connections between them evolve in time as results from numerical computations an initial diversity of cultural features in a community has an essential influence on an evolution of social network it was found that for a critical value of control parameter sigma c q there is a structural transition and a hierarchical network with small world topology of connections and a high clustering coefficient emerges the emergence of small world properties can be related to the creation of subculture groups in a community the power law relation between the clustering coefficient of a node and its connectivity c k approximately k beta was observed in the case of a scale free distribution of sociability ti and a high enough cultural diversity in a population

convex functions on symmetric spaces side lengths of polygons and the stability inequalities for weighted configurations at infinity in a symmetric space of noncompact type x g k oriented geodesic segments correspond modulo isometries to vectors in the euclidean weyl chamber we can hence assign vector valued lengths to segments our main result is a system of homogeneous linear inequalities which we call the generalized triangle inequalities or stability inequalities describing the restrictions on the vector valued side lengths of oriented polygons it is based on the mod 2 schubert calculus in the real grassmannians g p for maximal parabolic subgroups p the side lengths of polygons in euclidean buildings are studied in the related paper klm2 applications of the geometric results in both papers to algebraic group theory are given in klm3

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

prehomogeneous spaces associated with complex nilpotent orbits abstract we compute the irreducible components and relative invariants of certain prehomogeneous spaces arising in the theory of nilpotent orbits in complex reductive lie algebras

constructing narmax models using armax models this paper outlines how it is possible to decompose a complex non linear modelling problem into a set of simpler linear modelling problems local armax models valid within certain operating regimes are interpolated to construct a global narmax non linear narmax model knowledge of the system behaviour in terms of operating regimes is the primary basis for building such models hence it should not be considered as a pure black box approach but as an approach that utilizes a limited amount of a priori system knowledge it is shown that a large class of non linear systems can be modelled in this way and indicated how to decompose the systems range of operation into operating regimes standard system identification algorithms can be used to identify the narmax model and several aspects of the system identification problem are discussed and illustrated by a simulation example

fox h functions in fractional diffusion the h functions introduced by fox in 1961 are special functions of a very general nature which allow one to treat several phenomena including anomalous diffusion in a unified and elegant framework in this paper we express the fundamental solutions of the cauchy problem for the space time fractional diffusion equation in terms of proper fox h functions based on their mellin barnes integral representations we pay attention to the particular cases of space fractional time fractional and neutral fractional diffusion

local affinity in heterogeneous growing networks abstract in this paper we present a study of the influence of local affinity in heterogeneous preferential attachment pa networks heterogeneous pa models are a generalization of the barabasi albert model to heterogeneous networks where the affinity between nodes biases the attachment probability of links threshold models are a class of heterogeneous pa models where the affinity between nodes is inversely related to the distance between their states we propose a generalization of threshold models where network nodes have individual affinity functions which are then combined to yield the affinity of each potential interaction we analyze the influence of the affinity functions in the topological properties averaged over a network ensemble the network topology is evaluated through the distributions of connectivity degrees clustering coefficients and geodesic distances we show that the relaxation of the criterion of a single global affinity still leads to a reasonable power law scaling in the connectivity and clustering distributions under a wide spectrum of assumptions we also show that the richer behavior of the model often exhibits a better agreement with the empirical observations on real networks

state space control system design based on non minimal state variable feedback further generalization and unification results this paper shows how proportional integral plus linear quadratic pip lq control based on non minimal state space nmss control system design can be constrained to yield exactly the same control algorithm as both generalized predictive control gpc and standard minimal state linear quadratic gaussian lqg design methods however while nmss includes these other approaches as special cases it is less constrained and so more flexible in general terms for example while pip lq has the simplicity of gpc it is formulated like lqg in the powerful context of state variable feedback svf control which allows for ready access to modern robust control methods furthermore the paper suggests that the nmss approach provides the greater design freedom with a wider range of possible lq solutions than its minimal state space equivalent

classification of nilpotent elements in simple exceptional real lie algebras of inner type and description of their centralizers abstract let g0 be a connected lie group whose lie algebra g0 is a simple exceptional non compact real lie algebra of inner type up to isomorphism there are 10 such algebras we classify the nilpotent orbits of g0 in g0 for the adjoint action by using kostant s conjecture which was recently proved by j sekiguchi and the author independently for each nilpotent element e g0 e 0 we determine the dimension of the orbit and the levi factor of the centralizer of e in g0

extremum statistics in scale free network models we investigate the statistics of the most connected node in scale free networks for a scale free network model with homogeneous nodes we show by means of extensive simulations that the exponential truncation due to the finite size of the network of the degree distribution governs the scaling of the extreme values and that the distribution of maxima follows the gumbel statistics for a scale free network model with heterogeneous nodes we show that scaling no longer holds and that the truncation of the degree distribution no longer controls the maxima distribution

component groups of centralizers of nilpotents in complex symmetric spaces abstract let g be the adjoint group of a simple lie algebra g and let kc aut p c be the complexified isotropy representation at the identity coset of the corresponding symmetric space if e p c is nilpotent we consider the centralizer of e in kc we show that the conjugacy classes of the component group of this centralizer can be described in terms generalizing the bala carter classification of nilpotent orbits in the complexification of g

global culture a noise induced transition in finite systems we analyze the effect of cultural drift modeled as noise in axelrod s model for the dissemination of culture the disordered multicultural frozen configurations are found not to be stable this general result is proven rigorously in d51 where the dynamics is described in terms of a lyapunov potential in d52 the dynamics is governed by the average relaxation time t of perturbations noise at a rate r t 21 induces monocultural configurations whereasr t 21 sustains disorder in the thermodynamic limit the relaxation time diverges and global polarization persists in spite of a dynamics of local convergence

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

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

some remarks on richardson orbits in complex symmetric spaces roger w richardson proved that any parabolic subgroup of a complex semisimple lie group admits an open dense orbit in the nilradical of its corresponding parabolic subalgebra in the case of complex symmetric spaces we show that there exist some large classes of parabolic subgroups for which the analogous statement which fails in general is true our main contribution is the extension of a theorem of peter e trapa in 2005 to real semisimple exceptional lie groups

coevolution of glauber like ising dynamics and topology we study the coevolution of a generalized glauber dynamics for ising spins with tunable threshold and of the graph topology where the dynamics takes place this simple coevolution dynamics generates a rich phase diagram in the space of the two parameters of the model the threshold and the rewiring probability the diagram displays phase transitions of different types spin ordering percolation and connectedness at variance with traditional coevolution models in which all spins of each connected component of the graph have equal value in the stationary state we find that for suitable choices of the parameters the system may converge to a state in which spins of opposite sign coexist in the same component organized in compact clusters of like signed spins mean field calculations enable one to estimate some features of the phase diagram

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

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

parabolic higgs bundles and representations of the fundamental group of a punctured surface into a real group abstract we study parabolic g higgs bundles over a compact riemann surface with fixed punctures when g is a real reductive lie group and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in g with arbitrary holonomy around the punctures this generalizes simpson s results for gl n c to arbitrary complex and real reductive lie groups three interesting features are the relation between the parabolic degree and the tits geometry of the boundary at infinity of the symmetric space the treatment of the case when the logarithm of the monodromy is on the boundary of a weyl alcove and the correspondence of the orbits encoding the singularity via the kostant sekiguchi correspondence we also describe some special features of the moduli spaces when g is a split real form or a group of hermitian type

search in power law networks many communication and social networks have power law link distributions containing a few nodes that have a very high degree and many with low degree the high connectivity nodes play the important role of hubs in communication and networking a fact that can be exploited when designing efficient search algorithms we introduce a number of local search strategies that utilize high degree nodes in power law graphs and that have costs scaling sublinearly with the size of the graph we also demonstrate the utility of these strategies on the gnutella peer to peer network

identification of non linear stochastic systems by state dependent parameter estimation the paper outlines how improved estimates of time variable parameters in models of stochastic dynamic systems can be obtained using recursive filtering and fixed interval smoothing techniques with the associated hyper parameters optimized by maximum likelihood based on prediction error decomposition it then shows how by exploiting special data re ordering and back fitting procedures similar recursive parameter estimation techniques can be utilized to estimate much more rapid state dependent parameter sdp variations in this manner it is possible to identify and estimate a widely applicable class of nonlinear stochastic systems as illustrated by several examples that include simulated and real data from chaotic processes finally the paper points out that such sdp models can form the basis for new methods of signal processing automatic control and state estimation for nonlinear stochastic systems

memory effect in growing trees abstract we show that the structure of a growing tree preserves an information on the shape of an initial graph for the exponential trees evidence of this kind of memory is provided by means of the iterative equations derived for the moments of the node node distance distribution numerical calculations confirm the result and allow to extend the conclusion to the barabasi albert scale free trees the memory effect almost disappears if subsequent nodes are connected to the network with more than one link

comments on identification of non linear parametrically varying models using separable least squares by f previdi and m lovera black box or open box this note compares and contrasts the non linear parameter varying nlpv and state dependent parameter sdp model classes it shows that while they have similarities the two stage sdp modelling procedure involving non parametric identification followed by parametric estimation is quite different from the single stage nlpv procedure in particular the sdp procedure allows for the identification of the model structure and the nature of the non linearities prior to the estimation of the parameters that characterize this identified model structure in contrast to nlpv modelling therefore sdp estimation opens up the black box and reveals the inner nature of the non linear system

a hilbert mumford criterion for polystability in kaehler geometry consider a hamiltonian action of a compact lie group k on a kaehler manifold x with moment map x assume that the action of k extends to a holomorphic action of the complexification g of k we characterize which g orbits in x intersect 1 0 in terms of the maximal weights lim t e its x s where s we do not impose any a priori restriction on the stabilizer of x under some mild restrictions on the action k x we view the maximal weights as defining a collection of maps for each x x x k g where k g is the boundary at infinity of the symmetric space k g we prove that g x 1 0 0 if 1 x is everywhere nonnegative 2 any boundary point y such that x y 0 can be connected with a geodesic in k g to another boundary point y satisfying x y 0 we also prove that the maximal weight functions are g equivariant for any g g and any y k g we have g x y x y g

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

normal forms of elements of classical real and complex lie and jordan algebras elements of the classical complex and real lie and jordan algebras with involutions are classified into conjugacy classes under the action of the corresponding classical lie group normal forms of representatives of each conjugacy class are chosen so as to resemble the jordan normal forms of n n complex matrices for completeness similar results are given for gl n c gl n r and gl n h

surface group representations with maximal toledo invariant we develop the theory of maximal representations of the fundamental group 1 of a compact connected oriented surface possibly with boundary into lie groups g of hermitian type for any homomorphism p 1 g we define the toledo invariant t p a numerical invariant which has both topological and analytical interpretations we establish important properties of t among which continuity uniform boundedness on the representation variety additivity under connected sum of surfaces and congruence relations mod we thus obtain information about the representation variety as well as striking geometric properties of maximal representations that is representations whose toledo invariant achieves the maximum value moreover we establish properties of boundary maps associated to maximal representations which generalize naturally monotonicity properties of semiconjugations of the circle we define a rotation number function for general locally compact groups and study it in detail for groups of hermitian type properties of the rotation number together with the existence of boundary maps lead to additional invariants for maximal representations and show that the subset of maximal representations is always real semialgebraic

correlations in scale free networks tomography and percolation we discuss three related models of scale free networks with the same degree distribution but different correlation properties starting from the barabasi albert construction based on growth and preferential attachment we discuss two other networks emerging when randomizing it with respect to links or nodes we point out that the barabasi albert model displays dissortative behavior with respect to the nodes degrees while the node randomized network shows assortative mixing these kinds of correlations are visualized by discussing the shell structure of the networks around an arbitrary node in spite of different correlation behaviors all three constructions exhibit similar percolation properties this result for percolation is also detected for a network with finite second moment and its corresponding randomized models

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

parabolic higgs bundles and teichmuller spaces for punctured surfaces in this paper we study the relation between parabolic higgs vector bundles and irreducible representations of the fundamental group of punctured riemann surfaces established by simpson we generalize a result of hitchin identifying those parabolic higgs bundles that correspond to fuchsian representations we also study the higgs bundles that give representations whose image is contained after conjugation in sl k r we compute the real dimension of one of the components of this space of representations which in the absence of punctures is the generalized teichmuller space introduced by hitchin and which in the case of k 2 is the usual teichmuller space of the punctured surface

kleinberg navigation in fractal small world networks we study the kleinberg problem of navigation in small world networks when the underlying lattice is a fractal consisting of n ensuremath gg 1 nodes our extensive numerical simulations confirm the prediction that the most efficient navigation is attained when the length r of long range links is taken from the distribution p mathbf r ensuremath sim r ensuremath ensuremath alpha where ensuremath alpha d mathrm f is the fractal dimension of the underlying lattice we find finite size corrections to the exponent ensuremath alpha proportional to 1 mathrm ln phantom rule 0 2em 0ex n 2

statistics of changes in lead node in connectivity driven networks we study statistical properties of the highest degree or most popular nodes in growing networks we show that the number of lead changes increases logarithmically with network size n independent of the details of the growth mechanism the probability that the first node retains the lead approaches a finite constant for popularity driven growth and decays as n phi ln n 1 2 with phi 0 08607 em leader for growth with no popularity bias

nonlinear model validation using correlation tests new higher order correlation tests which use model residuals combined with system inputs and outputs are presented to check the validity of a general class of nonlinear models the new method is illustrated by testing both simple and complex nonlinear system models

sampling properties of random graphs the degree distribution we discuss two sampling schemes for selecting random subnets from a network random sampling and connectivity dependent sampling and investigate how the degree distribution of a node in the network is affected by the two types of sampling here we derive a necessary and sufficient condition that guarantees that the degree distributions of the subnet and the true network belong to the same family of probability distributions for completely random sampling of nodes we find that this condition is satisfied by classical random graphs for the vast majority of networks this condition will however not be met we furthermore discuss the case where the probability of sampling a node depends on the degree of a node and we find that even classical random graphs are no longer closed under this sampling regime we conclude by relating the results to real eschericia coli protein interaction network data

clustering properties of a generalized critical euclidean network many real world networks exhibit a scale free feature have a small diameter and a high clustering tendency we study the properties of a growing network which has all these features in which an incoming node is connected to its ith predecessor of degree k i with a link of length l using a probability proportional to k beta i l alpha for alpha 0 5 the network is scale free at beta 1 with the degree distribution p k proportional to k gamma and gamma 3 0 as in the barabasi albert model alpha 0 beta 1 we find a phase boundary in the alpha beta plane along which the network is scale free interestingly we find a scale free behavior even for beta 1 for alpha 0 5 where the existence of a different universality class is indicated from the behavior of the degree distribution and the clustering coefficients the network has a small diameter in the entire scale free region the clustering coefficients emulate the behavior of most real networks for increasing negative values of alpha on the phase boundary

finite time fluctuations in the degree statistics of growing networks this paper presents a comprehensive analysis of the degree statistics in models for growing networks where new nodes enter one at a time and attach to one earlier node according to a stochastic rule the models with uniform attachment linear attachment the barabasi albert model and generalized preferential attachment with initial attractiveness are successively considered the main emphasis is on finite size i e finite time effects which are shown to exhibit different behaviors in three regimes of the size degree plane stationary finite size scaling large deviations

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

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

representations of non linear systems the narmax model abstract input output representations of non linear discrete time systems are discussed it is shown that the narmax non linear autoregressive moving average with exogenous inputs model is a general and natural representation of non linear systems and contains as special cases several existing non linear models the problem of approximating non linear input output systems is also addressed and several properties of non linear models are highlighted using simple examples

phenomenological models of socioeconomic network dynamics we study a general set of models of social network evolution and dynamics the models consist of both a dynamics on the network and evolution of the network links are formed preferentially between similar nodes where the similarity is defined by the particular process taking place on the network the interplay between the two processes produces phase transitions and hysteresis as seen using numerical simulations for three specific processes we obtain analytic results using mean field approximations and for a particular case we derive an exact solution for the network in common with real world social networks we find coexistence of high and low connectivity phases and history dependence

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

components of springer fibers for the exceptional groups g2 and f4 abstract let g be the complex connected simply connected simple lie group of type g 2 or f 4 let k denote the fixed point subgroup relative to an involution of g that is lifted from a cartan involution this article gives a description of certain components of springer fibers associated to closed k orbits contained in the flag variety of g these components allow us to describe certain multiplicity polynomials associated to discrete series representations of the real form g 2 2 of g 2 and the two real forms f 4 4 and f 4 20 of f 4 the goals for this paper are motivated by the descriptions of springer fiber components and the associated multiplicity polynomials for type s u p q described in a paper of barchini and zierau

scale free networks from varying vertex intrinsic fitness a new mechanism leading to scale free networks is proposed in this letter it is shown that in many cases of interest the connectivity power law behavior is neither related to dynamical properties nor to preferential attachment assigning a quenched fitness value x i to every vertex and drawing links among vertices with a probability depending on the fitnesses of the two involved sites gives rise to what we call a good get richer mechanism in which sites with larger fitness are more likely to become hubs i e to be highly connected

scale free networks as preasymptotic regimes of superlinear preferential attachment we study the following paradox associated with networks growing according to superlinear preferential attachment superlinear preference cannot produce scale free networks in the thermodynamic limit but there are superlinearly growing network models that perfectly match the structure of some real scale free networks such as the internet we obtain an analytic solution supported by extensive simulations for the degree distribution in superlinearly growing networks with arbitrary average degree and confirm that in the true thermodynamic limit these networks are indeed degenerate i e almost all nodes have low degrees we then show that superlinear growth has vast preasymptotic regimes whose depths depend both on the average degree in the network and on how superlinear the preference kernel is we demonstrate that a superlinearly growing network model can reproduce in its preasymptotic regime the structure of a real network if the model captures some sufficiently strong structural constraints rich club connectivity for example these findings suggest that real scale free networks of finite size may exist in pre asymptotic regimes of network evolution processes that lead to degenerate network formations in the thermodynamic limit

structural transitions in scale free networks real growing networks such as the world wide web or personal connection based networks are characterized by a high degree of clustering in addition to the small world property and the absence of a characteristic scale appropriate modifications of the barabasi albert preferential attachment network growth capture all these aspects we present a scaling theory to describe the behavior of the generalized models and the mean field rate equation for clustering this is solved for a specific case with the result c k approximately 1 k for the clustering of a node of degree k this mean field exponent agrees with simulations and reproduces the clustering of many real networks

structure of a large social network we study a social network consisting of over 10 4 individuals with a degree distribution exhibiting two power scaling regimes separated by a critical degree k mathrm crit and a power law relation between degree and local clustering we introduce a growing random model based on a local interaction mechanism that reproduces the observed scaling features and their exponents we suggest that the double power law originates from two very different kinds of networks that are simultaneously present in the human social network

role of dimensionality in axelrod s model for the dissemination of culture we analyze a model of social interaction in one and two dimensional lattices for a moderate number of features we introduce an order parameter as a function of the overlap between neighboring sites in a one dimensional chain we observe that the dynamics is consistent with a second order transition where the order parameter changes continuously and the average domain diverges at the transition point however in a two dimensional lattice the order parameter is discontinuous at the transition point characteristic of a 4rst order transition between an ordered and a disordered state c

identification of non linear parametrically varying models using separable least squares in this paper a novel identification algorithm for a class of non linear possibly parameter varying models is proposed the algorithm is based on separable least squares ideas these models are given in the form of a linear fractional transformation lft where the forward part is represented by a conventional linear regression and the feedback part is given by a non linear map which can take into account scheduling variables available for measurement the non linear part of the model can be parameterized according to various paradigms like e g neural network nn or general nonlinear autoregressive exogenous narx models the estimation algorithm exploits the separability of the criterion used to estimate the parameters when using a nn it is possible the explicit computation of the frechet derivative needed to implement a separable least square algorithm

reduction of a class of fox wright psi functions for certain rational parameters abstract the fox wright psi function is a special case of fox s h function and a generalization of the generalized hypergeometric function in the present paper we show that the psi function reduces to a single generalized hypergeometric function when certain of its parameters are integers and to a finite sum of generalized hypergeometric functions when these parameters are rational numbers applications to the solution of algebraic trinomial equations and to a problem in information theory are provided a connection with meijer s g function is also discussed

a renormalization group theory of cultural evolution abstract we present a theory of cultural evolution based upon a renormalization group scheme we consider rational but cognitively limited agents who optimize their decision making process by iteratively updating and refining the mental representation of their natural and social environment these representations are built around the most important degrees of freedom of their world cultural coherence among agents is defined as the overlap of mental representations and is characterized using an adequate order parameter as the importance of social interactions increases or agents become more intelligent we observe and quantify a series of dynamic phase transitions by which cultural coherence advances in the society a similar phase transition may explain the so called cultural explosion in human evolution some 50 000 years ago

decentralized routing on spatial networks with stochastic edge weights we investigate algorithms to find short paths in spatial networks with stochastic edge weights our formulation of the problem of finding short paths differs from traditional formulations because we specifically do not make two of the usual simplifying assumptions 1 we allow edge weights to be stochastic rather than deterministic and 2 we do not assume that global knowledge of a network is available we develop a decentralized routing algorithm that provides en route guidance for travelers on a spatial network with stochastic edge weights without the need to rely on global knowledge about the network to guide a traveler our algorithm uses an estimation function that evaluates cumulative arrival probability distributions based on distances between pairs of nodes the estimation function carries a notion of proximity between nodes and thereby enables routing without global knowledge in testing our decentralized algorithm we define a criterion that makes it possible to discriminate among arrival probability distributions and we test our algorithm and this criterion using both synthetic and real networks

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

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

surface group representations and u p q higgs bundles using the l2 norm of the higgs field as a morse function we study the moduli spaces of u p q higgs bundles over a riemann surface we require that the genus of the surface be at least two but place no constraints on p q a key step is the identification of the function s local minima as moduli spaces of holomorphic triples in a companion paper 7 we prove that these moduli spaces of triples are nonempty and irreducible because of the relation between flat bundles and fundamental group representations we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple pu p q representations the topological invariants of the flat bundles are used to label subspaces these invariants are bounded by a milnor wood type inequality for each allowed value of the invariants satisfying a certain coprimality condition we prove that the corresponding subspace is nonempty and connected if the coprimality condition does not hold our results apply to the closure of the moduli space of irreducible representations 1 members of vbac vector bundles on algebraic curves which is partially supported by eager ec fp5 contract no hprn ct 2000 00099 and by edge ec fp5 contract no hprn ct 2000 00101 2 partially supported by the national science foundation under grant dms0072073 3 partially supported by the ministerio de ciencia y tecnolo ia spain under grant bfm2000 0024 4 partially supported by the fundacao para a ciencia e a tecnologia portugal through the centro de matematica da universidade do porto and through grant no sfrh bpd 1606 2000 5 partially supported by the portugal spain bilateral programme acciones integradas grant nos hp2000 0015 and ai 01 24 6 partially supported by a british epsrc grant october december 2001 received 11 27 2002

degree distribution rank size distribution and leadership persistence in mediation driven attachment networks abstract we investigate the growth of a class of networks in which a new node first picks a mediator at random and connects with m randomly chosen neighbors of the mediator at each time step we show that the degree distribution in such a mediation driven attachment mda network exhibits power law p k k m with a spectrum of exponents depending on m to appreciate the contrast between mda and barabasi albert ba networks we then discuss their rank size distribution to quantify how long a leader the node with the maximum degree persists in its leadership as the network evolves we investigate the leadership persistence probability f i e the probability that a leader retains its leadership up to time we find that it exhibits a power law f m with persistence exponent m 1 51 m in mda networks and m 1 53 exponentially with m in ba networks

conditional parametric models for storm sewer runoff 1 the method of conditional parametric modeling is introduced for flow prediction in a sewage system it is a well known fact that in hydrological modeling the response runoff to input precipitation varies depending on soil moisture and several other factors consequently nonlinear input output models are needed the model formulation described in this paper is similar to the traditional linear models like final impulse response fir and autoregressive exogenous arx except that the parameters vary as a function of some external variables the parameter variation is modeled by local lines using kernels for local linear regression as such the method might be referred to as a nearest neighbor method the results achieved in this study were compared to results from the conventional linear methods fir and arx the increase in the coefficient of determination is substantial furthermore the new approach conserves the mass balance better hence this new approach looks promising for various hydrological models and analysis

network growth by copying we introduce a growing network model in which a new node attaches to a randomly selected node as well as to all ancestors of the target node this mechanism produces a sparse ultrasmall network where the average node degree grows logarithmically with network size while the network diameter equals 2 we determine basic geometrical network properties such as the size dependence of the number of links and the in and out degree distributions we also compare our predictions with real networks where the node degree also grows slowly with time the internet and the citation network of all physical review papers

greedy connectivity of geographically embedded graphs we introduce a measure of greedy connectivity for geographical networks graphs embedded in space and where the search for connecting paths relies only on local information such as a node s location and that of its neighbors constraints of this type are common in everyday life applications greedy connectivity accounts also for imperfect transmission across established links and is larger the higher the proportion of nodes that can be reached from other nodes with a high probability greedy connectivity can be used as a criterion for optimal network design

brief paper a velocity algorithm for the implementation of gain scheduled controllers a new method is proposed to implement gain scheduled controllers for nonlinear plants given a family of linear feedback controllers designed for linearizations of a nonlinear plant about constant operating points a nonlinear gain scheduled controller is derived that preserves the input output properties of the linear closed loop systems locally about each equilibrium point the key procedures in the proposed method are to provide integral action at the inputs to the plant and differentiate some of the measured outputs before they are fed back to the scheduled controller for a fairly general class of systems the nonlinear gain scheduled controllers are easy to obtain and their structure is similar to that of the original linear controllers

why social networks are different from other types of networks we argue that social networks differ from most other types of networks including technological and biological networks in two important ways first they have nontrivial clustering or network transitivity and second they show positive correlations also called assortative mixing between the degrees of adjacent vertices social networks are often divided into groups or communities and it has recently been suggested that this division could account for the observed clustering we demonstrate that group structure in networks can also account for degree correlations we show using a simple model that we should expect assortative mixing in such networks whenever there is variation in the sizes of the groups and that the predicted level of assortative mixing compares well with that observed in real world networks

a convex characterization of gain scheduled h sub spl infin controllers an important class of linear time varying systems consists of plants where the state space matrices are fixed functions of some time varying physical parameters spl theta small gain techniques can be applied to such systems to derive robust time invariant controllers yet this approach is often overly conservative when the parameters spl theta undergo large variations during system operation in general higher performance can be achieved by control laws that incorporate available measurements of spl theta and therefore adjust to the current plant dynamics this paper discusses extensions of h sub spl infin synthesis techniques to allow for controller dependence on time varying but measured parameters when this dependence is linear fractional the existence of such gain scheduled h sub spl infin controllers is fully characterized in terms of linear matrix inequalities the underlying synthesis problem is therefore a convex program for which efficient optimization techniques are available the formalism and derivation techniques developed here apply to both the continuous and discrete time problems existence conditions for robust time invariant controllers are recovered as a special case and extensions to gain scheduling in the face of parametric uncertainty are discussed in particular simple heuristics are proposed to compute such controllers

random walks between leaves of random networks abstract motivated by the desire to model internet traffic we consider random walks that start and are absorbed on the leaves of random networks and study the length of such walks we present and test two techniques to analyse these walks on erd s renyi random graphs where the probability of a walk decays exponentially with its length the methods give indistinguishable results for the decay exponent this simple form of decay is not apparent on heterogeneous networks such as barabasi albert scale free networks and in this case each technique is demonstrated to have a different strength

highly dispersed networks by enhanced redirection we introduce a class of networks that grow by enhanced redirection nodes are introduced sequentially and each either attaches to a randomly chosen target node with probability 1 ensuremath r or to the parent of the target with probability r where r is an increasing function of the degree of the parent this mechanism leads to highly dispersed networks with unusual properties i existence of multiple macrohubs nodes whose degree is a finite fraction of the total number of network nodes n ii lack of self averaging and iii anomalous scaling in which n k the number of nodes of degree k scales as n k ensuremath sim n ensuremath nu ensuremath 1 k ensuremath nu with 1l ensuremath nu l2

designer nets from local strategies we propose a local strategy for constructing scale free networks of arbitrary degree distributions based on the redirection method of krapivsky and redner phys rev e 63 066123 2001 our method includes a set of external parameters that can be tuned at will to match detailed behavior at small degree k in addition to the scale free power law tail signature at large k once achieved the target distribution is maintained throughout the growth of the net the method is local in that addition of a new node requires knowledge of only the immediate environs of the randomly selected node to which it is attached global strategies require information on finite fractions of the growing net

twisted harmonic maps and the self duality equations si m est une surface de riemann compacte on utilise des resultats de la theorie des applications harmoniques pour trouver une solution aux equations d auto dualite associees a une representation irreductible p 1 m psl 2 c