thresholds for epidemic spreading in networks we study the threshold of epidemic models in quenched networks with degree distribution given by a power law forthesusceptible infected susceptiblemodeltheactivitythresholdc vanishesinthelargesize limitonanynetworkwhosemaximumdegree kmax divergeswiththesystemsize atoddswithheterogeneous mean field hmf theory the vanishing of the threshold has nothing to do with the scale free nature of the networkbutstemsinsteadfromthelargesthubinthesystembeingactiveforanyspreadingrate 1 ffiffiffiffiffiffiffiffiffimax p and playing the role of a self sustained source that spreads the infection to the rest of the system the susceptible infected removedmodel displays insteadagreement with hmftheory and afinite threshold for scale rich networks we conjecture that on quenched scale rich networks the threshold of generic epidemic models is vanishing or finite depending on the presence or absence of a steady state

symmetries of the energy momentum tensor of spherically symmetric lorentzian manifolds matter collineations of spherically symmetric lorentzian manifolds are considered these are investigated when the energy momentum tensor is nondegenerate and also when it is degenerate we have classified space times admitting higher symmetries and space times admitting so 3 as the maximal isometry group for the nondegenerate case we obtain either four six seven or ten independent matter collineations in which four are isometries and the rest are proper the results of the previous paper sharif and sehar gen relativ gravit 35 1091 2003 are recovered as a special case it is worth noting that we have also obtained two cases where the energy momentum tensor is degenerate but the group of matter collineations is finite dimensional i e four or ten

an inhomogeneous transference principle and diophantine approximation in a landmark paper 29 d y kleinbock and g a margulis established the fundamental baker sprindzuk conjecture on homogeneous diophantine approximation on manifolds subsequently there has been dramatic progress in this area of research however the techniques developed to date do not seem to be applicable to inhomogeneous approximation consequently the theory of inhomogeneous diophantine approximation on manifolds remains essentially non existent in this paper we develop an approach that enables us to transfer homogeneous statements to inhomogeneous ones this is rather surprising as the inhomogeneous theory contains the homogeneous theory and so is more general as a consequence we establish the inhomogeneous analogue of the baker sprindzuk conjecture furthermore we prove a complete inhomogeneous version of the profound theorem of kleinbock lindenstrauss weiss 27 on the extremality of friendly measures the results obtained in this paper constitute the first step towards developing a coherent inhomogeneous theory for manifolds in line with the homogeneous theory epsrc advanced research fellow grant ep c54076x 1 research supported by epsrc grant ep e061613 1 and intas grant 03 51 5070

message passing approach for recurrent state epidemic models on networks epidemic processes are common out of equilibrium phenomena of broad interdisciplinary interest recently dynamic message passing dmp has been proposed as an efficient algorithm for simulating epidemic models on networks and in particular for estimating the probability that a given node will become infectious at a particular time to date dmp has been applied exclusively to models with one way state changes as opposed to models like sis and sirs where nodes can return to previously inhabited states because many real world epidemics can exhibit such recurrent dynamics we propose a dmp algorithm for complex recurrent epidemic models on networks our approach takes correlations between neighboring nodes into account while preventing causal signals from backtracking to their immediate source and thus avoids echo chamber effects where a pair of adjacent nodes each amplify the probability that the other is infectious we demonstrate that this approach well approximates results obtained from monte carlo simulation and that its accuracy is often superior to the pair approximation which also takes second order correlations into account moreover our approach is more computationally efficient than the pair approximation especially for complex epidemic models the number of variables in our dmp approach grows as 2mk where m is the number of edges and k is the number of states as opposed to mk 2 for the pair approximation we suspect that the resulting reduction in computational effort as well as the conceptual simplicity of dmp will make it a useful tool in epidemic modeling especially for high dimensional inference tasks

epidemic thresholds of the susceptible infected susceptible model on networks a comparison of numerical and theoretical results recent work has shown that different theoretical approaches to the dynamics of the susceptible infectedsusceptible sis model for epidemics lead to qualitatively different estimates for the position of the epidemic threshold in networks here we present large scale numerical simulations of the sis dynamics on various types of networks allowing the precise determination of the effective threshold for systems of finite size n we compare quantitatively the numerical thresholds with theoretical predictions of the heterogeneous mean field theory and of the quenched mean field theory we show that the latter is in general more accurate scaling with n with the correct exponent but often failing to capture the correct prefactor

symmetries of the energy momentum tensor of cylindrically symmetric static space times we investigate matter symmetries of cylindrically symmetric static space times these are classified for both cases when the energy momentum tensor is nondegenerate and also when it is degenerate it is found that the nondegenerate energy momentum tensor gives either three four five six seven or ten independent matter collineations in which three are isometries and the rest are proper the worth mentioning cases are those where we obtain the group of matter collineations finite dimensional even the energy momentum tensor is degenerate these are either three four five or ten some examples are constructed satisfying the constraints on the energy momentum tensor

predicting epidemic threshold of correlated networks a comparison of methods abstract being able to theoretically predict the outbreak threshold of an epidemic is essential in disease control real world correlated networks are ubiquitous and their topological structure strongly affects the prediction accuracy of present day theoretical methods quantifying their accuracy and fitness in predicting outbreak thresholds of correlated networks is thus essential we use a susceptible infected removed sir model to examine four widely used theoretical methods the heterogeneous mean field h m f quenched mean field q m f dynamical message passing d m p and connectivity matrix c m methods to predict the outbreak threshold of a correlated network the potential topological structure of correlated network impacts on prediction accuracy of these four methods we emphasize that the quantitative changes of degree correlation degree distribution exponent and network size strongly affect the outbreak of sir spreading dynamics and compare the simulation results with the theoretical ones obtained from these four methods the extensive experiments in synthetic networks show that a the increasing degree correlation coefficients reduce outbreak threshold and suppress outbreak size b the increasing degree distribution exponents raise outbreak threshold but suppress outbreak size c the increasing network sizes decrease outbreak threshold but do not affect outbreak size d as for four theoretical methods c m and d m p are more likely to precisely predict outbreak threshold because they to some extent incorporate network topology with dynamical correlations the experimental results in 50 real world networks also prove the above conclusions

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

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

generation of uncorrelated random scale free networks uncorrelated random scale free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks we propose and analyze a model capable of generating random uncorrelated scale free networks with no multiple and self connections the model is based on the classical configuration model with an additional restriction on the maximum possible degree of the vertices we check numerically that the proposed model indeed generates scale free networks with no two and three vertex correlations as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree k respectively complex networks constitute a general framework for the topological characterization of many natural and technological systems whose complexity prevents a more detailed microscopic description f1 3g within this framework these systems are represented in terms of networks or graphs f4g in which vertices stand for the units composing the system while edges among vertices represent the interactions or relations between pairs of units the focus is thus shift to the topological characterization of the representative network a task which is largely more feasible and yields nevertheless a noticeable amount of information on the structure and properties of the original system the empirical analysis of many real complex networks has unveiled the presence of several typical properties widely found in systems belonging to a large variety of realms one of the most relevant is given by the scale free nature of the degree distribution pskd f1 3 5g defined as the probability that a randomly chosen vertex has degree k si e it is connected to other k verticesd in mathematical terms the scale free property translates into a power law function of the form psk d k g

special conformal killing vector space times and symmetry inheritance viscous heat conducting fluid and anisotropic fluid space times admitting a special conformal killing vector sckv are studied and some general theorems concerning the inheritance of the symmetry associated with the sckv are proved in particular for viscous fluid space times it is shown that i if the sckv maps fluid flow lines into fluid flow lines then all physical components of the energy momentum tensor inherit the sckv symmetry or ii if the lie derivative along a sckv of the shear viscosity term ab is zero then again we have symmetry inheritance all space times admitting a sckv and satisfying the dominant energy condition are found apart from the vacuum pp wave solutions which are the only vacuum solutions that can admit a sckv the energy momentum tensor associated with these space times is shown to admit at least one null eigenvector and can represent either a viscous fluid with heat conduction or an anisotropic fluid no perfect fluid space times can admit a sckv these sckv space t

addendum symmetries of the energy momentum tensor in recent papers j math phys 44 5142 2003 45 1518 2003 45 1532 2004 we have discussed matter symmetries of nonstatic spherically symmetric space times static plane symmetric space times and cylindrically symmetric static space times these have been classified for both cases when the energy momentum tensor is nondegenerate and also when it is degenerate here we add up some consequences and the missing references about the ricci tensor

geography in a scale free network model we offer an example of a network model with a power law degree distribution p k approximately k alpha for nodes but which nevertheless has a well defined geography and a nonzero threshold percolation probability for alpha 2 the range of real world contact networks this is different from p c 0 for alpha 3 results for the original well mixed scale free networks in our lattice based scale free network individuals link to nearby neighbors on a lattice even considerable additional small world links do not change our conclusion of nonzero thresholds when applied to disease propagation these results suggest that random immunization may be more successful in controlling human epidemics than previously suggested if there is geographical clustering

classification of static plane symmetric space times according to their matter collineations in this paper we classify static plane symmetric space times according to their matter collineations these have been studied for both cases when the energy momentum tensor is nondegenerate and also when it is degenerate it turns out that the nondegenerate case yields either four five six seven or ten independent matter collineations in which four are isometries and the rest are proper there exists three interesting cases where the energy momentum tensor is degenerate but the group of matter collineations is finite dimensional the matter collineations in these cases are either four six or ten

inhomogeneous diophantine approximation on planar curves in this paper we develop the inhomogeneous metric theory of simultaneous diophantine approximation on planar curves our results naturally extend the homogeneous khintchine and jarnik type theorems established in beresnevich et al ann math 166 2 367 426 2007 and vaughan and velani invent math 166 103 124 2006 and are the first of their kind the key lies in obtaining essentially the best possible results regarding the distribution of shifted rational points near planar curves

reshuffling scale free networks from random to assortative to analyze the role of assortativity in networks we introduce an algorithm which produces assortative mixing to a desired degree this degree is governed by one parameter p changing this parameter one can construct networks ranging from fully random p 0 to totally assortative p 1 we apply the algorithm to a barabasi albert scale free network and show that the degree of assortativity is an important parameter governing the geometrical and transport properties of networks thus the average path length of the network increases dramatically with the degree of assortativity moreover the concentration dependences of the size of the giant component in the node percolation problem for uncorrelated and assortative networks are strongly different the behavior of the clustering coefficient is also discussed

matter collineations the inverse symmetry inheritance problem matter collineations as a symmetry property of the energy momentum tensor tab are studied from the point of view of the lie algebra of vector fields generating them most attention is given to space times with a degenerate energy momentum tensor some examples of matter collineations are found for dust fluids including szekeres s space times and null fluid space times

information dimension and the degrees of freedom of the interference channel the degrees of freedom dofs of the k user gaussian interference channel determine the asymptotic growth of the maximal sum rate as a function of the signal to noise ratio subject to a very general sufficient condition on the cross channel gains we give a formula for the dofs of the scalar interference channel as a function of the deterministic channel matrix which involves maximization of a sum of information dimensions over k scalar input distributions known special cases are recovered and even generalized in certain cases with unified proofs

inhomogeneous theory of dual diophantine approximation on manifolds the theory of inhomogeneous diophantine approximation on manifolds is developed in particular the notion of nice manifolds is introduced and the divergence part of the groshev type theory is established for all such manifolds our results naturally incorporate and generalize the homogeneous measure and dimension theorems for non degenerate manifolds established to date the results have natural applications beyond the standard inhomogeneous theory such as diophantine approximation by algebraic integers

epidemic spreading in a variety of scale free networks we have shown that the epidemic spreading in scale free networks is very sensitive to the statistics of degree distribution characterized by the index gamma the effective spreading rate lambda the social strategy used by individuals to choose a partner and the policy of administrating a cure to an infected node depending on the interplay of these four factors the stationary fractions of infected population f gamma as well as the epidemic threshold properties can be essentially different we have given an example of the evolutionary scale free network which is disposed to the spreading and the persistence of infections at any spreading rate lambda 0 for any gamma probably it is impossible to obtain a simple immunization program that can be simultaneously effective for all types of scale free networks we have also studied the dynamical solutions for the evolution equation governed by the epidemic spreading in scale free networks and found that for the case of vanishingly small cure rate delta 1 the initial configuration of infected nodes would feature the solution for very long times

flows on homogeneous spaces and diophantine approximation on manifolds we present a new approach to metric diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces this approach yields a new proof of a conjecture of mahler originally settled by v g sprindzuk in 1964 we also prove several related hypotheses of baker and sprindzuk formulated in 1970s the core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices

localization and centrality in networks eigenvector centrality is a common measure of the importance of nodes in a network here we show that under common conditions the eigenvector centrality displays a localization transition that causes most of the weight of the centrality to concentrate on a small number of nodes in the network in this regime the measure is no longer useful for distinguishing among the remaining nodes and its efficacy as a network metric is impaired as a remedy we propose an alternative centrality measure based on the nonbacktracking matrix which gives results closely similar to the standard eigenvector centrality in dense networks where the latter is well behaved but avoids localization and gives useful results in regimes where the standard centrality fails

modified shrinking target problem in beta dynamical systems abstract in the paper we investigate a modified version of the shrinking target problem in beta dynamical systems which is induced by the transformation t x x mod 1 the hausdorff dimension of the lim sup set w f x 0 1 t n x f x n for infinitely many n n is determined where f 0 1 0 1 is a lipschitz function and is a positive function defined on n the set w f can be viewed as a dynamical analogue of the inhomogeneous diophantine approximation in which the inhomogeneous factor is allowed to vary