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

percolation and epidemics in random clustered networks the social networks that infectious diseases spread along are typically clustered because of the close relation between percolation and epidemic spread the behavior of percolation in such networks gives insight into infectious disease dynamics a number of authors have studied percolation or epidemics in clustered networks but the networks often contain preferential contacts in high degree nodes we introduce a class of random clustered networks and a class of random unclustered networks with the same preferential mixing percolation in the clustered networks reduces the component sizes and increases the epidemic threshold compared to the unclustered networks

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

improved numerical method for the traction boundary integral equation by application of stokes theorem this paper concerns the direct numerical evaluation of singular integrals arising in boundary integral equations for displacement bie and displacement gradients bide and the formulation of a traction boundary integral equation tbie for solving general elastostatic crack problems subject to certain continuity conditions concerning displacements and tractions at the source point singular integrals in the bie and the bide corresponding to coefficients of displacement and displacement gradients at the source point are shown to be of a form that allows application of stokes theorem all the singular integrals in 3 d bie and bide are reduced to non singular line integrals and those in 2 d bie and bide are evaluated in closed form remaining terms involve regular integrals and no references to cauchy or hadamard principal values are required continuous isoparametric interpolations used on continuous elements local to the source point are modified to include unique displacement gradients at the source point which are compatible with all local tractions the resulting numerical bide is valid for source points located arbitrarily on the boundary including corners and a procedure is given for constructing a tbie from the bide some example solutions obtained using the present numerical method for the tbie in 2 d and 3 d are presented british crown copyright 1997 dera int j numer meth engng 40 3141 3161 1997 no of figures 5 no of tables 5 no of references 13

the unreasonable effectiveness of tree based theory for networks with clustering we demonstrate that a tree based theory for various dynamical processes operating on static undirected networks yields extremely accurate results for several networks with high levels of clustering we find that such a theory works well as long as the mean intervertex distance l is sufficiently small that is as long as it is close to the value of l in a random network with negligible clustering and the same degree degree correlations we support this hypothesis numerically using both real world networks from various domains and several classes of synthetic clustered networks we present analytical calculations that further support our claim that tree based theories can be accurate for clustered networks provided that the networks are sufficiently small worlds

hypersingular integrals how smooth must the density be hypersingular integrals are guaranteed to exist at a point x only if the density function f in the integrand satises certain conditions in a neighbourhood of x it is well known that a su cient condition is that f has a holder continuous rst derivative this is a stringent condition especially when it is incorporated into boundary element methods for solving hypersingular in tegral equations this paper is concerned with nding weaker conditions for the existence of one dimensional hadamard nite part integrals it is shown that it is sucien t for the even part of f with respect to x to have a holder continuous rst derivative the odd part is al lowed to be discontinuous a similar condition is obtained for cauchy principal value integrals these simple results have non trivial consequences they are applied to the calculation of the tangential derivative of a single layer potential and to the normal derivative of a double layer potential particular attention is paid to discontinuous densities and to discontinuous boundary conditions also despite the weaker sucien t conditions it is rearmed that for hypersingular integral equations collocation at a point x at the junction between two standard conforming boundary elements is not permissible theoretically various modications to the denition of nite part integral are explored

nonbacktracking expansion of finite graphs message passing equations yield a sharp percolation transition in finite graphs as an artifact of the locally treelike approximation for an arbitrary finite connected undirected graph we construct an infinite tree having the same local structural properties as this finite graph when observed by a nonbacktracking walker formally excluding the boundary this infinite tree is a generalization of the bethe lattice we indicate an infinite locally treelike random network whose local structure is exactly given by this infinite tree message passing equations for various cooperative models on this construction are the same as for the original finite graph but here they provide the exact solutions of the corresponding cooperative problems these solutions are good approximations to observables for the models on the original graph when it is sufficiently large and not strongly correlated we show how to express these solutions in the critical region in terms of the principal eigenvector components of the nonbacktracking matrix as representative examples we formulate the problems of the random and optimal destruction of a connected graph in terms of our construction the nonbacktracking expansion we analyze the limitations and the accuracy of the message passing algorithms for different classes of networks and compare the complexity of the message passing calculations to that of direct numerical simulations notably in a range of important cases simulations turn out to be more efficient computationally than the message passing

on the non singular traction bie in elasticity the work reported herein develops a generalized traction bie formulation which involves only weakly singular integrals in the three dimensional problem or totally regular integrals in the two dimensional problem the first step deals with the terms in the somigliana displacement identity and then the derivatives of these terms the only conditions required for the existence of the traction bie and the related somigliana stress identity are weak continuity of the in plane derivatives of the surface displacements and of the surface tractions it is shown that the cauchy principal value cpv interpretations so commonly used in bie developments are unnecessary the formulation is established not only at a smooth boundary point but also at a corner point the extension of the non singular formulation to discontinuous boundary tractions and tangential derivatives of the boundary displacements applicable to a generalized problem statement as well as the usual bem implementations is also shown in the demonstrated formulation the source points are located directly at the boundary nodes and non conformal elements are not needed

tight lower bound for percolation threshold on an infinite graph we construct a tight lower bound for the site percolation threshold on an infinite graph which becomes exact for an infinite tree the bound is given by the inverse of the maximal eigenvalue of the hashimoto matrix used to count nonbacktracking walks on the original graph our bound always exceeds the inverse spectral radius of the graph s adjacency matrix and it is also generally tighter than the existing bound in terms of the maximum degree we give a constructive proof for existence of such an eigenvalue in the case of a connected infinite quasitransitive graph a graph theoretic analog of a translationally invariant system

predicting percolation thresholds in networks we consider different methods which do not rely on numerical simulations of the percolation process to approximate percolation thresholds in networks we perform a systematic analysis on synthetic graphs and a collection of 109 real networks to quantify their effectiveness and reliability as prediction tools our study reveals that the inverse of the largest eigenvalue of the nonbacktracking matrix of the graph often provides a tight lower bound for true percolation threshold however in more than 40 of the cases this indicator is less predictive than the naive expectation value based solely on the moments of the degree distribution we find that the performance of all indicators becomes worse as the value of the true percolation threshold grows thus none of them represents a good proxy for the robustness of extremely fragile networks

network cloning unfolds the effect of clustering on dynamical processes we introduce network l cloning a technique for creating ensembles of random networks from any given real world or artificial network each member of the ensemble is an l cloned network constructed from l copies of the original network the degree distribution of an l cloned network and more importantly the degree degree correlation between and beyond nearest neighbors are identical to those of the original network the density of triangles in an l cloned network and hence its clustering coefficient is reduced by a factor of l compared to those of the original network furthermore the density of loops of any fixed length approaches zero for sufficiently large values of l other variants of l cloning allow us to keep intact the short loops of certain lengths as an application we employ these network cloning methods to investigate the effect of short loops on dynamical processes running on networks and to inspect the accuracy of corresponding tree based theories we demonstrate that dynamics on l cloned networks with sufficiently large l are accurately described by the so called adjacency tree based theories examples of which include the message passing technique some pair approximation methods and the belief propagation algorithm used respectively to study bond percolation si epidemics and the ising model

on the treatment of corners in the boundary element method abstract a new technique for treating surface discontinuities within boundary element calculations is proposed multiple nodes are used to represent the geometry and the necessary additional equations are obtained by differentiating the usual boundary element integral equation in deriving expressions for the resulting singular integrals it is found that constraints must be placed on the functional approximation at the discontinuity an algorithm for incorporating these constraints is developed and numerical tests for an exactly solvable three dimensional laplace equation problem are presented

central units in blocks and the odd zp theorem abstract we construct some units of order p in certain p adic group rings of finite groups which have relevance to the odd analogue of glauberman s z theorem but whose construction may have wider applicability and interest

corner treatment by assigning dual tractions to every node for elastodynamics in td bem in this study a corner treatment on the boundary for time domain boundary element method td bem by assigning dual tractions to every node is proposed the proposed corner treatment on boundary mimics the real mechanical situation on the physical boundary for elastodynamic problem and the computation accuracy is good taking the rectangular boundary as an example the numerical implementation of the proposed corner treatment on boundary by assigning dual tractions to every node is demonstrated two verification examples one dimensional rod in finite medium and two dimensional circular cavity in infinite medium under dynamic loads are conducted in which the results from the td bem based on the proposed corner treatment on boundary are compared with the results from the analytical solutions good agreement between the analytical results and the td bem results based on the proposed corner treatment on boundary for td bem is achieved for the two verification examples

beyond the locally treelike approximation for percolation on real networks theoretical attempts proposed so far to describe ordinary percolation processes on real world networks rely on the locally treelike ansatz such an approximation however holds only to a limited extent because real graphs are often characterized by high frequencies of short loops we present here a theoretical framework able to overcome such a limitation for the case of site percolation our method is based on a message passing algorithm that discounts redundant paths along triangles in the graph we systematically test the approach on 98 real world graphs and on synthetic networks we find excellent accuracy in the prediction of the whole percolation diagram with significant improvement with respect to the prediction obtained under the locally treelike approximation residual discrepancies between theory and simulations do not depend on clustering and can be attributed to the presence of loops longer than three edges we present also a method to account for clustering in bond percolation but the improvement with respect to the method based on the treelike approximation is much less apparent

error analysis and adaptivity in three dimensional linear elasticity by the usual and hypersingular boundary contour method abstract two related topics are addressed in this article the first part of the article proves that for a certain admissible class of problems in linear elasticity the hypersingular boundary contour method hbcm can be collocated at all boundary points on the surface of a three dimensional 3 d body including those on boundary contours edges and corners because the hbcm shape functions satisfy a priori all the smoothness requirements for collocation at these points in contrast the hypersingular boundary element method needs in general relaxation of some of these smoothness requirements for its shape functions even for collocation at regular points that lie on the boundaries of boundary elements a hypersingular residual obtained from the standard and hypersingular boundary integral equations hbies has been recently proposed as a local error estimator for a boundary element for the boundary integral equation the second part in the present article is concerned with a definition of an analogous local error estimator for the boundary contour method for 3 d linear elasticity this error estimator is then used to drive an h adaptive meshing procedure numerical results are presented to demonstrate adaptive meshing for selected example problems

non singular boundary integral equation implementation the basis of the boundary integral equation bie development is the singular field solution for the kelvin point load problem the presence of the mathematical singularity in an otherwise non singular problem has created over the years a variety of numerical coping strategies the current paper extends some of the earlier works in a simple manner that eliminates the mathematical singularity through analytical integration of the critical terms the resulting formulation is fully non singular and is shown to be amenable to simple low order gaussian integration the critical class of problems involving thin sections and or highly graded bie meshes now can be solved without previous problems and concerns the accuracy of the numerical integrations used in creating the linear bie equations extension of the formulation approach to remove hypersingular formulation problems in fracture mechanics is seen to be a future beneficiary of the semi analytical bie implementation reported herein

on the extraction technique in boundary integral equations in this paper we develop and analyze a bootstrapping algorithm for the extraction of potentials and arbitrary derivatives of the cauchy data of regular three dimensional second order elliptic boundary value problems in connection with corresponding boundary integral equations the method rests on the derivatives of the generalized green s representation formula which are expressed in terms of singular boundary integrals as hadamard s finite parts their regularization together with asymptotic pseudohomogeneous kernel expansions yields a constructive method for obtaining generalized jump relations these expansions are obtained via composition of taylor expansions of the local surface representation the density functions differential operators and the fundamental solution of the original problem together with the use of local polar coordinates in the parameter domain for boundary integral equations obtained by the direct method this method allows the recursive numerical extraction of potentials and their derivatives near and up to the boundary surface

how clustering affects the bond percolation threshold in complex networks the question of how clustering nonzero density of triangles in networks affects their bond percolation threshold has important applications in a variety of disciplines recent advances in modeling highly clustered networks are employed here to analytically study the bond percolation threshold in comparison to the threshold in an unclustered network with the same degree distribution and correlation structure the presence of triangles in these model networks is shown to lead to a larger bond percolation threshold i e clustering increases the epidemic threshold or decreases resilience of the network to random edge deletion

a simple local smoothing scheme in strongly singular boundary integral representation of potential gradient abstract a new approach for computation of potential gradient at and near boundary is introduced a strongly singular boundary integral representation of potential gradient whose integral density is the potential gradient is derived and analysed applying the concept of the osculating circle a local smoothing procedure which computes a continuous approximation of potential gradient from the results of a 2d boundary element method bem analysis using linear elements is proposed and evaluated this approximation is used in the integral representation derived as an integral density which fulfills the continuity requirements numerical experiments demonstrate for quasiuniform meshes an o h 2 accuracy of potential gradient computed by both the local smoothing procedure on smooth parts of the boundary and by the integral representation on smooth boundary parts and near smooth boundary parts for points inside the domain a consequence of the latter result is that no significant increase in the error appears near the boundary boundary layer effect thus being eliminated in this approach

percolation on dense graph sequences in this paper we determine the percolation threshold for an arbitrary sequence of dense graphs g n let n be the largest eigenvalue of the adjacency matrix of g n and let g n p n be the random subgraph of g n obtained by keeping each edge independently with probability p n we show that the appearance of a giant component in g n p n has a sharp threshold at p n 1 n in fact we prove much more if g n converges to an irreducible limit then the density of the largest component of g n c n tends to the survival probability of a multi type branching process defined in terms of this limit here the notions of convergence and limit are those of borgs chayes lovasz s6s and vesztergombi in addition to using basic properties of convergence we make heavy use of the methods of bollobas janson and riordan who used multi type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs

some notes on singular integral techniques in boundary element analysis the paper presented here reviews the numerical techniques used currently to calculate the singular integrals and nearly singular integrals in the boundary element analysis some incorrect algorithms published before are discussed and a new numerical technique to calculate the nearly singular integral is developed the numerical results show a significant improvement in both accuracy and efficiency compared to the traditional adaptive gaussian quadrature and subdivision techniques

units of p power order in principal p blocks of p constrained groups abstract let g be a finite group having a normal p subgroup n that contains its centralizer c g n and let r be a p adic ring it is shown that any finite p group of units of augmentation one in rg which normalizes n is conjugate to a subgroup of g by a unit of rg and if it centralizes n it is even contained in n

the dual boundary element method effective implementation for crack problems summary the present paper is concerned with the effective numerical implementation of the two dimensional dual boundary element method for linear elastic crack problems the dual equations of the method are the displacement and the traction boundary integral equations when the displacement equation is applied on one of the crack surfaces and the traction equation on the other general mixed mode crack problems can be solved with a single region formulation both crack surfaces are discretized with discontinuous quadratic boundary elements this strategy not only automatically satisfies the necessary conditions for the existence of the finite part integrals which occur naturally but also circumvents the problem of collocation at crack tips crack kinks and crack edge corners examples of geometries with edge and embedded crack are analysed with the present method highly accurate results are obtained when the stress intensity factor is evaluated with the j integral technique the accuracy and efficiency of the implementation described herein make this formulation ideal for the study of crack growth problems under mixed mode conditions

observability transition in real networks we consider the observability model in networks with arbitrary topologies we introduce a system of coupled nonlinear equations valid under the locally treelike ansatz to describe the size of the largest observable cluster as a function of the fraction of directly observable nodes present in the network we perform a systematic analysis on 95 real world graphs and compare our theoretical predictions with numerical simulations of the observability model our method provides almost perfect predictions in the majority of the cases even for networks with very large values of the clustering coefficient potential applications of our theory include the development of efficient and scalable algorithms for real time surveillance of social networks and monitoring of technological networks

hypersingular integrals in boundary element fracture analysis a new general purpose boundary element method for domains with cracks has been recently developed this technique avoids the use of a multi domain decomposition by including an additional integral equation expressing the boundary condition on the crack the principal requirement of this technique is the analytic determination of certain hypersingular integrals of the green s function which arise from this equation in order to establish the applicability of this method for fracture these integrals are evaluated herein for the kelvin solution of the three dimensional navier equations of linear elasticity numerical results for fracture problems using the single domain boundary element analysis are also presented

the traction boundary contour method for linear elasticity this paper presents a further development of the boundary contour method the boundary contour method is extended to cover the traction boundary integral equation a traction boundary contour method is proposed for linear elastostatics the formulation of traction boundary contour method is regular for points except the ends of the boundary element and corners the present approach only requires line integrals for three dimensional problems and function evaluations at the ends of boundary elements for two dimensional cases the implementation of the traction boundary contour method with quadratic boundary elements is presented for two dimensional problems numerical results are given for some two dimensional examples and these are compared with analytical solutions this method is shown to give excellent results for illustrative examples copyright 1999 john wiley sons ltd

on the computation of the derivatives of potentials on a boundary by using boundary integral equations this work presents a new recursion scheme to compute the cartesian derivatives of potentials on the smooth surface of a connected solid the advocated strategy solely appeals to boundary integral equations and a very few informations regarding the surface geometry the whole algorithm is carefully tested against analytical solutions both for interior and exterior problems by implementing a collocation points method

boundary element formulation for 3d transversely isotropic cracked bodies the boundary traction integral representation is obtained in elasticity when the classical displacement representation is differentiated and combined according to hooke s law the use of both traction and displacement integral representations leads to a mixed or dual formulation of the bem where the discretization effort for crack problems is much smaller than in the classical formulation a boundary element analysis of three dimensional fracture mechanics problems of transversely isotropic solids based on the mixed formulation is presented in this paper the hypersingular and strongly singular kernels appearing in the formulation are regularized by using two terms of the displacement series expansion and one term of the traction expansion at the collocation point all the remaining integrals are analytically evaluated or transformed by means of stokes theorem into regular or weakly singular integrals which are numerically computed the method is general and can be used for elements of any shape including quarter point crack front elements no change of co ordinates is required for the integration the formulation as presented in this paper is something as clear general and easy to handle as the classical be formulation it is used in combination with three dimensional quadratic and quarter point elements to obtain accurate results for several different crack problems cracks in boundless and finite transversely isotropic domains are studied the meshes are simple and include only discretization of the crack and the external boundary the obtained results are in good agreement with those existing in the literature copyright 2004 john wiley sons ltd

a single domain boundary element method for 3 d elastostatic crack analysis using continuous elements summary a boundary element method is presented for single domain analysis of cracked three dimensional isotropic elastostatic solids a numerical treatment for the hypersingular boundary integro differential equation bide for displacement derivatives is described in which continuous boundary elements may be used hadamard principal values of the hypersingular integrals arising in the formulation are evaluated using polar co ordinates defined on the tangent planes at the source point and the free term coefficients are calculated directly using a numerical technique the forms of the boundary integral equation bie and the bide are considered for a source point on the coincident surfaces of a crack and a scheme is given for defining the traction boundary integral equation tbie so that it optimally incorporates the traction information deficient in its complementary partner the bie numerical results for some example mixedmode crack problems are presented

clustering in complex networks ii percolation properties the percolation properties of clustered networks are analyzed in detail in the case of weak clustering we present an analytical approach that allows us to find the critical threshold and the size of the giant component numerical simulations confirm the accuracy of our results in more general terms we show that weak clustering hinders the onset of the giant component whereas strong clustering favors its appearance this is a direct consequence of the differences in the k core structure of the networks which are found to be totally different depending on the level of clustering an empirical analysis of a real social network confirms our predictions

use of simple solutions for boundary integral methods in elasticity and fracture analysis the simple solutions or indirect method of analysing cauchy and hypersingular integrands in the gradient flux or traction boundary integral equation bie is applied to linear elastic fracture analysis because of the geometric singularity of the crack surface application of the simple solutions formulas on the crack face requires integration over a temporary closure surface rather than the remainder of the body closure surface constructions are exhibited for crack surfaces allowing the gradient bie to be applied as a constraint equation on a crack surface where the primary bie is degenerate computational results are given for two benchmark fracture problems

bond percolation on a class of clustered random networks analytical results are derived for the bond percolation threshold and the size of the giant connected component in a class of random networks with nonzero clustering the network s degree distribution and clustering spectrum may be prescribed and theoretical results match well with numerical simulations on both synthetic and real world networks