closed walks for community detection abstract in this paper we propose a novel measure that integrates both the concept of closed walks and clustering coefficients to replace the edge betweenness in the well known divisive hierarchical clustering algorithm the girvan and newman method gn the edges with the lowest value are removed iteratively until the network is degenerated into isolated nodes the experimental results on computer generated networks and real world networks showed that our method makes a better tradeoff of accuracy and runtime based on the analysis of the results we observe that the nontrivial closed walks of order three and four can be considered as the basic elements in constructing community structures meanwhile we discover that those nontrivial closed walks outperform trivial closed walks in the task of analyzing the structure of networks the double peak structure problem is mentioned in the last part of the article we find that our proposed method is a novel way to solve the double peak structure problem our work can provide us with a new perspective for understanding community structure in complex networks

dominating communities for hierarchical control of complex networks abstract well selected nodes driven by external signals in complex networks are of great importance for a dynamic system s structural controllability which can also be achieved by selecting nodes in the minimum dominating set mds of the networks as driver nodes however hierarchical structures such as communities are widely observed in complex networks which motivates us to develop the hierarchical control of complex networks by dominating communities and the control over whole networks is transformed into that within communities i e a non driver node is effectively controlled by its driver node if they belong to the same community for the hierarchical control we propose the structure driven minimum dominating set sd mds which assumes that a driver node can control a non driver node if the link between them belongs to a cyclic structure of the networks we further investigate the impact of community dynamics on our framework the results show that it is easier to control the networks with stronger communities and more heterogeneous community sizes and the sd mds achieves more effective control for non driver nodes than the mds this work also indicates that the number of driver nodes for the sd mds is associated with the degree distribution of the network

anti modularity and anti community detecting in complex networks abstract many networks of interest in sciences and social research can be divided naturally into anti communities the problem of detecting and characterizing such anti community structure has attracted recent attention in this paper we first define the anti modularity as a quantitative measure of anti community partitioning on a network we also theoretically and empirically show the reliability of anti modularity as a measurement of the quality of an anti community partitioning a label propagation algorithm lpad for anti community detection is proposed experimental results on synthetic and real world networks show that our algorithm lpad can obtain higher quality anti community partitioning than other methods

on the pre lie algebra of specified feynman graphs we study the pre lie algebra of specified feynman graphs v t and we define a pre lie structure on its doubling space f t we prove that f t is pre lie module on v t and we find some relations between the two pre lie structures also we study the enveloping algebras of two pre lie algebras denoted respectively by d t and h t and we prove that d t is a module bialgebra on h t we study the pre lie algebra of specified feynman graphs v t and we define a pre lie structure on its doubling space f t we prove that f t is pre lie module on v t and we find some relations between the two pre lie structures also we study the enveloping algebras of two pre lie algebras denoted respectively by d t and h t and we prove that d t is a module bialgebra on h t

rota baxter paired modules and their constructions from hopf algebras abstract in this paper we introduce the concept of a rota baxter paired module to study rota baxter modules without necessarily a rota baxter operator we obtain two characterizations of rota baxter paired modules and give some basic properties of rota baxter paired modules beginning with the connection between the notion of integrals in the representations of hopf algebra and of the notion of an integral algebra as a motivation and special case of rota baxter algebra of weight zero we obtain a large number of rota baxter paired modules from hopf related algebras and modules including semisimple hopf algebras weak hopf algebras long bialgebras quasitriangular hopf algebras weak hopf modules dimodules and doi hopf modules some of them give new examples of rota baxter algebras

noncommutative poisson bialgebras abstract in this paper we introduce the notion of a noncommutative poisson bialgebra and establish the equivalence between matched pairs manin triples and noncommutative poisson bialgebras using quasi representations and the corresponding cohomology theory of noncommutative poisson algebras we study coboundary noncommutative poisson bialgebras which leads to the introduction of the poisson yang baxter equation a skew symmetric solution of the poisson yang baxter equation naturally gives a coboundary noncommutative poisson bialgebra rota baxter operators more generally o operators on noncommutative poisson algebras and noncommutative pre poisson algebras are introduced by which we construct skew symmetric solutions of the poisson yang baxter equation in some special noncommutative poisson algebras obtained from these structures

algebraic renormalisation of regularity structures we give a systematic description of a canonical renormalisation procedure of stochastic pdes containing nonlinearities involving generalised functions this theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms this subgroup is sufficiently large to be able to implement a version of the bphz renormalisation prescription in this context this is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations our construction is based on bialgebras of decorated coloured forests in cointeraction more precisely we have two hopf algebras in cointeraction coacting jointly on a vector space which represents the generalised functions of the theory two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic birkhoff factorisation that arises naturally in perturbative quantum field theory

preordered forests packed words and contraction algebras abstract we introduce the notions of preordered and heap preordered forests generalizing the construction of ordered and heap ordered forests we prove that the algebras of preordered and heap preordered forests are hopf for the cut coproduct and we construct a hopf morphism to the hopf algebra of packed words moreover we define another coproduct on the preordered forests given by the contraction of edges finally we give a combinatorial description of morphism defined on hopf algebras of forests with values in the hopf algebras of shuffles or quasi shuffles

integrable renormalization i the ladder case in recent years a hopf algebraic structure underlying the process of renormalization in quantum field theory was found it led to a birkhoff factorization for regularized hopf algebra characters i e for feynman rules in this work we would like to show that this birkhoff factorization finds its natural formulation in terms of a classical r matrix coming from a rota baxter structure underlying the target space of the regularized hopf algebra characters working in the rooted tree hopf algebra the simple case of the hopf subalgebra of ladder trees is treated in detail the extension to the general case i e the full hopf algebra of rooted trees or feynman graphs is briefly outlined

statistical mechanics of community detection starting from a general ansatz we show how community detection can be interpreted as finding the ground state of an infinite range spin glass our approach applies to weighted and directed networks alike it contains the ad hoc introduced quality function from j reichardt and s bornholdt phys rev lett 93 218701 2004 and the modularity q as defined by newman and girvan phys rev e 69 026113 2004 as special cases the community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices we elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study further we show how hierarchies and overlap in the community structure can be detected computationally efficient local update rules for optimization procedures to find the ground state are given we show how the ansatz may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension finally we give expectation values for the modularity of random graphs which can be used in the assessment of statistical significance of community structure

a lie theoretic approach to renormalization motivated by recent work of connes and marcolli based on the connes kreimer approach to renormalization we augment the latter by a combinatorial lie algebraic point of view our results rely both on the properties of the dynkin idempotent one of the fundamental lie idempotents in the theory of free lie algebras and on properties of hopf algebras encapsulated in the notion of associated descent algebras besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures the new techniques give rise to an algebraic approach to the galois theory of renormalization in particular they do not depend on the geometry underlying the case of dimensional regularization and the riemann hilbert correspondence this is illustrated with a discussion of the bphz renormalization scheme

rota baxter operators on 4 dimensional complex simple associative algebras rota baxter operators or relations were introduced to solve certain analytic and combinatorial problems and then applied to many fields in mathematics and mathematical physics in this paper we commence to study the rota baxter operators of weight zero on a 4 dimensional complex simple associative algebra which is isomorphic to the algebra of all 2x2 matrices over the field of complex numbers such operators satisfy the operator form of the classical yang baxter equation on the general linear lie algebra gl 2 c we provide two different solving methods one by hand to show some tricks for solving a large nonlinear system and another one by computer algebra to show the possibility of solving higher dimensional cases

hopf algebras of rooted forests cocyles and free rota baxter algebras the hopf algebra and the rota baxter algebra are the two algebraic structures underlying the algebraic approach of connes and kreimer to renormalization of perturbative quantum field theory in particular the hopf algebra of rooted trees serves as the baby model of feynman graphs in their approach and can be characterized by certain universal properties involving a hochschild 1 cocycle decorated rooted trees have also been applied to study feynman graphs we will continue the study of universal properties of various spaces of decorated rooted trees with such a 1 cocycle leading to the concept of a cocycle hopf algebra we further apply the universal properties to equip a free rota baxter algebra with the structure of a cocycle hopf algebra

parametric representation of rank d tensorial group field theory abelian models with kinetic term sum_s p_s mu we consider the parametric representation of the amplitudes of abelian models in the so called framework of rank d tensorial group field theory these models are called abelian because their elds live on

methods to find community based on edge centrality abstract divisive algorithms are of great importance for community detection in complex networks one algorithm proposed by girvan and newman gn based on an edge centrality named betweenness is a typical representative of this field here we studied three edge centralities based on network topology walks and paths respectively to quantify the relevance of each edge in a network and proposed a divisive algorithm based on the rationale of gn algorithm for finding communities that removes edges iteratively according to the edge centrality values in a certain order in addition we gave a comparison analysis of these measures with the edge betweenness and information centrality we found the principal difference among these measures in the partition procedure is that the edge centrality based on walks first removes the edge connected with a leaf vertex but the others first delete the edge as a bridge between communities it indicates that the edge centrality based on walks is harder to uncover communities than other edge centralities we also tested these measures for community detection the results showed that the edge information centrality outperforms other measures the edge centrality based on walks obtains the worst results and the edge betweenness gains better performance than the edge centrality based on network topology we also discussed our method s efficiency and found that the edge centrality based on walks has a high time complexity and is not suitable for large networks

benchmark graphs for testing community detection algorithms community structure is one of the most important features of real networks and reveals the internal organization of the nodes many algorithms have been proposed but the crucial issue of testing i e the question of how good an algorithm is with respect to others is still open standard tests include the analysis of simple artificial graphs with a built in community structure that the algorithm has to recover however the special graphs adopted in actual tests have a structure that does not reflect the real properties of nodes and communities found in real networks here we introduce a class of benchmark graphs that account for the heterogeneity in the distributions of node degrees and of community sizes we use this benchmark to test two popular methods of community detection modularity optimization and potts model clustering the results show that the benchmark poses a much more severe test to algorithms than standard benchmarks revealing limits that may not be apparent at a first analysis

renormalization in quantum field theory and the riemann hilbert problem i the hopf algebra structure of graphs and the main theorem this paper gives a complete selfcontained proof of our result announced in 6 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the riemann hilbert problem we shall first show that for any quantum field theory the combinatorics of feynman graphs gives rise to a hopf algebra which is commutative as an algebra it is the dual hopf algebra of the enveloping algebra of a lie algebra whose basis is labelled by the one particle irreducible feynman graphs the lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa the corresponding lie group g is the group of characters of we shall then show that using dimensional regularization the bare unrenormalized theory gives rise to a loop where c is a small circle of complex dimensions around the integer dimension d of space time our main result is that the renormalized theory is just the evaluation at z d of the holomorphic part of the birkhoff decomposition of we begin to analyse the group g and show that it is a semi direct product of an easily understood abelian group by a highly non trivial group closely tied up with groups of diffeomorphisms the analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title

controllability and modularity of complex networks two primary properties of complex networks controllability and modularity not only are closely related to each other but also play an important role in understanding the networks characteristics in this paper we discuss an elastic model to enhance the control of undirected networks and study the controllability of random networks with built in gradually varied modularity as well as real world networks to elaborate the correlation between the networks controllability and modularity the results show that it is easier to control the networks with stronger modularity than that of weaker modularity and the networks with larger size communities probably need more driver nodes to control than that of smaller size communities when we fix both the number of nodes and the number of links of the networks in addition the robustness analysis indicates that the model enhances the resistance of networks against link failure this work shows that the controllability of complex networks is highly associated with the networks degree distribution as well as the networks modularity which gives a new insight into the understanding of controllability and modularity of complex networks

renormalization of gauge fields a hopf algebra approach we study the connes kreimer hopf algebra of renormalization in the case of gauge theories we show that the ward identities and the slavnov taylor identities in the abelian and non abelian case respectively are compatible with the hopf algebra structure in that they generate a hopf ideal consequently the quotient hopf algebra is well defined and has those identities built in this provides a purely combinatorial and rigorous proof of compatibility of the slavnov taylor identities with renormalization

a combinatorial hopf algebra for nonlinear output feedback control systems abstract in this work a combinatorial description is provided of a faa di bruno type hopf algebra which naturally appears in the context of fliess operators in nonlinear feedback control theory it is a connected graded commutative and non cocommutative hopf algebra defined on rooted circle trees a cancellation free forest formula for its antipode is given

non commutative hopf algebra of formal diffeomorphisms abstract this paper deals with two hopf algebras which are the non commutative analogues of two different groups of formal power series the first group is the set of invertible series with the group law being multiplication of series while the second is the set of formal diffeomorphisms with the group law being a composition of series the motivation to introduce these hopf algebras comes from the study of formal series with non commutative coefficients invertible series with non commutative coefficients still form a group and we interpret the corresponding new non commutative hopf algebra as an alternative to the natural hopf algebra given by the co ordinate ring of the group which has the advantage of being functorial in the algebra of coefficients for the formal diffeomorphisms with non commutative coefficients this interpretation fails because in this case the composition is not associative anymore however we show that for the dual non commutative algebra there exists a natural co associative co product defining a non commutative hopf algebra moreover we give an explicit formula for the antipode which represents a non commutative version of the lagrange inversion formula and we show that its coefficients are related to planar binary trees then we extend these results to the semi direct co product of the previous hopf algebras and to series in several variables finally we show how the non commutative hopf algebras of formal series are related to some renormalization hopf algebras which are combinatorial hopf algebras motivated by the renormalization procedure in quantum field theory and to the renormalization functor given by the double tensor algebra on a bi algebra

nonabelian generalized lax pairs the classical yang baxter equation and postlie algebras we generalize the classical study of generalized lax pairs the related mathcal o operators and the modified classical yang baxter equation by introducing the concepts of nonabelian generalized lax pairs extended mathcal o operators and the extended classical yang baxter equation we study in this context the nonabelian generalized r matrix ansatz and the related double lie algebra structures relationship between extended mathcal o operators and the extended classical yang baxter equation is established especially for self dual lie algebras this relationship allows us to obtain an explicit description of the manin triples for a new class of lie bialgebras furthermore we show that a natural structure of postlie algebra is behind mathcal o operators and fits in a setup of the triple lie algebra that produces self dual nonabelian generalized lax pairs

perturbative quantum field theory via vertex algebras in this paper we explain how perturbative quantum field theory can be formulated in terms of a version of vertex algebras our starting point is the wilson zimmermann operator product expansion ope following ideas of a previous paper s hollands e print arxiv 0802 2198 we consider a consistency essentially associativity condition satisfied by the coefficients in this expansion we observe that the information in the ope coefficients can be repackaged straightforwardly into vertex operators and that the consistency condition then has essentially the same form as the key condition in the theory of vertex algebras we develop a general theory of perturbations of the algebras that we encounter similar in nature to the hochschild cohomology describing the deformation theory of ordinary algebras the main part of the paper is devoted to the question how one can calculate the perturbations corresponding to a given interaction lagrangian such as 4 in practice using the consistency condition an

holder continuous rough paths by fourier normal ordering we construct in this article an explicit geometric rough path over arbitrary d dimensional paths with finite 1 variation for any alpha in 0 1 the method may be coined as fourier normal ordering since it consists in a regularization obtained after permuting the order of integration in iterated integrals so that innermost integrals have highest fourier frequencies in doing so there appear non trivial tree combinatorics which are best understood by using the structure of the hopf algebra of decorated rooted trees in connection with the chen or multiplicative property and of the hopf shuffle algebra in connection with the shuffle or geometric property holder continuity is proved by using besov norms the method is well suited in particular in view of applications to probability theory see the companion article 34 for the construction of a rough path over multidimensional fractional brownian motion with hurst index 1 4 or 35 for a short survey in that case

feynman integrals and motives of configuration spaces we formulate the problem of renormalization of feynman integrals and its relation to periods of motives in configuration space instead of momentum space the algebro geometric setting is provided by the wonderful compactifications overline conf gamma x of arrangements of subvarieties associated to the subgraphs of a feynman graph with x a quasi projective variety the motive and the class in the grothendieck ring are computed explicitly for these wonderful compactifications in terms of the motive of x and the combinatorics of the feynman graph using recent results of li li the pullback to the wonderful compactification of the form defined by the unrenormalized feynman amplitude has singularities along a hypersurface whose real locus is contained in the exceptional divisors of the iterated blowup that gives the wonderful compactification a regularization of the feynman integrals can be obtained by modifying the cycle of integration by replacing the divergent locus with a leray coboundary the ambiguities are then defined by poincare residues while these residues give periods associated to the cohomology of the exceptional divisors and their intersections the regularized integrals give rise to periods of the hypersurface complement in the wonderful compactification

normal coordinates and primitive elements in the hopf algebra of renormalization we introduce normal coordinates on the infinite dimensional group g introduced by connes and kreimer in their analysis of the hopf algebra of rooted trees we study the primitive elements of the algebra and show that they are generated by a simple application of the inverse poincare lemma given a closed left invariant 1 form on g for the special case of the ladder primitives we find a second description that relates them to the hopf algebra of functionals on power series with the usual product either approach shows that the ladder primitives are given by the schur polynomials the relevance of the lower central series of the dual lie algebra in the process of renormalization is also discussed leading to a natural concept of k primitiveness which is shown to be equivalent to the one already in the literature

renormalization in quantum field theory and the riemann hilbert problem ii the β function diffeomorphisms and the renormalization group we showed in part i that the hopf algebra of feynman graphs in a given qft is the algebra of coordinates on a complex infinite dimensional lie group g and that the renormalized theory is obtained from the unrenormalized one by evaluating at 0 the holomorphic part of the riemann hilbert decomposition 1 of the loop g provided by dimensional regularization we show in this paper that the group g acts naturally on the complex space x of dimensionless coupling constants of the theory more precisely the formula g 0 gz 1 z 3 3 2 for the effective coupling constant when viewed as a formal power series does define a hopf algebra homomorphism between the hopf algebra of coordinates on the group of formal diffeomorphisms to the hopf algebra this allows first of all to read off directly without using the group g the bare coupling constant and the renormalized one from the riemann hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms this shows that renormalization is intimately related with the theory of non linear complex bundles on the riemann sphere of the dimensional regularization parameter it also allows to lift both the renormalization group and the function as the asymptotic scaling in the group g this exploits the full power of the riemann hilbert decomposition together with the invariance of under a change of unit of mass this not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group g for the full higher pole structure of minimal subtracted counterterms in terms of the residue

from dynamical systems to renormalization in this paper we study logarithmic derivatives associated to derivations on completed graded lie algebra as well as the existence of inverses these logarithmic derivatives when invertible generalize the exp log correspondence between a lie algebra and its lie group such correspondences occur naturally in the study of dynamical systems when dealing with the linearization of vector fields and the non linearizability of a resonant vector fields corresponds to the non invertibility of a logarithmic derivative and to the existence of normal forms these concepts stemming from the theory of dynamical systems can be rephrased in the abstract setting of lie algebra and the same difficulties as in perturbative quantum field theory pqft arise here surprisingly one can adopt the same ideas as in pqft with fruitful results such as new constructions of normal forms with the help of the birkhoff decomposition the analogy goes even further locality of counter terms choice of a renormalization scheme and shall lead to more interactions between dynamical systems and quantum field theory

identification of overlapping and non overlapping community structure by fuzzy clustering in complex networks this paper proposes a novel method based on fuzzy clustering to detect community structure in complex networks in contrast to previous studies our method does not focus on a graph model but rather on a fuzzy relation model which uses the operations of fuzzy relation to replace a traversal search of the graph for identifying community structure in our method we first use a fuzzy relation to describe the relation between vertices as well as the similarity in network topology to determine the membership grade of the relation then we transform this fuzzy relation into a fuzzy equivalence relation finally we map the non overlapping communities as equivalence classes that satisfy a certain equivalence relation because most real world networks are made of overlapping communities e g in social networks people may belong to multiple communities we can consider the equivalence classes above as the skeletons of overlapping communities and extend our method by adding vertices to the skeletons to identify overlapping communities we evaluated our method on artificial networks with built in communities and real world networks with known and unknown communities the experimental results show that our method works well for detecting these communities and gives a new understanding of network division and community formation

weighting links based on edge centrality for community detection abstract link weights have the equally important position as links in complex networks and they are closely associated with each other for the emergence of communities how to assign link weights to make a clear distinction between internal links of communities and external links connecting communities is of vital importance for community detection edge centralities provide a powerful approach for distinguishing internal links from external ones here we first use edge centralities such as betweenness information centrality and edge clustering coefficient to weight links of networks respectively to transform unweighted networks into weighted ones and then a weighted function that both considers links and link weights is adopted on the weighted networks for community detection we evaluate the performance of our approach on random networks as well as real world networks better results are achieved on weighted networks with stronger weights of internal links of communities and the results on unweighted networks outperform that of weighted networks with weaker weights of internal links of communities the availability of our findings is also well supported by the study of granovetter that the weak links maintain the global integrity of the network while the strong links maintain the communities especially in the karate club network all the nodes are correctly classified when we weight links by edge betweenness the results also give us a more comprehensive understanding on the correlation between links and link weights for community detection

o operators on hom lie algebras o operators also known as relative rota baxter operators on lie algebras have several applications in integrable systems and the classical yang baxter equations in this article we study o operators on hom lie algebras we define a cochain complex for o operators on hom lie algebras with respect to a representation any o operator induces a hom pre lie algebra structure we express the cochain complex of an o operator in terms of the specific hom lie algebra cochain complex if the structure maps in a hom lie algebra and its representation are invertible then we can extend the above cochain complex to a deformation complex for o operators by adding the space of zero cochains subsequently we study formal deformations of o operators on regular hom lie algebras in terms of the deformation cohomology in the end we deduce deformations of s rota baxter operators of weight 0 and skew symmetric r matrices on hom lie algebras as particular cases of o operators on hom lie algebras

dimensional regularization in position space and a forest formula for epstein glaser renormalization we reformulate dimensional regularization as a regularization method in position space and show that it can be used to give a closed expression for the renormalized time ordered products as solutions to the induction scheme of epstein glaser this closed expression which we call the epstein glaser forest formula is analogous to zimmermann s forest formula for bph renormalization for scalar fields the resulting renormalization method is always applicable we compute several examples we also analyze the hopf algebraic aspects of the combinatorics our starting point is the main theorem of renormalization of stora and popineau and the arising renormalization group as originally defined by stuckelberg and petermann

on the invariance of residues of feynman graphs we use simple iterated one loop graphs in massless yukawa theory and qed to pose the following question what are the symmetries of the residues of a graph under a permutation of places to insert subdivergences the investigation confirms partial invariance of the residue under such permutations the highest weight transcendental is invariant under such a permutation for qed this result is gauge invariant i e the permutation invariance holds for any gauge computations are done making use of the hopf algebra structure of graphs and employing ginac to automate the calculations

rota baxter operators on sl 2 c and solutions of the classical yang baxter equation we explicitly determine all rota baxter operators of weight zero on sl 2 c under the cartan weyl basis for the skew symmetric operators we give the corresponding skew symmetric solutions of the classical yang baxter equation in sl 2 c confirming the related study by semenov tian shansky in general these rota baxter operators give a family of solutions of the classical yang baxter equation in the six dimensional lie algebra sl 2 c ad sl 2 c they also give rise to three dimensional pre lie algebras which in turn yield solutions of the classical yang baxter equation in other six dimensional lie algebras

the signed permutation group on feynman graphs the feynman rules assign to every graph an integral which can be written as a function of a scaling parameter l assuming l for the process under consideration is very small so that contributions to the renormalization group are small we can expand the integral and only consider the lowest orders in the scaling the aim of this article is to determine specific combinations of graphs in a scalar quantum field theory that lead to a remarkable simplification of the first non trivial term in the perturbation series it will be seen that the result is independent of the renormalization scheme and the scattering angles to achieve that goal we will utilize the parametric representation of scalar feynman integrals as well as the hopf algebraic structure of the feynman graphs under consideration moreover we will present a formula which reduces the effort of determining the first order term in the perturbation series for the specific combination of graphs to a minimum

hopf algebras cyclic cohomology and the transverse index theorem in this paper we solve a longstanding internal problem of noncommutative geometry namely the computation of the index of transversally elliptic operators on foliations we show that the computation of the local index formula for transversally hypoelliptic operators can be settled thanks to a very specific hopf algebra associated to each integer codimension this hopf algebra reduces transverse geometry to a universal geometry of affine nature the structure of this hopf algebra its relation with the lie algebra of formal vector fields as well as the computation of its cyclic cohomology are done in the present paper in which we also show that under a suitable unimodularity condition the cosimplicial space underlying the hochschild cohomology of a hopf algebra carries a highly nontrivial cyclic structure

free tri dendriform family algebras abstract in this paper we first prove that a rota baxter family algebra indexed by a semigroup amounts to an ordinary rota baxter algebra structure on the tensor product with the semigroup algebra we show that the same phenomenon arises for dendriform and tridendriform family algebras then we construct free dendriform family algebras in terms of typed decorated planar binary trees finally we generalize typed decorated rooted trees to typed valently decorated schroder trees and use them to construct free tridendriform family algebras

decomposition spaces incidence algebras and mobius inversion i basic theory abstract this is the first in a series of papers devoted to the theory of decomposition spaces a general framework for incidence algebras and mobius inversion where algebraic identities are realised by taking homotopy cardinality of equivalences of groupoids a decomposition space is a simplicial groupoid satisfying an exactness condition weaker than the segal condition expressed in terms of active and inert maps in just as the segal condition expresses composition the new exactness condition expresses decomposition and there is an abundance of examples in combinatorics after establishing some basic properties of decomposition spaces the main result of this first paper shows that to any decomposition space there is an associated incidence coalgebra spanned by the space of 1 simplices and with coefficients in groupoids we take a functorial viewpoint throughout emphasising conservative ulf functors these induce coalgebra homomorphisms reduction procedures in the classical theory of incidence coalgebras are examples of this notion and many are examples of decalage of decomposition spaces an interesting class of examples of decomposition spaces beyond segal spaces is provided by hall algebras the waldhausen s construction of an abelian or stable infinity category is shown to be a decomposition space in the second paper in this series we impose further conditions on decomposition spaces to obtain a general mobius inversion principle and to ensure that the various constructions and results admit a homotopy cardinality in the third paper we show that the lawvere menni hopf algebra of mobius intervals is the homotopy cardinality of a certain universal decomposition space two further sequel papers deal with numerous examples from combinatorics note the notion of decomposition space was arrived at independently by dyckerhoff and kapranov 17 who call them unital 2 segal spaces our theory is quite orthogonal to theirs the definitions are different in spirit and appearance and the theories differ in terms of motivation examples and directions

qed hopf algebras on planar binary trees abstract in this paper we describe the hopf algebras on planar binary trees used to renormalize the feynman propagators of quantum electrodynamics and the coaction which describes the renormalization procedure both structures are related to some semi direct coproduct of hopf algebras

optimizing controllability of complex networks by minimum structural perturbations to drive a large complex networked dynamical system toward some desired state using as few external signals as possible is a fundamental issue in the emerging field of controlling complex networks optimal control is referred to the situation where such a network can be fully controlled using only one driving signal we propose a general approach to optimizing the controllability of complex networks by judiciously perturbing the network structure the principle of our perturbation method is validated theoretically and demonstrated numerically for homogeneous and heterogeneous random networks and for different types of real networks as well the applicability of our method is discussed in terms of the relative costs of establishing links and imposing external controllers besides the practical usage of our approach its implementation elucidates interestingly the intricate relationship between certain structural properties of the network and its controllability

etude de l algebre de lie double des arbres enracines decores the double lie algebra ld of rooted trees decorated by a set d is introduced generalising the construction of connes and kreimer it is shown that it is a simple lie algebra its derivations and its automorphisms are described as well as some central extensions finally the category of lowest weight modules is introduced and studied

from rota baxter algebras to pre lie algebras rota baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics rota baxter algebras provide a construction of pre lie algebras from associative algebras in this paper we give all rota baxter operators of weight 1 on complex associative algebras in dimension 3 and their corresponding pre lie algebras

renormalization of multiple zeta values abstract multiple zeta values mzvs in the usual sense are the special values of multiple variable zeta functions at positive integers their extensive studies are important in both mathematics and physics with broad connections and applications in contrast very little is known about the special values of multiple zeta functions at non positive integers since the values are usually undefined we define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory and following the hopf algebra approach of connes and kreimer this definition of renormalized mzvs agrees with the convergent mzvs and extends the work of ihara kaneko zagier on renormalization of mzvs with positive arguments we further show that the important quasi shuffle stuffle relation for usual mzvs remains true for the renormalized mzvs

deformations of associative rota baxter operators abstract rota baxter operators and more generally o operators on associative algebras are important in probability combinatorics associative yang baxter equation and splitting of algebras using a method of uchino we construct an explicit graded lie algebra whose maurer cartan elements are given by o operators this allows us to construct cohomology for an o operator this cohomology can also be seen as the hochschild cohomology of a certain algebra with coefficients in a suitable bimodule next we study linear and formal deformations of an o operator which are governed by the above defined cohomology we introduce nijenhuis elements associated with an o operator which give rise to trivial deformations as applications we conclude deformations of weight zero rota baxter operators associative r matrices and averaging operators

birkhoff type decompositions and the baker campbell hausdorff recursion we describe a unification of several apparently unrelated factorizations arising from quantum field theory vertex operator algebras combinatorics and numerical methods in differential equations the unification is given by a birkhoff type decomposition that was obtained from the baker campbell hausdorff formula in our study of the hopf algebra approach of connes and kreimer to renormalization in perturbative quantum field theory there we showed that the birkhoff decomposition of connes and kreimer can be obtained from a certain baker campbell hausdorff recursion formula in the presence of a rota baxter operator we will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for lie algebras that arose in vertex operator algebra and conformal field theory and the even odd decomposition of combinatorial hopf algebra characters as well as the lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations

method to find community structures based on information centrality community structures are an important feature of many social biological and technological networks here we study a variation on the method for detecting such communities proposed by girvan and newman and based on the idea of using centrality measures to define the community boundaries m girvan and m e j newman proc natl acad sci u s a 99 7821 2002 we develop an algorithm of hierarchical clustering that consists in finding and removing iteratively the edge with the highest information centrality we test the algorithm on computer generated and real world networks whose community structure is already known or has been studied by means of other methods we show that our algorithm although it runs to completion in a time o n4 is very effective especially when the communities are very mixed and hardly detectable by the other methods

on the stability of some groups of formal diffeomorphisms by the birkhoff decomposition abstract let g be the group of one parameter identity tangent diffeomorphisms on the line whose coefficients are formal laurent series in the parameter e with a pole of finite order at 0 it is well known that the birkhoff decomposition can be defined in such a group we investigate the stability of the birkhoff decomposition in subgroups of g and give a formula for this decomposition these results are strongly related to renormalization in quantum field theory since it was proved by a connes and d kreimer that after dimensional regularization the unrenormalized effective coupling constants are the image by a formal identity tangent diffeomorphism of the coupling constants of the theory in the massless 6 3 theory this diffeomorphism is in g and its birkhoff decomposition gives directly the bare coupling constants and the renormalized coupling constants

maximizing modularity intensity for community partition and evolution most previous studies of community partition have often focused on the network topology the topology and link weights are in fact closely associated with each other for community formation in complex networks this paper proposes a function modularity intensity a variation of modularity density d value for evaluating the cohesiveness of a community which considers links between vertices as well as link weights the results showed that maximizing the modularity intensity not only can resolve the resolution limits problem but also achieve better performance for community partition to further evaluate the function and clarify the weight topology correlation with communities we give a simple model to simulate the topology and link weights development for community evolution and use the modularity intensity to capture communities of networks in each step of this process in this model a network is treated as a fuzzy relation and two operations of the fuzzy relation are used to make the link weights stronger and weaker respectively with the growth of the network in each evolutionary step by simulation experiments we found that in the model the modularity intensity catches communities of networks that undergo a gradual transition from faintness to clearness our model also reproduces the finding of granovetter that strong links are confined mainly in tight communities and the links between communities are predominantly weak from the results above we believe that the modularity intensity gives a more comprehensive evaluation for communities and by studying the weight topology correlation this model provides a new view for community evolution

the structure of the ladder insertion elimination lie algebra we continue our investigation into the insertion elimination lie algebra open image in new window of feynman graphs in the ladder case emphasizing the structure of this lie algebra relevant for future applications in the study of dyson schwinger equations we work out the relation to the classical infinite dimensional lie algebra open image in new window and we determine the cohomology of open image in new window

deformations and homotopy theory of relative rota baxter lie algebras we determine the l infty algebra that controls deformations of a relative rota baxter lie algebra and show that it is an extension of the dg lie algebra controlling deformations of the underlying mathsf lie mathsf rep pair by the dg lie algebra controlling deformations of the relative rota baxter operator consequently we define the cohomology of relative rota baxter lie algebras and relate it to their infinitesimal deformations a large class of relative rota baxter lie algebras is obtained from triangular lie bialgebras and we construct a map between the corresponding deformation complexes next the notion of a homotopy relative rota baxter lie algebra is introduced we show that a class of homotopy relative rota baxter lie algebras is intimately related to pre lie infty algebras

an algebraic birkhoff decomposition for the continuous renormalization group this paper aims at presenting the first steps towards a formulation of the exact renormalization group equation in the hopf algebra setting of connes and kreimer it mostly deals with some algebraic preliminaries allowing us to formulate perturbative renormalization within the theory of differential equations the relation between renormalization formulated as a change of boundary condition for a differential equation and an algebraic birkhoff decomposition for rooted trees is explicited