scaling properties of random walks on small world networks using both numerical simulations and scaling arguments we study the behavior of a random walker on a one dimensional small world network for the properties we study we find that the random walk obeys a characteristic scaling form these properties include the average number of distinct sites visited by the random walker the mean square displacement of the walker and the distribution of first return times the scaling form has three characteristic time regimes at short times the walker does not see the small world shortcuts and effectively probes an ordinary euclidean network in d dimensions at intermediate times the properties of the walker shows scaling behavior characteristic of an infinite small world network finally at long times the finite size of the network becomes important and many of the properties of the walker saturate we propose general analytical forms for the scaling properties in all three regimes and show that these analytical forms are consistent with our numerical simulations

orbifold index and equivariant k homology let g be countable group and m be a proper cocompact even dimensional g manifold with orbifold quotient bar m let d be a g invariant dirac operator on m it induces an equivariant k homology class d in k g 0 m and an orbifold dirac operator bar d on bar m composing the assembly map k g 0 m rightarrow k 0 c g with the homomorphism k 0 c g rightarrow mathbb z given by the representation c g rightarrow mathbb c of the maximal group c algebra induced from the trivial representation of g we define index d in mathbb z in the second section of the paper we show that index bar d index d and obtain explicit formulas for this integer in the third section we review the decomposition of k 0 g m in terms of the contributions of fixed point sets of finite cyclic subgroups of g obtained by w luck in particular the class d decomposes in this way in the last section we derive an explicit formula for the contribution to d associated to a finite cyclic subgroup of g

return times of random walk on generalized random graphs random walks are used for modeling various dynamics in for example physical biological and social contexts furthermore their characteristics provide us with useful information on the phase transition and critical phenomena of even broader classes of related stochastic models abundant results are obtained for random walk on simple graphs such as the regular lattices and the cayley trees however random walks and related processes on more complex networks which are often more relevant in the real world are still open issues possibly yielding different characteristics in this paper we investigate the return times of random walks on random graphs with arbitrary vertex degree distributions we analytically derive the distributions of the return times the results are applied to some types of networks and compared with numerical data

noncommutative tachyons and k theory we show that the relation between d branes and noncommutative tachyons leads very naturally to the relation between d branes and k theory we also discuss some relations between d branes and k homology provide a noncommutative generalization of the abs construction and give a simple physical interpretation of bott periodicity in addition a framework for constructing neveu schwarz fivebranes as noncommutative solitons is proposed

parallel demodulation of multidimensional signals the design of constellations that allow a high degree of parallelism in the staged decoder structure and in addition allow soft decoding of the component algebraic codes based on a systolic algorithm amenable to vlsi implementation is considered combination of these multidimensional constellations with trellis coded modulation tcm is also considered set partitioning is based on the partition of a linear code into a subcode and its cosets and the author shows that a highly parallel demodulator structure can also be applied when tcm is used

geometric k homology of flat d branes we use the baum douglas construction of k homology to explicitly describe various aspects of d branes in type ii superstring theory in the absence of background supergravity form fields we rigorously derive various stability criteria for states of d branes and show how standard bound state constructions are naturally realized directly in terms of topological k cycles we formulate the mechanism of flux stabilization in terms of the k homology of non trivial fibre bundles along the way we derive a number of new mathematical results in topological k homology of independent interest

on the generalized hamming weights of geometric goppa codes we study the generalized hamming weights of geometric goppa codes giving a translation of these weights into arithmetical terms concerning the arithmetic of the curve used to construct the code we derive some results and bounds that we apply to some known codes

random walk and trapping processes on scale free networks in this work i investigate the dynamics of random walk processes on scale free networks in a short to moderate time scale i perform extensive simulations for the calculation of the mean squared displacement the network coverage and the survival probability on a network with a concentration c of static traps it is shown that the random walkers remain close to their origin but cover a large part of the network at the same time this behavior is markedly different than usual random walk processes in the literature for the trapping problem i numerically compute phi n c the survival probability of mobile species at time n as a function of the concentration of trap nodes c comparison of these results to the rosenstock approximation indicate that this is an adequate description for networks with 2 3 the behavior is more complicated and one needs to employ a truncated cumulant expansion

anomalous electrical and frictionless flow conductance in complex networks we study transport properties such as electrical and frictionless flow conductance on scale free and erd s renyi networks we consider the conductance g between two arbitrarily chosen nodes where each link has the same unit resistance our theoretical analysis for scale free networks predicts a broad range of values of g with a power law tail distribution where gg 2 1 where is the decay exponent for the scale free network degree distribution we confirm our predictions by simulations of scale free networks solving the kirchhoff equations for the conductance between a pair of nodes the power law tail in leads to large values of g thereby significantly improving the transport in scale free networks compared to erd s renyi networks where the tail of the conductivity distribution decays exponentially based on a simple physical transport backbone picture we suggest that the conductances of scale free and erd s renyi networks can be approximated by ckakb ka kb for any pair of nodes a and b with degrees ka and kb thus a single quantity c which depends on the average degree of the network characterizes transport on both scale free and erd s renyi networks we determine that c tends to 1 for increasing and it is larger for scale free networks we compare the electrical results with a model for frictionless transport where conductance is defined as the number of link independent paths between a and b and find that a similar picture holds the effects of distance on the value of conductance are considered for both models and some differences emerge finally we use a recent data set for the as autonomous system level of the internet and confirm that our results are valid in this real world example

ramond ramond fields fractional branes and orbifold differential k theory we study d branes and ramond ramond fields on global orbifolds of type ii string theory with vanishing h flux using methods of equivariant k theory and k homology we illustrate how bredon equivariant cohomology naturally realizes stringy orbifold cohomology we emphasize its role as the correct cohomological tool which captures known features of the low energy effective field theory and which provides new consistency conditions for fractional d branes and ramond ramond fields on orbifolds we use an equivariant chern character from equivariant k theory to bredon cohomology to define new ramond ramond couplings of d branes which generalize previous examples we propose a definition for groups of differential characters associated to equivariant k theory we derive a dirac quantization rule for ramond ramond fluxes and study flat ramond ramond potentials on orbifolds

bounds on the weight hierarchy of generalized concatenated codes we determine upper and lower bounds to the generalized hamming weights of generalized concatenated codes they are derived from the generalized hamming weights of the outer codes and certain properties of the inner code and its partitions

quadratic functions in geometry topology and m theory we describe an interpretation of the kervaire invariant of a riemannian manifold of dimension 4k 2 in terms of a holomorphic line bundle on the abelian variety h 2k 1 m otimes r z our results are inspired by work of witten on the fivebrane partition function in m theory hep th 9610234 hep th 9609122 our construction requires a refinement of the algebraic topology of smooth manifolds better suited to the needs of mathematical physics and is based on our theory of differential functions these differential functions generalize the differential characters of cheeger simons and the bulk of this paper is devoted to their study

chern characters for proper equivariant homology theories and applications to k and l theory we construct for an equivariant homology theory for proper equiv ariant cw complexes an equivariant chern character provided that certain conditions are satisfied this applies for instance to the sources of the assembly maps in the farrell jones conjecture with respect to the family f of finite subgroups and in the baum connes conjecture thus we get an explicit calculation in terms of group homology of qn z knorgu and qn z lnorgu for a commutative ring r with qh r provided the farrell jones conjecture with respect to f is true and of qnz k top n c r og fu for fa r c provided the baum connes conjecture is true

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

on the generalized hamming weights of product codes the rth generalized hamming weight of a linear code is the minimum support size of any r dimensional subcode it has been found useful in the studies of cryptography and trellis coding the authors derive several results on expressing the generalized hamming weights of a product code in terms of those of its component codes they also formulate a general conjecture

anomalous biased diffusion in networks we study diffusion with a bias toward a target node in networks this problem is relevant to efficient routing strategies in emerging communication networks like optical networks bias is represented by a probability p of the packet or particle to travel at every hop toward a site that is along the shortest path to the target node we investigate the scaling of the mean first passage time mfpt with the size of the network we find by using theoretical analysis and computer simulations that for random regular rr and erd ifmmode mbox h o else h o fi s r enyi networks there exists a threshold probability p mathrm th such that for pl p mathrm th the mfpt scales anomalously as n ensuremath alpha where n is the number of nodes and ensuremath alpha depends on p for pg p mathrm th the mfpt scales logarithmically with n the threshold value p mathrm th of the bias parameter for which the regime transition occurs is found to depend only on the mean degree of the nodes an exact solution for every value of p is given for the scaling of the mfpt in rr networks the regime transition is also observed for the second moment of the probability distribution function the standard deviation for the case of scale free sf networks we present analytical bounds and simulations results showing that the mfpt scales at most as mathrm ln n to a positive power for any finite bias which means that in sf networks even a very small bias is considerably more efficient in comparison to unbiased walk

stratified fiber bundles quinn homology and brane stability of hyperbolic orbifolds we revisit the problem of stability of string vacua involving hyperbolic orbifolds using methods from homotopy theory and k homology we propose a definition of type ii string theory on such backgrounds that further carry stratified systems of fiber bundles which generalize the more conventional orbifold and symmetric string backgrounds together with a classification of wrapped branes by a suitable generalized homology theory for spaces stratified fibered over hyperbolic orbifolds we use the algebraic k theory of their fundamental groups and quinn homology to derive criteria for brane stability in terms of an atiyah hirzebruch type spectral sequence with its lift to k homology stable d branes in this setting carry stratified charges which induce new additive structures on the corresponding k homology groups we extend these considerations to backgrounds which support h flux where we use k groups of twisted group algebras of the fundamental groups to analyze stability of locally symmetric spaces with k amenable isometry groups and derive stability conditions for branes wrapping the fibers of an eilenberg maclane spectrum functor

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

dimension length profiles and trellis complexity of linear block codes this semi tutorial paper discusses the connections between the dimension length profile dlp of a linear code which is essentially the same as its generalized hamming weight hierarchy and the complexity of its minimal trellis diagram these connections are close and deep dlp duality is closely related to trellis duality the dlp of a code gives tight bounds on its state and branch complexity profiles under any coordinate ordering these bounds can often be met a maximum distance separable mds code is characterized by a certain extremal dlp from which the main properties of mds codes are easily derived the simplicity and generality of these interrelationships are emphasized

transport in weighted networks partition into superhighways and roads transport in weighted networks is dominated by the minimum spanning tree mst the tree connecting all nodes with the minimum total weight we find that the mst can be partitioned into two distinct components having significantly different transport properties characterized by centrality the number of times a node or link is used by transport paths one component superhighways is the infinite incipient percolation cluster for which we find that nodes or links with high centrality dominate for the other component roads which includes the remaining nodes low centrality nodes dominate we find also that the distribution of the centrality for the infinite incipient percolation cluster satisfies a power law with an exponent smaller than that for the entire mst the significance of this finding is that one can improve significantly the global transport by improving a tiny fraction of the network the superhighways

gradient mechanism in a communication network we study the efficiency of the gradient mechanism of message transfer in a two dimensional communication network of regular nodes and randomly distributed hubs each hub on the network is assigned some randomly chosen capacity and hubs with lower capacities are connected to the hubs with maximum capacity the average travel times of single messages traveling on the lattice decrease rapidly as the number of hubs increase the functional dependence of the average travel times on the hub density shows q exponential behavior with a power law tail we also study the relaxation behavior of the network when a large number of messages are created simultaneously at random locations and travel on the network toward their designated destinations for this situation in the absence of the gradient mechanism the network can show congestion effects due to the formation of transport traps we show that if hubs of high betweenness centrality are connected by the gradient mechanism efficient decongestion can be achieved the gradient mechanism is less prone to the formation of traps than other decongestion schemes we also study the spatial configurations of transport traps and propose minimal strategies for their elimination

phase transition and diffusivity in social hierarchies with attractive sites we study the effects of including a distribution of valuable or attractive sites in a two dimensional lattice in selforganizing social hierarchies agents move aleatorily except in the case where an attractive site is located in their neighborhood we find that the transition between an egalitarian society at low population density and a hierarchical one at high population density strongly depends on the distribution and percolation of strategic sites also it is shown how agent diffusivity is closely related to the amount of inequality the proposed model introduces an optimization aspect to the problem of social hierarchies since the system tends to maximize the occupation of attractive sites wealth per capita however when the density of attractive sites is small the system fails to reach this state and is trapped in a local minimum as in a glass or jam transition r 2006 elsevier b v all rights reserved

twisted orbifold k theory we use equivariant methods to define and study the orbifold k theory of an orbifold x adapting techniques from equivariant k theory we construct a chern character and exhibit a multiplicative decomposition for k orb x in particular showing that it is additively isomorphic to the orbifold cohomology of x a number of examples are provided we then use the theory of projective representations to define the notion of twisted orbifold k theory in the presence of discrete torsion an explicit expression for this is obtained in the case of a global quotient

explicit determination of mean first passage time for random walks on deterministic uniform recursive trees the determination of mean first passage time mfpt for random walks in networks is a theoretical challenge and is a topic of considerable recent interest within the physics community in this paper according to the known connections between mfpt effective resistance and the eigenvalues of graph laplacian we first study analytically the mfpt between all node pairs of a class of growing treelike networks which we term deterministic uniform recursive trees durts since one of its particular cases is a deterministic version of the famous uniform recursive tree the interesting quantity is determined exactly through the recursive relation of the laplacian spectra obtained from the special construction of durts the analytical result shows that the mfpt between all couples of nodes in durts varies as n text text ln text n for large networks with node number n second we study trapping on a particular network of durts focusing on a special case with the immobile trap positioned at a node having largest degree we determine exactly the average trapping time att that is defined as the average of fpt from all nodes to the trap in contrast to the scaling of the mfpt the leading behavior of att is a linear function of n interestingly we show that the behavior for att of the trapping problem is related to the trapping location which is in comparison with the phenomenon of trapping on fractal t graph although both networks exhibit tree structure finally we believe that the methods could open the way to exactly calculate the mfpt and att in a wide range of deterministic media

on the optimum bit orders with respect to the state complexity of trellis diagrams for binary linear codes it was shown earlier that for a punctured reed muller rm code or a primitive bch code which contains a punctured rm code of the same minimum distance as a large subcode the state complexity of the minimal trellis diagrams is much greater than that for an equivalent code obtained by a proper permutation of the bit positions the problem of finding a permutation of the bit positions for a given code that minimizes the state complexity of its minimal trellis diagram is related to the generalized hamming weight hierarchy of a code and it is shown that for rm codes the standard binary order of bit positions is optimum at every bit position with respect to the state complexity of a minimal trellis diagram by using a theorem due to v k wei 1991 the state complexity of the trellis diagram for the extended and permuted 64 24 bch code is discussed

a highly clustered scale free network evolved by random walking abstract in present paper we propose a highly clustered weighted network model that incorporates the addition of a new node with some links new links between existing nodes and the edge s weight dynamical evolution based on weight dependent walks at each time step the analytical approach and numerical simulation show that the system grows into a weighted network with the power law distributions of strength weight and degree the weight dependent walk length l will not influence the strength distribution but the clustering coefficient of the network is sensitive to l particularly the clustering coefficient is especially high and almost independent of the network size when l 2

ring structures and mean first passage time in networks in this paper we address the problem of the calculation of the mean first passage time on generic graphs we focus in particular on the mean first passage time on a node s for a random walker starting from a generic unknown node x we introduce an approximate scheme of calculation which maps the original process in a markov process in the space of the so called rings described by a transition matrix of size o ln n ln k ln n ln k where n is the size of the graph and k the average degree in the graph in this way one has a drastic reduction of degrees of freedom with respect to the size n of the transition matrix of the original process corresponding to an extremely low computational cost we first apply the method to the erdos renyi random graphs for which the method allows for almost perfect agreement with numerical simulations then we extend the approach to the barabasi albert graph as an example of scale free graph for which one obtains excellent results finally we test the method with two real world graphs internet and a network of the brain for which we obtain accurate results

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

first passage properties of the erdos renyi random graph we study the mean time for a random walk to traverse between two arbitrary sites of the erd s renyi random graph we develop an effective medium approximation that predicts that the mean first passage time between pairs of nodes as well as all moments of this first passage time are insensitive to the fraction p of occupied links this prediction qualitatively agrees with numerical simulations away from the percolation threshold near the percolation threshold the statistically meaningful quantity is the mean transit rate namely the inverse of the first passage time this rate varies non monotonically with p near the percolation transition much of this behaviour can be understood by simple heuristic arguments

scaling of distances in correlated complex networks abstract the influence of node node degree correlations on distances in complex networks has been studied we have found that even the presence of strong correlations in complex networks does not break a universal scaling of distances between vertices of such networks as science collaboration networks biological networks internet autonomous systems and public transport systems a mean distance between two nodes of degrees k i and k j in such networks equals to l ij a b log k i k j for a fixed value of the product k i k j the scaling is valid over several decades parameters a and b depend on the mean value of a node degree k nn calculated for the nearest neighbors we have found that extending our simple theory basing on a random branching tree by the first order node degree correlations improves theoretical predictions for parameters a and b in assortative networks while it fails in disassortative ones

static pairwise annihilation in complex networks we study static annihilation on complex networks in which pairs of connected particles annihilate at a constant rate during time through a mean field formalism we compute the temporal evolution of the distribution of surviving sites with an arbitrary number of connections this general formalism which is exact for disordered networks is applied to kronecker erd os r enyi i e poisson and scale free networks we compare our theoretical results with extensive numerical simulations obtaining excellent agreement although the mean field approach applies in an exact way neither to ordered lattices nor to small world networks it qualitatively describes the annihilation dynamics in such structures our results indicate that the higher the connectivity of a given network element the faster it annihilates this fact has dramatic consequences in scale free networks for which once the hubs have been annihilated the network disintegrates and only isolated sites are left