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

constrained spin dynamics description of random walks on hierarchical scale free networks we study a random walk problem on the hierarchical network which is a scale free network grown deterministically the random walk problem is mapped onto a dynamical ising spin chain system in one dimension with a nonlocal spin update rule which allows an analytic approach we show analytically that the characteristic relaxation time scale grows algebraically with the total number of nodes n as t n z from a scaling argument we also show the power law decay of the autocorrelation function c sigma t t alpha which is the probability to find the ising spins in the initial state sigma after t time steps with the state dependent nonuniversal exponent alpha it turns out that the power law scaling behavior has its origin in a quasiultrametric structure of the configuration space

total diameter and area of closed submanifolds the total diameter of a closed planar curve c subset mathbf r 2 is the integral of its antipodal chord lengths we show that this quantity is bounded below by twice the area of c furthermore when c is convex or centrally symmetric the lower bound is twice as large both inequalities are sharp and the equality holds in the convex case only when c is a circle we also generalize these results to m dimensional submanifolds of mathbf r n where the area will be defined in terms of the mod 2 winding numbers of the submanifold about the n m 1 dimensional affine subspaces of mathbf r n

on the non existence of certain classes of perfect p ary sequences and perfect almost p ary sequences we obtain new non existence results of perfect p ary sequences with period n called type p n the first case contains a class with type p equiv 5 pmod 8 p a qn the second case contains five types p equiv 3 pmod 4 p a q l n for certain p q and l moreover we also obtain similar non existence results of perfect almost p ary sequences

relaxation properties of small world networks recently watts and strogatz introduced the so called small world networks in order to describe systems that combine simultaneously properties of regular and random lattices in this work we study diffusion processes defined on such structures by considering explicitly the probability for a random walker to be present at the origin the results are intermediate between the corresponding ones for fractals and cayley trees

a new notion of angle between three points in a metric space we give a new notion of angle in general metric spaces more precisely given a triple a points p x q in a metric space x d we introduce the notion of angle cone pxq as being an interval pxq pxq pxq where the quantities pxq are defined in terms of the distance functions from p and q via a duality construction of differentials and gradients holding for locally lipschitz functions on a general metric space our definition in the euclidean plane gives the standard angle between three points and in a riemannian manifold coincides with the usual angle between the geodesics if x is not in the cut locus of p or q we show that in general the angle cone is not single valued even in case the metric space is a smooth riemannian manifold if x is in the cut locus of p or q but if we endow the metric space with a positive borel measure m obtaining the metric measure space x d m then under quite general assumptions which include many fundamental examples as riemannian manifolds finite dimensional alexandrov spaces with curvature bounded from below gromov hausdorff limits of riemannian manifolds with ricci curvature bounded from below and normed spaces with strictly convex norm fixed p q x the angle cone at x is single valued for m a e x x we prove some basic properties of the angle cone such as the invariance under homotheties of the space and we analyze in detail the case x d m is a measured gromov hausdorff limit of a sequence of riemannian manifolds with ricci curvature bounded from below showing the consistency of our definition with a recent construction of honda 24

total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below ii we prove as our main theorem the finiteness of topological type of a complete open riemannian manifold m with a base point p m whose radial curvature at p is bounded from below by that of a non compact model surface of revolution m which admits a finite total curvature and has no pair of cut points in a sector here a sector is by definition a domain cut off by two meridians emanating from the base point p m notice that our model m does not always satisfy the diameter growth condition introduced by abresch and gromoll in order to prove the main theorem we need a new type of the toponogov comparison theorem as an application of the main theorem we present a partial answer to milnor s open conjecture on the fundamental group of complete open manifolds

synchronization landscapes in small world connected computer networks motivated by a synchronization problem in distributed computing we studied a simple growth model on regular and small world networks embedded in one and two dimensions we find that the synchronization landscape corresponding to the progress of the individual processors exhibits kardar parisi zhang like kinetic roughening on regular networks with short range communication links although the processors on average progress at a nonzero rate their spread the width of the synchronization landscape diverges with the number of nodes desynchronized state hindering efficient data management when random communication links are added on top of the one and two dimensional regular networks resulting in a small world network large fluctuations in the synchronization landscape are suppressed and the width approaches a finite value in the large system size limit synchronized state in the resulting synchronization scheme the processors make close to uniform progress with a nonzero rate without global intervention we obtain our results by simulating the simulations based on the exact algorithmic rules supported by coarse grained arguments

computing igusa class polynomials we give an algorithm that computes the genus two class polynomials of a primitive quartic cm eld k and we give a running time bound and a proof of correctness of this algorithm this is the rst proof of correctness and the rst running time bound of any algorithm that computes these polynomials our algorithm is based on the complex analytic method of spallek and van wamelen and runs in time e o 7 2 where is the discriminant of k

many endpoints and few interior points of geodesics in the sense of all except those in a set of first baire category the space of all convex surfaces in ir endowed with hausdorfl s metric is a baire space we shall see how abnormal convex surfaces may be by proving that most of them are so

farthest points on convex surfaces in the very enjoyable book 3 p 44 of craft falconer and guy we read we take a convex surface c in r3 steinhaus asked what can be said qualitatively about the set of all farthest points from a point x our aim here is to investigate this question let s be the space of all closed convex surfaces i e boundaries of open bounded convex sets in r3 and denote for any s s and x s by fx the set of all farthest points from x and by cx the set of all points joined with x by at least two segments i e shortest paths we shall see that to any point x on a closed convex surface we may associate in a natural way a point or a jordan arc jx fx lying in the cut locus of x this will provide the topological characterization of fx further we shall prove an easily stated remarkable geometric property of fx any three of its points if it contains at least three form an obtuse or right geodesic triangle moreover the preceding result is shown to hold for a large class of geodesic triangles with vertices in the cut locus interestingly the extreme case of a right triangle does not imply the degeneracy of the surface for the reader to be familiar with aleksandrov s book 1 see also 2 4 would be of considerable help the usual intrinsic metric of s s induced by the euclidean distance in r3 will be denoted by let x s and denote by ex the cut locus of x i e the set of all endpoints different from x of maximal by inclusion segments starting at x also let e be the set of all endpoints of s i e points not interior to any segment of s the set

first passage time of multiple brownian particles on networks with applications abstract in this article we study some theoretical and technological problems with relation to multiple brownian particles on networks we are especially interested in the behavior of the first arriving brownian particle when all the brownian particles start out from the source s simultaneously and head to the destination h randomly we analyze the first passage time fpt y s h z and the mean first passage time mfpt y s h z of multiple brownian particles on complex networks equations of y s h z and y s h z are obtained on a variety of commonly encountered networks we observe first passage properties of multiple brownian particles from different aspects we find that y s h z drops substantially when particle number z increases at the first stage and converges to d s h the distance between the source and the destination when z the distribution of fpt prob y s h z t t 0 1 2 is also analyzed in these networks the distribution curve peaks up towards t d s h when z increases consequently if particle number z is set appropriately large the first arriving brownian particle will go along the shortest or near shortest paths between the source and the destination with high probability simulations confirm our analysis based on theoretical studies we also investigate some practical problems using multiple brownian particles such as communication on p2p networks optimal routing in small world networks phenomenon of asymmetry in scale free networks information spreading in social networks pervasion of viruses on the internet and so on our analytic and experimental results on multiple brownian particles provide useful evidence for further understanding and properly tackling these problems

deterministic scale free small world networks of arbitrary order abstract in many real life networks both the scale free distribution of degree and small world behavior are important features there are many random or deterministic models of networks to simulate these features separately however there are few models that combine the scale free effect and small world behavior especially in terms of deterministic versions what is more all the existing deterministic algorithms running in the iterative mode generate networks with only several discrete numbers of nodes this contradicts the purpose of creating a deterministic network model on which we can simulate some dynamical processes as widely as possible according to these facts this paper proposes a deterministic network generation algorithm which can not only generate deterministic networks following a scale free distribution of degree and small world behavior but also produce networks with arbitrary number of nodes our scheme is based on a complete binary tree and each newly generated leaf node is further linked to its full brother and one of its direct ancestors analytical computation and simulation results show that the average degree of such a proposed network is less than 5 the average clustering coefficient is high larger than 0 5 even for a network of size 2 million and the average shortest path length increases much more slowly than logarithmic growth for the majority of small world network models

spectra of real world graphs beyond the semicircle law results on the spectra of adjacency matrices corresponding to models of real world graphs we find that when the number of links grows as the number of nodes the spectral density of uncorrelated random matrices does not converge to the semicircle law furthermore the spectra of real world graphs have specific features depending on the details of the corresponding models in particular scale free graphs develop a trianglelike spectral density with a power law tail while small world graphs have a complex spectral density consisting of several sharp peaks these and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks

with respect to whom are you critical abstract for any compact riemannian surface s and any point y in s q y 1 denotes the set of all points in s for which y is a critical point and q y 1 its cardinality we proved 2 together with imre barany that q y 1 1 and that equality for all y s characterizes the surfaces homeomorphic to the sphere here we show for any orientable surface s and any point y s the following two main results there exists an open and dense set of riemannian metrics g on s for which y is critical with respect to an odd number of points in s and this is sharp if s is the torus then q y 1 5 and if s has genus g 2 then q y 1 8 g 5 properties involving points at globally maximal distance on s are eventually presented

a volume convergence theorem for alexandrov spaces with curvature bounded above we obtain a volume convergence theorem for alexandrov spaces with curvature bounded above with respect to the gromov hausdorff distance as one of the main tools proving this we construct an almost isometry between alexandrov spaces with curvature bounded above with weak singularities which are close to each other furthermore as an application of our researches of convergence phenomena for given positive integer n we prove that if a compact geodesically complete n dimensional cat 1 space has the volume sufficiently close to that of the unit n sphere then it is bi lipschitz homeomorphic to the unit n sphere

dynamic model of time dependent complex networks the characterization of the most connected nodes in static or slowly evolving complex networks has helped in understanding and predicting the behavior of social biological and technological networked systems including their robustness against failures vulnerability to deliberate attacks and diffusion properties however recent empirical research of large dynamic networks characterized by irregular connections that evolve rapidly has demonstrated that there is little continuity in degree centrality of nodes over time even when their degree distributions follow a power law this unexpected dynamic centrality suggests that the connections in these systems are not driven by preferential attachment or other known mechanisms we present an approach to explain real world dynamic networks and qualitatively reproduce these dynamic centrality phenomena this approach is based on a dynamic preferential attachment mechanism which exhibits a sharp transition from a base pure random walk scheme

sobolev spaces laplacian and heat kernel on alexandrov spaces we prove the compactness of the imbedding of the sobolev space w 1 2 0 omega into l 2 omega for any relatively compact open subset omega of an alexandrov space as a corollary the generator induced from the dirichlet energy form has discrete spectrum the generator can be approximated by the laplacian induced from the dc structure on the alexandrov space we also prove the existence of the locally holder continuous heat kernel

exploring complex networks by walking on them we carry out a comparative study of the problem of a walker searching several typical complex networks the search efficiency is evaluated for various strategies having no knowledge of the global properties of the underlying networks and the optimal path between any two given nodes it is found that the best search strategy is the self avoiding random walk the preferentially self avoiding random walk does not help in improving the search efficiency further in return topological information of the underlying networks may be drawn by comparing the results of the different search strategies

on typical degenerate convex surfaces various properties are given concerning geodesics on and distance functions from points in typical degenerate convex surfaces i e surfaces obtained by gluing together two isometric copies of typical in the sense of baire category convex bodies by identifying the corresponding points of their boundaries

parallel translation of curvature along geodesics soient m et m des varietes de riemann connexes completes de dimension 2 et certaines conditions si m est simplement connexe et si chaque sous ensemble compact du lieu de coupure de p a une mesure de hansdorff fini a 1 dimension alors il y a une immersion isometrique f m m avec f p p et df i en p

every point is critical abstract we show that for any compact alexandrov surface s without boundary and any point y in s there exists a point x in s for which y is a critical point moreover we prove that uniqueness characterizes the surfaces homeomorphic to the sphere among smooth orientable surfaces

scale free network provides an optimal pattern for knowledge transfer abstract we study numerically the knowledge innovation and diffusion process on four representative network models such as regular networks small world networks random networks and scale free networks the average knowledge stock level as a function of time is measured and the corresponding growth diffusion time is defined and computed on the four types of networks the growth diffusion times all depend linearly on the network size n as n while the slope for scale free network is minimal indicating the fastest growth and diffusion of knowledge the calculated variance and spatial distribution of knowledge stock illustrate that optimal knowledge transfer performance is obtained on scale free networks we also investigate the transient pattern of knowledge diffusion on the four networks and a qualitative explanation of this finding is proposed

extreme points of the distance function on convex surfaces we first see that in the sense of baire categories many convex surfaces have quite large cut loci and infinitely many relative maxima of the distance function from a point then we find that on any convex surface all these extreme points lie on a single subtree of the cut locus with at most three endpoints finally we confirm both in the sense of measure and in the sense of baire categories steinhaus conjecture that almost all points admit a single farthest point on the surface

continuous time quantum walks on nonorientable surfaces analytical solutions for mobius strips and klein bottles in this paper we present an extensive analytical study of continuous time quantum walks ctqws occurring on two dimensional lattices under various boundary conditions focusing on the m n lattice wrapped on mobius strips and klein bottles which are featured by the twisted boundary conditions we find that the eigenvalue spectra and transport efficiency both quantum mechanical and classical for the two structures show comparable results with those for other well known two dimensional lattices including rectangles cylinders and tori we also demonstrate that the behaviors of ctqws depend on the initial node of the excitation and the network size both on short and long timescales in addition we discover the asymmetric behaviors of limiting probabilities for mobius strips and klein bottles which are quite different from each other and are also compared to those discovered in other two dimensional networks our work provides a comprehensive understanding of recent results about ctqws on two dimensional lattices and sheds light on quantum dynamics on lattices especially those with different boundary conditions

scaling of random spreading in small world networks in this study we have carried out computer simulations of random walks on watts strogatz type small world networks and measured the mean number of visited sites and the return probabilities these quantities were found to obey scaling behavior with intuitively reasoned exponents as long as the probability p of having a long range bond was sufficiently low

multiple scale analysis of complex networks using the empirical mode decomposition method abstract empirical mode decomposition emd method can decompose any complicated data into finite intrinsic mode functions imfs in this paper we use emd method to analyze and discuss the structural properties of complex networks a random walk method is used to collect the data series of network systems utilizing the emd method we decompose the obtained data into finite imfs under different spatial scales the analysis results show that emd method is an effective tool for capturing the topological properties of network systems under different spatial scales such as the modular structures of network systems and their energy densities