topological properties of citation and metabolic networks topological properties of scale free networks are investigated by determining their spectral dimensions d s which reflect a diffusion process in the corresponding graphs data bases for citation networks and metabolic networks together with simulation results from the growing network model a l barabasi and r albert science 286 509 1999 are probed for completeness and comparisons lattice random and small world models are also investigated we find that d s is around 3 for citation and metabolic networks which is significantly different from the growing network model for which d s is approximately 7 5 this signals a substantial difference in network topology despite the observed similarities in vertex order distributions in addition the diffusion analysis indicates that the citation networks are treelike in structure whereas the metabolic networks contain many loops

synchronization of globally coupled phase oscillators singularities and scaling for general couplings abstract the onset of collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied for phase dynamical models with arbitrary coupling the effect of a stochastic temporal variation in the frequencies is also included the fokker planck equation for the coupled langevin system is reduced to a kinetic equation for the oscillator distribution function instabilities of the phase incoherent state are studied by center manifold reduction to the amplitude dynamics of the unstable modes depending on the coupling the coefficients in the normal form can be singular in the limit of weak instability when the diffusive effect of the noise is neglected a detailed analysis of these singularities to all orders in the normal form expansion is presented physically the singularities are interpreted as predicting an altered scaling of the entrained component near the onset of synchronization these predictions are verified by numerically solving the kinetic equation for various couplings and frequency distributions

uniqueness and nonuniqueness for positive radial solutions of δu f u r 0 on considere le probleme de l unicite pour les solutions positives du probleme de dirichlet non lineaire u f u x 0 dans u 0 est une boule ou un anneau de r n et f 0 est superlineaire en u et satisfaisait f 0 x 0

detection of topological patterns in complex networks correlation profile of the internet abstract a general scheme for detecting and analyzing topological patterns in large complex networks is presented in this scheme the network in question is compared with its properly randomized version that preserves some of its low level topological properties statistically significant deviation of any topological property of a network from this null model likely reflects its design principles and or evolutionary history we illustrate this basic scheme using the example of the correlation profile of the internet quantifying correlations between degrees of its neighboring nodes this profile distinguishes the internet from previously studied molecular networks with a similar scale free degree distribution we finally demonstrate that the clustering in a network is very sensitive to both the degree distribution and its correlation profile and compare the clustering in the internet to the appropriate null model

existence of many positive solutions of semilinear elliptic equations on annulus abstract we study the existence of many nonradial positive solutions in an annulus of rn it is an improvement of results in c v coffman a nonlinear boundary value problem with many positive solutions j differential equations 54 1984 429 437 an annulus is invariant under many group actions and these group actions act naturaly on the functional and leave the functional invariant therefore when we look for critical points of the functional we can restrict the functional to the manifold consisting of all fixed points under the group action we are going to use different group actions to obtain critical points and furthermore we can distinguish them

on non radially symmetric bifurcation in the annulus we discuss the radially symmetric solutions and the non radially symmetric bifurcation of the semilinear elliptic equation u 2 eu 0 in and u 0 on where x r2 a2 0 a the upper branch of radial solutions has a non radially symmetric bifurcation symmetry breaking at each k a k 1 as a 0 the radial solutions will tend to the radial solutions on the disk and 0 a 1 the critical number on the disk

a morse index formula for radial solutions of lane emden problems abstract we consider the semilinear lane emden problem e p u u p 1 u in b u 0 on b where b is the unit ball of r n n 3 centered at the origin and 1 p p s p s n 2 n 2 we prove that for any radial solution u p of e p with m nodal domains its morse index m u p is given by the formula m u p m n m 1 if p is sufficiently close to p s

stationary and nonstationary properties of evolving networks with preferential linkage networks evolving by preferential attachment of both external and internal links are investigated the rate of adding an external link is assumed to depend linearly on the degree of a preexisting node to which a new node is connected the process of creating an internal link between a pair of existing vertices is assumed to be controlled entirely by the vertex that has more links than the other vertex in the pair and the rate of creation of such a link is assumed to be in general nonlinear in the degree of the more strongly connected vertex it is shown that degree distributions of networks evolving only by creating internal links display for large degrees a nonstationary power law decay with a time dependent scaling exponent nonstationary power law behaviors are numerically shown to persist even when the number of nodes is not fixed and both external and internal connections are introduced provided that the rate of preferential attachment of internal connections is nonlinear it is argued that nonstationary effects are not unlikely in real networks although these effects may not be apparent especially in networks with a slowly varying mean degree

existence of positive nonradial solutions for nonlinear elliptic equations in annular domains we study the existence of positive nonradial solutions of equation a f u 0 in sia u 0 on d la where s x 6 w a 2 and is positive and superlinear at both 0 and oo we use a bifurcation method to show that there is a nonradial bifurcation with mode k at ak 0 1 for any positive integer k if is subcritical and for large k if is supercritical when is subcritical then a nehari type variational method can be used to prove that there exists a e 0 1 such that for any a e a 1 the equation has a nonradial solution on 0 a

measuring the dimension of partially embedded networks a b s t r a c t scaling phenomena have been intensively studied during the past decade in the context of complex networks as part of these works recently novel methods have appeared to measure the dimension of abstract and spatially embedded networks in this paper we pro pose a new dimension measurement method for networks which does not require global knowledge on the embedding of the nodes instead it exploits link wise information link lengths link delays or other physical quantities our method can be regarded as a general ization of the spectral dimension that grasps the network s large scale structure through local observations made by a random walker while traversing the links we apply the pre sented method to synthetic and real world networks including road maps the internet infrastructure and the gowalla geosocial network we analyze the theoretically and empir ically designated case when the length distribution of the links has the form p 1 we show that while previous dimension concepts are not applicable in this case the new dimension measure still exhibits scaling with two distinct scaling regimes our observa tions suggest that the link length distribution is not sufficient in itself to entirely control the dimensionality of complex networks and we show that the proposed measure provides information that complements other known measures

positive solutions for the p laplacian in annuli we consider the dirichlet problem for the equation p u q x f u in an annulus r n n 1 where p u div u p 2 u is the p laplacian operator p 1 with no assumption on the behaviour of the nonlinearity f either at zero or at infinity we prove existence and localization of positive radial solutions for this problem by applying schauder s fixed point theorem precisely we show the existence of at least one such solution each time the graph of f passes through an appropriate tunnel so it is easy to exhibit multiple or even infinitely many positive solutions moreover upper and lower bounds for the maximum value of the solution are obtained our results are easily extended to the exterior of a ball when n p

what is the dimension of citation space abstract citation networks represent the flow of information between agents they are constrained in time and so form directed acyclic graphs which have a causal structure here we provide novel quantitative methods to characterise that structure by adapting methods used in the causal set approach to quantum gravity by considering the networks to be embedded in a minkowski spacetime and measuring its dimension using myrheim meyer and midpoint scaling estimates we illustrate these methods on citation networks from the arxiv supreme court judgements from the usa and patents and find that otherwise similar citation networks have measurably different dimensions we suggest that these differences can be interpreted in terms of the level of diversity or narrowness in citation behaviour

random walks along the streets and canals in compact cities spectral analysis dynamical modularity information and statistical mechanics different models of random walks on the dual graphs of compact urban structures are considered analysis of access times between streets helps to detect the city modularity the statistical mechanics approach to the ensembles of lazy random walkers is developed the complexity of city modularity can be measured by an informationlike parameter which plays the role of an individual fingerprint of genius loci global structural properties of a city can be characterized by the thermodynamic parameters calculated in the random walk problem

nonradial sign changing solutions to lane emden problem in an annulus abstract in this paper we prove the existence of continua of nonradial solutions for the lane emden equation in the annulus in a first result we show that there are infinitely many global continua detaching from the curve of radial solutions with any prescribed number of nodal zones next using the fixed point index in cone we produce nonradial solutions with a new type of symmetry this result also applies to solutions with fixed signed showing that the set of solutions to the lane emden problem has a very rich and complex structure

stability of incoherence in a population of coupled oscillators we analyze a mean field model of coupled oscillators with randomly distributed frequencies this system is known to exhibit a transition to collective oscillations for small coupling the system is incoherent with all the oscillators running at their natural frequencies but when the coupling exceeds a certain threshold the system spontaneously synchronizes we obtain the first rigorous stability results for this model by linearizing the fokker planck equation about the incoherent state an unexpected result is that the system has pathological stability properties the incoherent state is unstable above threshold butneutrally stable below threshold we also show that the system is singular in the sense that its stability properties are radically altered by infinitesimal noise

collective synchronization in populations of globally coupled phase oscillators with drifting frequencies we generalize the kuramoto model for coupled phase oscillators by allowing the frequencies to drift in time according to ornstein uhlenbeck dynamics such drifting frequencies were recently measured in cellular populations of circadian oscillator and inspired our work linear stability analysis of the fokker planck equation for an infinite population is amenable to exact solution and we show that the incoherent state is unstable past a critical coupling strength k c gamma sigma f where gamma is the inverse characteristic drifting time and sigma f the asymptotic frequency dispersion expectedly k c agrees with the noisy kuramoto model in the large gamma schmolukowski limit but increases slower as gamma decreases asymptotic expansion of the solution for gamma 0 shows that the noiseless kuramoto model with gaussian frequency distribution is recovered in that limit thus varying a single parameter allows us to interpolate smoothly between two regimes one dominated by the frequency dispersion and the other by phase diffusion

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

superlinear problems without ambrosetti and rabinowitz growth condition superlinear elliptic boundary value problems without ambrosetti and rabinowitz growth condition are considered existence of nontrivial solution result is established by combining some arguments used by struwe and tarantello and schechter and zou also by wang and wei firstly by using the mountain pass theorem due to ambrosetti and rabinowitz is constructed a solution for almost every parameter by varying the parameter then it is considered the continuation of the solutions

spontaneous synchronization of coupled oscillator systems with frequency adaptation we study the synchronization of kuramoto oscillators with all to all coupling in the presence of slow noisy frequency adaptation in this paper we develop a new model for oscillators which adapt both their phases and frequencies it is found that this model naturally reproduc es some observed phenomena that are not qualitatively produced by the standard kuramoto model such as long waiting times before the synchronization of clapping audiences by assuming a self consistent steady state solution we find three stability regimes for the coupling constant k separated by critical points k1 and k2 i for k k2 the incoherent state becomes unstable and only the synchronized state exists iii for k1 k2 the average waiting time grows logarithmically with the number of oscillators

asymptotically radial solutions in expanding annular domains in this paper we consider the problem left begin array ll delta u u p quad rm in omega r u 0 quad quad quad rm on partial omega r quad quad quad 0 1 end array right where p 1 and r is a smooth bounded domain with a hole which is diffeomorphic to an annulus and expands as r longrightarrow infty the main goal of the paper is to prove for large r the existence of a positive solution to 0 1 which is close to the positive radial solution in the corresponding diffeomorphic annulus the proof relies on a careful analysis of the spectrum of the linearized operator at the radial solution as well as on a delicate analysis of the nondegeneracy of suitable approximating solutions

elliptic equations of order 2m in annular domains in this paper we study the existence of positive radial solutions for some semilinear elliptic problems of order 2m in an annulus with dirichlet boundary conditions we consider a nonlinearity which is either sublinear or the sum of a sublinear and a superlinear term

existence of radial solutions for nonlinear elliptic equations with gradient terms in annular domains abstract this paper is concerned with the problem of elliptic equations in annular domains involving derivative terms using nonlinear analysis methods some results regarding existence of solutions are established more precisely at least one radial solution is obtained basing on schauder s fixed point theorem and contraction mapping theorem respectively some iterative technique is also applied which allows us to overcome difficulties that the nonlinearity is related to derivative terms