periodic orbits near bifurcations of codimension two classical mechanics semiclassics and stokes transitions we investigate classical and semiclassical aspects of codimension two bifurcations of periodic orbits in hamiltonian systems a classification of these bifurcations in autonomous systems with two degrees of freedom or time periodic systems with one degree of freedom is presented we derive uniform approximations to be used in semiclassical trace formulae and determine also certain global bifurcations in conjunction with stokes transitions that become important in the ensuing diffraction catastrophe integrals

reversals and palindromes in continued fractions several results on continued fractions expansions are on indirect consequences of the mirror formula we survey occurrences of this formula for sturmian real numbers for simultaneous diophantine approximation and for formal power series

on quantisation using periodic classical orbits two approaches to semiclassical quantisation of integrable systems using periodic classical orbits are considered they both lead to approximate formulae for the density of states function a delta function at each energy level the first due to gutzwiller 1971 involves a sum over isolated stable periodic orbits of the system and leads to the harmonic approximation to the eigenvalues the second due to berry and tabor 1976 involves a sum over families of periodic orbits and leads to the ebk torus approximation to the eigenvalues here the author extracts a modified form of the gutzwiller series from the berry tabor series by using a uniform approximation and hence show that the complete spectrum involves both these series the analysis demonstrates that genuine semiclassical quantisation rules for generic systems using periodic orbits will involve uniform approximation which more closely reflects the underlying classical structure than do the existing stationary phase approximations

coauthorship networks and patterns of scientific collaboration by using data from three bibliographic databases in biology physics and mathematics respectively networks are constructed in which the nodes are scientists and two scientists are connected if they have coauthored a paper we use these networks to answer a broad variety of questions about collaboration patterns such as the numbers of papers authors write how many people they write them with what the typical distance between scientists is through the network and how patterns of collaboration vary between subjects and over time we also summarize a number of recent results by other authors on coauthorship patterns

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

semiclassical trace formulas for pitchfork bifurcation sequences in non integrable hamiltonian systems with mixed phase space and discrete symmetries sequences of pitchfork bifurcations of periodic orbits pave the way from integrability to chaos in extending the semiclassical trace formula for the spectral density we develop a uniform approximation for the combined contribution of pitchfork bifurcation pairs for a two dimensional double well potential and the familiar h enon heiles potential we obtain very good agreement with exact quantum mechanical calculations we also consider the integrable limit of the scenario which corresponds to the bifurcation of a torus from an isolated periodic orbit for the separable version of the h enon heiles system we give an analytical uniform trace formula which also yields the correct harmonic oscillator su 2 limit at low energies and obtain excellent agreement with the slightly coarse grained quantum mechanical density of states

attack vulnerability of complex networks we study the response of complex networks subject to attacks on vertices and edges several existing complex network models as well as real world networks of scientific collaborations and internet traffic are numerically investigated and the network performance is quantitatively measured by the average inverse geodesic length and the size of the largest connected subgraph for each case of attacks on vertices and edges four different attacking strategies are used removals by the descending order of the degree and the betweenness centrality calculated for either the initial network or the current network during the removal procedure it is found that the removals by the recalculated degrees and betweenness centralities are often more harmful than the attack strategies based on the initial network suggesting that the network structure changes as important vertices or edges are removed furthermore the correlation between the betweenness centrality and the degree in complex networks is studied

uniform approximation for bifurcations of periodic orbits with high repetition numbers the semiclassical contribution of a periodic orbit to the quantum density of states diverges when the orbit bifurcates in this case one has to apply approximations which are uniformly valid both in and a parameter which describes the distance to the bifurcation the form of the approximation depends on the repetition number m of the orbit that bifurcates in a two dimensional system the approximations are different for m 1 up to m 5 and for they have the same form as for m 5 in this article we consider the case which occurs first when an integrable system is perturbed a uniform approximation for the contribution to the spectral density is derived which in the limit of large reduces to a sum of semiclassical contributions of isolated periodic orbits

tunnelling in near integrable systems tunnelling rates are qualitatively different for integrable and near integrable systems here a uniform result is derived which interpolates between the two regimes and is applied successfully to two dimensional double well and pendulum potentials when the underlying symmetry is reflection in a single coordinate the splitting remains positive after perturbation but for potentials whose underlying symmetry is reflection through the origin the splitting is predicted to oscillate as a function of a perturbation parameter with regularly spaced zeros where tunnelling switches off

centrality properties of directed module members in social networks several recent studies of complex networks have suggested algorithms for locating network communities also called modules or clusters which are mostly defined as groups of nodes with dense internal connections along with the rapid development of these clustering techniques the ability of revealing overlaps between communities has become very important as well an efficient search technique for locating overlapping modules is provided by the clique percolation method cpm and its extension to directed graphs the cpmd algorithm here we investigate the centrality properties of directed module members in social networks obtained from e mail exchanges and from sociometric questionnaires our results indicate that nodes in the overlaps between modules play a central role in the studied systems furthermore the two different types of networks show interesting differences in the relation between the centrality measures and the role of the nodes in the directed modules

entropy optimization of scale free networks robustness to random failures abstract many networks are characterized by highly heterogeneous distributions of links which are called scale free networks and the degree distributions follow p k ck we study the robustness of scale free networks to random failures from the character of their heterogeneity entropy of the degree distribution can be an average measure of a network s heterogeneity optimization of scale free networks robustness to random failures with average connectivity constant is equivalent to maximizing the entropy of the degree distribution by examining the relationship of the entropy of the degree distribution scaling exponent and the minimal connectivity we get the optimal design of scale free networks to random failures we conclude that the entropy of the degree distribution is an effective measure of network s resilience to random failures

spatially self organized resilient networks by a distributed cooperative mechanism abstract the robustness of connectivity and the efficiency of paths are incompatible in many real networks we propose a self organization mechanism for incrementally generating onion like networks with positive degree degree correlations whose robustness is nearly optimal as a spatial extension of the generation model based on cooperative copying and adding shortcut we show that the growing networks become more robust and efficient through enhancing the onion like topological structure on a space the reasonable constraint for locating nodes on the perimeter in typical surface growth as a self propagation does not affect these properties of the tolerance and the path length moreover the robustness can be recovered in the random growth damaged by insistent sequential attacks even without any remedial measures

invariant measures and the set of exceptions to littlewood s conjecture we classify the measures on sl k r sl k z which are invariant and ergodic under the action of the group a of positive diagonal matrices with positive entropy we apply this to prove that the set of exceptions to littlewood s conjecture has hausdorff dimension zero

universal twinkling exponents for spectral fluctuations associated with mixed chaology for systems that are neither fully integrable nor fully chaotic bifurcations of periodic orbits give rise to semiclassically emergent singularities in the fluctuating part n fl of the energy level counting function the bifurcations dominate the spectral moments m m n fl 2 m in the limit 0 we argue that m m const vm and calculate the twinkling exponents v m as the result of a competition between bifurcations with different codimensions and repetition numbers

asymptotic estimates for efficiency vulnerability and cost for random networks an investigation both analytical and numerical of the values of efficiency vulnerability and cost functions is provided several comparisons among these functions are made following a pairwise strategy and taking into account different efficiency vulnerability and cost functions it is shown in all cases that these magnitudes display strong correlations which are typically piecewise linear in log space as well as universal i e independent of the number of nodes n to a great extent the analytical part of the work develops explicit bounds linking all pairs of such parameters these results are numerically compared with the result of a random testing for erdos renyi networks

on a problem in simultaneous diophantine approximation schmidt s conjecture for any i j 0 with i j 1 let bad i j denote the set of points x y 2 r 2 for which maxfkqxk 1 i kqyk 1 j g c q for all q 2 n here c c x y is a positive constant our main result implies that any nite intersection of such sets has full dimension this settles a conjecture of wolfgang m schmidt in the theory of simultaneous diophantine approximation

the work of einsiedler katok and lindenstrauss on the littlewood conjecture this document is intended as a slightly expanded writeup of my anticipated talk at the ams current events bulletin in new orleans january 2007 it is a brief report on the work of einsiedler katok and lindenstrauss on the littlewood conjecture 5 it is not intended in any sense for specialists and is indeed aimed at readers without any specific background either in measure theory dynamics or number theory any reader with any background in ergodic theory will be better served by consulting either the original paper or one of the surveys written by those authors see 7 and 12

efficient calculation of the robustness measure r for complex networks abstract in a recent work schneider et al 2011 proposed a new measure r for network robustness where the value of r is calculated within the entire process of malicious node attacks in this paper we present an approach to improve the calculation efficiency of r in which a computationally efficient robustness measure r is introduced when the fraction of failed nodes reaches to a critical threshold q c simulation results on three different types of network models and three real networks show that these networks all exhibit a computationally efficient robustness measure r the relationships between r and the network size n and the network average degree k are also explored it is found that the value of r decreases with n while increases with k our results would be useful for improving the calculation efficiency of network robustness measure r for complex networks

preferential network perturbation abstract we study network robustness to perturbation for a pair of non directional networks motivated by applications in cancer network modelling the susceptibility of single networks to attack and failure has been well studied but a similar problem for a pair of networks with non zero similarity has so far not been considered in the current approach we wish to maximise the perturbation of one network while minimising that of a second network thus achieving a preferential perturbation we focus on network pairs derived from a single underlying network which we call correlated networks and use model networks to generate results and principles for general network topologies we investigate a number of vertex ranking methods to maximise the preferential perturbation each involving successive vertex removal and we show that novel robustness or susceptibility metrics are required for preferential perturbation we propose that the average perturbation gradient up to maximum perturbation is generally the most suitable metric which accounts for both the extent of perturbation and fraction of vertices removed vertex betweenness is found to be the optimum vertex ranking method for a network ensemble according to this metric which requires no knowledge of the second network for a particular network pair however none of the methods studied is guaranteed to be optimum the maximum perturbation score is an alternative susceptibility metric which is suitable if there is no cost associated with vertex removal preferential perturbation is strongly influenced by network similarity and we investigate other topological dependencies

when is a scale free graph ultra small in this paper we study typical distances in the configuration model when the degrees have asymptotically infinite variance we assume that the empirical degree distribution follows a power law with exponent tau in 2 3 up to value n beta n for some beta n gg log n gamma and gamma in 0 1 this assumption is satisfied for power law i i d degrees and also includes truncated power law empirical degree distributions where the possibly exponential truncation happens at n beta n these examples are commonly observed in many real life networks we show that the graph distance between two uniformly chosen vertices centers around 2 log log n beta n log tau 2 1 beta n 3 tau with tight fluctuations thus the graph is an ultrasmall world whenever 1 beta n o log log n we determine the distribution of the fluctuations around this value in particular we prove these form a sequence of tight random variables with distributions that show log log periodicity and as a result it is non converging we describe the topology and number of shortest paths we show that the number of shortest paths is of order n f n beta n where f n in 0 1 is a random variable that oscillates with n we decompose shortest paths into three segments two end segments starting at each of the two uniformly chosen vertices and a middle segment the two end segments of any shortest path have length log log n beta n log tau 2 tight and the total degree is increasing towards the middle of the path on these segments the connecting middle segment has length 1 beta n 3 tau tight and it contains only vertices with degree at least of order n 1 f n beta n thus all the degrees on this segment are comparable to the maximal degree our theorems also apply when instead of truncating the degrees we start with a configuration model and we remove every vertex with degree at least n beta n and the edges attached to these vertices this sheds light on the attack vulnerability of the configuration model with infinite variance degrees

stability and continuity of centrality measures in weighted graphs this paper presents a formal definition of stability for node centrality measures in weighted graphs it is shown that the commonly used measures of degree closeness and eigenvector centrality are stable whereas betweenness centrality is not an alternative definition of the latter that preserves the same centrality notion while satisfying the stability criteria is introduced continuity is presented as a less stringent alternative to stability betweenness centrality is shown to be not only unstable but discontinuous numerical experiments in synthetic random networks and real world data show that in practice stability and continuity imply different levels of robustness in the presence of noisy data in particular the stable betweenness centrality is shown to exhibit resilience against noise that is absent in the discontinuous and unstable standard betweenness centrality while preserving a similar notion of centrality

bifurcations of periodic orbits and uniform approximations we derive uniform approximations for contributions to gutzwiller s periodic orbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space there orbits lie close together and give collective contributions while the individual contributions of gutzwiller s type would diverge at the bifurcation new results for the tangent the period doubling and the period tripling bifurcation are given they are obtained by going beyond the local approximation and including higher order terms in the normal form of the action the uniform approximations obtained are tested on the kicked top and are found to be in excellent agreement with exact quantum results

spin glass model for the c dismantling problem c dismantling cd problem aims at finding the minimum vertex set d of a graph g v e after removing which the remaining graph will break into connected components with the size not larger than c in this paper we introduce a spin glass model with c 1 integer value states into the cd problem and then study the properties of this spin glass model by the belief propagation bp equations under the replica symmetry ansatz we give the lower bound rho c of the relative size of d with finite c on regular random graphs and erdos renyi random graphs we find rho c will decrease gradually with growing c and it converges to rho infty as c to infty the cd problem is called dismantling problem when c is a small finite fraction of v therefore rho infty is also the lower bound of the dismantling problem when v to infty to reduce the computation complexity of the bp equations taking the knowledge of the probability of a random selected vertex belonging to a remaining connected component with the size a the original bp equations can be simplified to one with only three states when c to infty the simplified bp equations are very similar to the bp equations of the feedback vertex set spin glass model h j zhou eur phys j b 86 455 2013 at last we develop two practical belief propagation guide decimation algorithms based on the original bp equations cd bpd and the simplified bp equations scd bpd to solve the cd problem on a certain graph our bpd algorithms and two other state of art heuristic algorithms are applied on various random graphs and some real world networks computation results show that the cd bpd is the best in all tested algorithms in the case of small c but considering the performance and computation consumption we recommend using scd bpd for the network with small clustering coefficient when c is large

average hopcount of the shortest path in tree like components with finite size abstract an exact formula for computing the average hopcount of the shortest path in finite size tree like components of undirected unweighted random networks is proposed in a tree like component with size s there exists virtually only one shortest path between two arbitrary nodes the summation of hopcounts of all shortest paths can be calculated approximately by the summation of the betweenness of all nodes and the difference between them is only a constant s s 1 therefore the average hopcount can be calculated by further dividing the summation by the number of all shortest paths in this paper we first derive the conditional probability p k s of the degree distribution of finite components with size s and the summation of all nodal betweenness respectively by means of these results we obtain the exact formula for calculating the average hopcount also we confirm the proposed formula by simulations for networks with poisson and power law degree distributions respectively

diophantine approximation for products of linear maps logarithmic improvements this paper is devoted to the study of a problem of cassels in multiplicative diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points it was previously known that values of this product become arbitrary close to zero and we establish that in fact they approximate zero with an explicit rate our approach is based on investigating quantitative density of orbits of higher rank abelian groups

occurrence of periodic lame functions at bifurcations in chaotic hamiltonian systems we investigate cascades of isochronous pitchfork bifurcations of straight line librating orbits in some two dimensional hamiltonian systems with mixed phase space we show that the new bifurcated orbits which are responsible for the onset of chaos are given analytically by the periodic solutions of the lame equation as classified in 1940 by ince in hamiltonians with c2v symmetry they occur alternatingly as lame functions of period 2k and 4k respectively where 4k is the period of the jacobi elliptic function appearing in the lame equation we also show that the two pairs of orbits created at period doubling bifurcations of island chain type are given by two different linear combinations of algebraic lame functions with period 8k

optimization of random searches on defective lattice networks we study the general problem of how to search efficiently for targets randomly located on defective lattice networks i e regular lattices which have some fraction of its nodes randomly removed we consider large but finite triangular lattices and assume for the search dynamics that the walker chooses steps lengths lj from the power law distribution p lj approximately lj mu with the exponent mu regulating the strategy of the search process at each step lj the searcher moves in straight lines and constantly looks within a detection radius of vision rv for the targets along the way if there is contact with a defect the movement stops and a new step length is chosen hence the presence of defects decreases the efficiency of the overall process we study numerically how three different aspects of the lattice influence the optimization of the search efficiency i the type of boundary conditions ii the concentration of targets and defects and iii the category or class of search destructive nondestructive or regenerative motivated by the results we develop a type of mean field model for the problem and obtain an analytical approximation for the search efficiency function finally we discuss in the context of searches how defective lattices compare with perfect lattices and with continuous environments

enhancing robustness and immunization in geographical networks we find that different geographical structures of networks lead to varied percolation thresholds although these networks may have similar abstract topological structures thus strategies for enhancing robustness and immunization of a geographical network are proposed using the generating function formalism we obtain an explicit form of the percolation threshold q c for networks containing arbitrary order cycles for three cycles the dependence of q c on the clustering coefficients is ascertained the analysis substantiates the validity of the strategies with analytical evidence

fingerprinting networks correlations of local and global network properties in complex networks a common task is to identify the most important or central nodes there are several definitions often called centrality measures which often lead to different results here we introduce fingerprints of networks which we define as correlation plots of local and global network properties we show that these fingerprints are suitable tools for characterizing networks beyond single quantity distributions in particular we study the correlations between four local and global measures namely the degree the shortest path betweenness the random walk betweenness and the subgraph centrality on different random network models like erd s renyi small world and barabasi albert as well as on different real networks like metabolic pathways social collaborations and computer networks and compare those fingerprints to determine the quality of those basic models the correlation fingerprints are quite different between the real networks and the model networks questioning whether the models really reflect all important properties of the real world

uniform approximation for period quadrupling bifurcations we derive a uniform approximation for semiclassical contributions of periodic orbits to the spectral density which is valid for generic period quadrupling bifurcations in systems with a mixed phase space these bifurcations involve three periodic orbits which coalesce at the bifurcation in the vicinity of the bifurcation the three orbits give a collective contribution to the spectral density while the individual contributions of gutzwiller s type would diverge at the bifurcation the uniform approximation is obtained by mapping the action function onto the normal form corresponding to the bifurcation this article is a continuation of previous work in which uniform approximations for generic period m tupling bifurcations with were derived

the effect of pitchfork bifurcations on the spectral statistics of hamiltonian systems we present a quantitative semiclassical treatment of the effects of bifurcations on the spectral rigidity and the spectral form factor of a hamiltonian quantum system defined by two coupled quartic oscillators which on the classical level exhibits mixed phase space dynamics we show that the signature of a pitchfork bifurcation is two fold beside the known effect of an enhanced periodic orbit contribution due to its peculiar hbar dependence at the bifurcation we demonstrate that the orbit pair born at the bifurcation gives rise to distinct deviations from universality slightly above the bifurcation this requires a semiclassical treatment beyond the so called diagonal approximation our semiclassical predictions for both the coarse grained density of states and the spectral rigidity are in excellent agreement with corresponding quantum mechanical results

effect of edge removal on topological and functional robustness of complex networks abstract we study the robustness of several network models subject to edge removal the robustness is measured by the statistics of network breakdowns where a breakdown is defined as the destroying of the total connectedness of a network rather than the disappearance of the giant component we introduce a simple traffic dynamics as the function of a network topology and the total connectedness can be destroyed in the sense of either the topology or the function the overall effect of the topological breakdown and the functional breakdown as well as the relative importance of the topological robustness and the functional robustness are studied under two edge removal strategies

identifying influential nodes in complex networks abstract identifying influential nodes that lead to faster and wider spreading in complex networks is of theoretical and practical significance the degree centrality method is very simple but of little relevance global metrics such as betweenness centrality and closeness centrality can better identify influential nodes but are incapable to be applied in large scale networks due to the computational complexity in order to design an effective ranking method we proposed a semi local centrality measure as a tradeoff between the low relevant degree centrality and other time consuming measures we use the susceptible infected recovered sir model to evaluate the performance by using the spreading rate and the number of infected nodes simulations on four real networks show that our method can well identify influential nodes

effect of random edge failure on the average path length we study the effect of random removal of edges on the average path length apl in a large class of uncorrelated random networks in which vertices are characterized by hidden variables controlling the attachment of edges between pairs of vertices a formula for approximating the apl of networks suffering random edge removal is derived first then the formula is confirmed by simulations for classical er erdos and renyi random graphs ba barabasi and albert networks networks with exponential degree distributions as well as random networks with asymptotic power law degree distributions with exponent 2

robustness of weighted networks abstract complex network response to node loss is a central question in different fields of network science because node failure can cause the fragmentation of the network thus compromising the system functioning previous studies considered binary networks where the intensity weight of the links is not accounted for i e a link is either present or absent however in real world networks the weights of connections and thus their importance for network functioning can be widely different here we analyzed the response of real world and model networks to node loss accounting for link intensity and the weighted structure of the network we used both classic binary node properties and network functioning measure introduced a weighted rank for node importance node strength and used a measure for network functioning that accounts for the weight of the links weighted efficiency we find that i the efficiency of the attack strategies changed using binary or weighted network functioning measures both for real world or model networks ii in some cases removing nodes according to weighted rank produced the highest damage when functioning was measured by the weighted efficiency iii adopting weighted measure for the network damage changed the efficacy of the attack strategy with respect the binary analyses our results show that if the weighted structure of complex networks is not taken into account this may produce misleading models to forecast the system response to node failure i e consider binary links may not unveil the real damage induced in the system last once weighted measures are introduced in order to discover the best attack strategy it is important to analyze the network response to node loss using nodes rank accounting the intensity of the links to the node

generalized theory for node disruption in finite size complex networks after a failure or attack the structure of a complex network changes due to node removal here we show that the degree distribution of the distorted network under any node disturbances can be easily computed through a simple formula based on this expression we derive a general condition for the stability of noncorrelated finite complex networks under any arbitrary attack we apply this formalism to derive an expression for the percolation threshold fc under a general attack of the form fk k where fk stands for the probability of a node of degree k of being removed during the attack we show that fc of a finite network of size n exhibits an additive correction which scales as n 1 with respect to the classical result for infinite networks

a comparative analysis of network robustness against different link attacks abstract recently the study of optimizing network robustness has attracted increasing attentions and the constraint that every node s degree cannot be changed is considered although this constraint maintains the node degree distribution consistently in order to reserve the structure of networks it makes the network structure be lack of flexibility since many network structure always transform in the modern society given this consideration in this paper we analyze the robustness of networks through setting a new constraint that is only the number of edges should be unchanged then we use the link robustness index r l as the measure of the network robustness against either random failures or intentional attacks and make a comparative analysis of network robustness against different types of link attacks moreover we use four types of networks as initial networks namely scale free networks random networks regular networks and small world networks the experimental results show that the values of robustness measures for the optimized networks starting from different initial networks are similar under different link attacks but the network topologies may be different that is to say networks with different topologies may have similar robustness in terms of the robustness measures we also find that the optimized networks obtained by one link attack may not robust against other link attacks sometimes even weaker than the original networks therefore before building networks it is better to study which type of link attacks may happen

new attack strategies for complex networks abstract the invulnerability of complex networks is an important issue in that the behavior of scale free network differs from that of exponential network according to the structural characteristics of the networks we propose two new attack strategies named idb initial degree and betweenness and rdb recalculated degree and betweenness the strategies are originated from id initial degree distribution and rd recalculated degree distribution strategies in which attacks are based on initial structural information of a network the probability of node removals depends on a new metric combining degree centrality and betweenness centrality we evaluate the efficiency of the proposed strategies on one real world network and three network models experimental results indicate that the proposed strategies are more efficient than the traditional id and rd strategies specially the ws small world network behaves more sensitive to the proposed strategies the attack efficiency of rdb strategy is improved by 20 to rd strategy and idb strategy is improved by 40 to id strategy

current flow efficiency of networks abstract many real world networks from infrastructure networks to social and communication networks can be formulated as flow networks how to realistically measure the transport efficiency of these networks is of fundamental importance the shortest path based efficiency measurement has limitations as it assumes that flow travels only along those shortest paths here we propose a new metric named current flow efficiency in which we calculate the average reciprocal effective resistance between all pairs of nodes in the network this metric takes the multipath effect into consideration and is more suitable for measuring the efficiency of many real world flow equilibrium networks moreover this metric can handle a disconnected graph and can thus be used to identify critical nodes and edges from the efficiency loss perspective we further analyze how the topological structure affects the current flow efficiency of networks based on some model and real world networks our results enable a better understanding of flow networks and shed light on the design and improvement of such networks with higher transport efficiency

networked characteristics of the urban rail transit networks abstract urban rail transit networks urtns have experienced rapid development and have been receiving much attention recently in this paper we comprehensively analyze the topological characteristics of urban rail transit networks and we find that the average degrees of nodes of urban rail transit networks lie in the interval 2 2 45 most of the average shortest path lengths between pairs of nodes belong to the interval 10 16 and the average betweenness of nodes and edges linearly increase with the increase of the number of stations moreover the cumulative probability distributions of the degree and shortest path length can be fitted by exponential distribution and gauss distribution respectively the network failures of the urban rail transit networks are discussed and we also discover that the highest betweenness node based attack is the most effective method to destroy the network

vertex similarity in networks we consider methods for quantifying the similarity of vertices in networks we propose a measure of similarity based on the concept that two vertices are similar if their immediate neighbors in the network are themselves similar this leads to a self consistent matrix formulation of similarity that can be evaluated iteratively using only a knowledge of the adjacency matrix of the network we test our similarity measure on computer generated networks for which the expected results are known and on a number of real world networks

radial structure of the internet the structure of the internet at the autonomous system as level has been studied by the physics mathematics and computer science communities we extend this work to include features of the core and the periphery taking a radial perspective on as network structure new methods for plotting as data are described and they are used to analyse datasets that have been extended to contain edges missing from earlier collections the average distance from one vertex to the rest of the network is used as the baseline metric for investigating radial structure common vertex specific quantities are plotted against this metric to reveal distinctive characteristics of central and peripheral vertices two datasets are analysed

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

algorithm to determine the percolation largest component in interconnected networks interconnected networks have been shown to be much more vulnerable to random and targeted failures than isolated ones raising several interesting questions regarding the identification and mitigation of their risk the paradigm to address these questions is the percolation model where the resilience of the system is quantified by the dependence of the size of the largest cluster on the number of failures numerically the major challenge is the identification of this cluster and the calculation of its size here we propose an efficient algorithm to tackle this problem we show that the algorithm scales as o nlogn where n is the number of nodes in the network a significant improvement compared to o n 2 for a greedy algorithm which permits studying much larger networks our new strategy can be applied to any network topology and distribution of interdependencies as well as any sequence of failures

network structure classification and features of water distribution systems the network connectivity structure of water distribution systems wdss represents the domain where hydraulic processes occur driving the emerging behavior of such systems for example with respect to robustness and vulnerability in complex network theory cnt a common way of classifying the network structure and connectivity is the association of the nodal degree distribution to specific probability distribution models and during the last decades researchers classified many real networks using the poisson or pareto distributions in spite of the fact that degree based network classification could play a crucial role to assess wds vulnerability this task is not easy because the network structure of wdss is strongly constrained by spatial characteristics of the environment where they are constructed the consequence of these spatial constraints is that the nodal degree spans very small ranges in wdss hindering a reliable classification by the standard approach based on the nodal degree distribution this work investigates the classification of the network structure of twenty two real wdss built in different environments demonstrating that the poisson distribution generally models the degree distributions very well in order to overcome the problem of the reliable classification based on the standard nodal degree we define the neighborhood degree equal to the sum of the nodal degrees of the nearest topological neighbors i e the adjacent nodes this definition of neighborhood degree is consistent with the fact that the degree of a single node is not significant for analysis of wdss this article is protected by copyright all rights reserved

vulnerability of labeled networks abstract we propose a metric for vulnerability of labeled graphs that has the following two properties 1 when the labeled graph is considered as an unlabeled one the metric reduces to the corresponding metric for an unlabeled graph and 2 the metric has the same value for differently labeled fully connected graphs reflecting the notion that any arbitrarily labeled fully connected topology is equally vulnerable as any other a vulnerability analysis of two real world networks the power grid of the european union and an autonomous system network has been performed the networks have been treated as graphs with node labels the analysis consists of calculating characteristic path lengths between labels of nodes and determining largest connected cluster size under two node and edge attack strategies results obtained are more informative of the networks vulnerability compared to the case when the networks are modeled with unlabeled graphs

correlations in complex networks under attack for any initially correlated network after any kind of attack where either nodes or edges are removed we obtain general expressions for the degree degree probability matrix and degree distribution we show that the proposed analytical approach predicts the correct topological changes after the attack by comparing the evolution of the assortativity coefficient for different attack strategies and intensities in theory and simulations we find that it is possible to turn an initially assortative network into a disassortative one and vice versa by fine tuning removal of either nodes or edges for an initially uncorrelated network on the other hand we discover that only a targeted edge removal attack can induce such correlations

graph partitioning induced phase transitions we study the percolation properties of graph partitioning on random regular graphs with n vertices of degree k optimal graph partitioning is directly related to optimal attack and immunization of complex networks we find that for any partitioning process even if nonoptimal that partitions the graph into essentially equal sized connected components clusters the system undergoes a percolation phase

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

semiclassical transition from an elliptical to an oval billiard semiclassical approximations often involve the use of stationary phase approximations this method can be applied when is small in comparison with relevant actions or action differences in the corresponding classical system in many situations however action differences can be arbitrarily small and then uniform approximations are more appropriate in this paper we examine different uniform approximations for describing the spectra of integrable systems and systems with mixed phase space this is done on the example of two billiard systems an elliptical billiard and a deformation of it an oval billiard we derive a trace formula for the ellipse which involves a uniform approximation for the maslov phases near the separatrix and a uniform approximation for tori of periodic orbits close to a bifurcation we then examine how the trace formula is modified when the ellipse is deformed into an oval this involves uniform approximations for the break up of tori and uniform approximations for bifurcations of periodic orbits relations between different uniform approximations are discussed

tunable path centrality quantifying the importance of paths in networks abstract centrality is a fundamental measure in network analysis specifically centrality of a path describes the importance of the path with respect to the remaining part of the network in this paper we propose a tunable path centrality tpc measure which quantifies the centrality of a path by integrating the path degree pd number of neighbors of the path and the path bridge pb number of bridges in the path with a control parameter considering the complexity of large scale and dynamical topologies of many real world networks both pd and pb are computed with only the local topological structure of a path we demonstrate the distribution of the three path centralities tpc pd and pb in computer generated networks and real world networks furthermore we apply the three path centralities to the network fragility problem and exploit the distribution of the optimal control parameter o p t through simulation and analysis finally the simulation results show that generally tpc is more efficient than pd and pb in the network fragility problem these path centralities are also applicable in many other network problems including spread control prediction and so on

fourier transforms of gibbs measures for the gauss map we investigate under which conditions a given invariant measure mu for the dynamical system defined by the gauss map x mapsto 1 x mathrm mod 1 is a rajchman measure with polynomially decaying fourier transform begin aligned widehat mu xi o xi eta quad text as quad xi rightarrow infty end aligned we show that this property holds for any gibbs measure mu of hausdorff dimension greater than 1 2 with a natural large deviation assumption on the gibbs potential in particular we obtain the result for the hausdorff measure and all gibbs measures of dimension greater than 1 2 on badly approximable numbers which extends the constructions of kaufman and queffelec ramare our main result implies that the fourier stieltjes coefficients of the minkowski s question mark function decay to 0 polynomially answering a question of salem from 1943 as an application of the davenport erd s leveque criterion we obtain an equidistribution theorem for gibbs measures which extends in part a recent result by hochman shmerkin our proofs are based on exploiting the nonlinear and number theoretic nature of the gauss map and large deviation theory for hausdorff dimension and lyapunov exponents

a vibrational approach to node centrality and vulnerability in complex networks we propose a new measure of vulnerability of a node in a complex network the measure is based on the analogy in which the nodes of the network are represented by balls and the links are identified with springs we define the measure as the node displacement or the amplitude of vibration of each node under fluctuation due to the thermal bath in which the network is supposed to be submerged we prove exact relations among the thus defined node displacement the information centrality and the kirchhoff index the relation between the first two suggests that the node displacement has a better resolution of the vulnerability than the information centrality because the latter is the sum of the local node displacement and the node displacement averaged over the entire network

resonant periodic orbits and the semiclassical energy spectrum the semiclassical density of states depends according to the periodic orbit sum formula on the linear stability of the orbits this means however that contributions from the marginally stable or resonant orbits which necessarily accompany stable ones diverge unphysically the remedy for a system of two degrees of freedom is found to lie in the classical non linear normal forms for periodic orbits which describe how satellite periodic orbits coalesce with the central one as resonance is approached in to 0 through these forms the resonant contributions are expressed as diffraction integrals the first few being diffraction catastrophes uniformly valid in in and h cross and finite even for in to 0 provided h cross 0 an extension is proposed to incorporate jointly multiple resonances found in repetitions of orbits

the dynamic correlation between degree and betweenness of complex network under attack abstract complex networks are often subjected to failure and attack recent work has addressed the resilience of complex networks to either random or intentional deletion of nodes or links here we simulate the breakdown of the small world network and the scale free network under node failure or attacks we analyze and discuss the dynamic correlation between degree and betweenness in the process of attack the simulation results show that the correlation for scale free network obeys a power law distribution until the network collapses while it represents irregularly for small world network

perfectly ordered quasicrystals and the littlewood conjecture linearly repetitive cut and project sets are mathematical models for perfectly ordered quasicrystals in a previous paper we presented a characterization of linearly repetitive cut and project sets in this paper we extend the classical definition of linear repetitivity to try to discover whether or not there is a natural class of cut and project sets which are models for quasicrystals which are better than perfectly ordered in the positive direction we demonstrate an uncountable collection of such sets in fact a collection with large hausdorff dimension for every choice of dimension of the physical space on the other hand we show that for many natural versions of the problems under consideration the existence of these sets turns out to be equivalent to the negation of a well known open problem in diophantine approximation the littlewood conjecture

efficiency of attack strategies on complex model and real world networks we investigated the efficiency of attack strategies to network nodes when targeting several complex model and real world networks we tested 5 attack strategies 3 of which were introduced in this work for the first time to attack 3 model networks erdos and renyi barabasi and albert preferential attachment network and scale free network configuration models and 3 real networks gnutella peer to peer network email network of the university of rovira i virgili and immunoglobulin interaction network nodes were removed sequentially according to the importance criterion defined by the attack strategy and we used the size of the largest connected component lcc as a measure of network damage we found that the efficiency of attack strategies fraction of nodes to be deleted for a given reduction of lcc size depends on the topology of the network although attacks based on either the number of connections of a node or betweenness centrality were often the most efficient strategies sequential deletion of nodes in decreasing order of betweenness centrality was the most efficient attack strategy when targeting real world networks the relative efficiency of attack strategies often changed during the sequential removal of nodes especially for networks with power law degree distribution

using mapping entropy to identify node centrality in complex networks abstract the problem of finding the best strategy to attack a network or immunize a population with a minimal number of nodes has attracted much current research interest the assessment of node importance has been a fundamental issue in the research of complex networks in this paper we propose a new concept called mapping entropy me to identify the importance of a node in the complex network the concept is established according to the local information which considers the correlation among all neighbors of a node we evaluate the efficiency of the centrality by static and dynamic attacks on standard network models and real world networks the simulation result shows that the new centrality is more efficient than traditional attack strategies whether it is static or dynamic

efficient algorithms for game theoretic betweenness centrality betweenness centrality measures the ability of different nodes to control the flow of information in a network in this article we extend the standard definition of betweenness centrality using semivalues a family of solution concepts from cooperative game theory that includes among others the shapley value and the banzhaf power index any semivalue based betweenness centrality measure such as for example the shapley value based betweenness centrality measure has the advantage of evaluating the importance of individual nodes by considering the roles they each play in different groups of nodes our key result is the development of a general polynomial time algorithm to compute the semivalue based betweenness centrality measure and an even faster algorithm to compute the shapley value based betweenness centrality measure both for weighted and unweighted networks interestingly for the unweighted case our algorithm for computing the shapley value based centrality has the same complexity as the best known algorithm for computing the standard betweenness centrality due to brandes 15 we empirically evaluate our measures in a simulated scenario where nodes fail simultaneously we show that compared to the standard measure the ranking obtained by our measures reflects more accurately the influence that different nodes have on the functionality of the network we propose a betweenness centrality based on the shapley value and semivalue we develop polynomial algorithms for our game theoretic metrics we evaluate our measures in scenarios where simultaneous node failures occur our measures obtain better results than standard betweenness centrality we provide an empirical evaluation of algorithms on real life and random graphs

optimal network topology for structural robustness based on natural connectivity abstract the structural robustness of the infrastructure of various real life systems which can be represented by networks is of great importance thus we have proposed a tabu search algorithm to optimize the structural robustness of a given network by rewiring the links and fixing the node degrees the objective of our algorithm is to maximize a new structural robustness measure natural connectivity which provides a sensitive and reliable measure of the structural robustness of complex networks and has lower computation complexity we initially applied this method to several networks with different degree distributions for contrast analysis and investigated the basic properties of the optimal network we discovered that the optimal network based on the power law degree distribution exhibits a roughly eggplant like topology where there is a cluster of high degree nodes at the head and other low degree nodes scattered across the body of eggplant additionally the cost to rewire links in practical applications is considered therefore we optimized this method by employing the assortative rewiring strategy and validated its efficiency

link prediction based on local information considering preferential attachment abstract link prediction in complex networks has attracted much attention in many fields in this paper a common neighbors plus preferential attachment index is presented to estimate the likelihood of the existence of a link between two nodes based on local information of the nearest neighbors numerical experiments on six real networks demonstrated the high effectiveness and efficiency of the new index compared with five well known and widely accepted indices the common neighbors resource allocation index preferential attachment index local path index and katz index the new index provides competitively accurate prediction with local path index and katz index while has less computational complexity and is more accurate than the other two indices

network reachability of real world contact sequences we use real world contact sequences time ordered lists of contacts from one person to another to study how fast information or disease can spread across network of contacts specifically we measure the reachability time the average shortest time for a series of contacts to spread information between a reachable pair of vertices a pair where a chain of contacts exists leading from one person to the other and the reachability ratio the fraction of reachable vertex pairs these measures are studied using conditional uniform graph tests we conclude among other things that the network reachability depends much on a core where the path lengths are short and communication frequent that clustering of the contacts of an edge in time tends to decrease the reachability and that the order of the contacts really does make sense for dynamical spreading processes

link prediction based on path entropy abstract information theory has been taken as a prospective tool for quantifying the complexity of complex networks in this paper first we study the information entropy or uncertainty of a path using the information theory after that we apply the path entropy to the link prediction problem in real world networks specifically we propose a new similarity index namely path entropy pe index which considers the information entropies of shortest paths between node pairs with penalization to long paths empirical experiments demonstrate that pe index outperforms the mainstream of link predictors

network growth with arbitrary initial conditions degree dynamics for uniform and preferential attachment this paper provides time dependent expressions for the expected degree distribution of a given network that is subject to growth we consider both uniform attachment where incoming nodes form links to existing nodes selected uniformly at random and preferential attachment where probabilities are assigned proportional to the degrees of the existing nodes we consider the cases of single and multiple links being formed by each newly introduced node the initial conditions are arbitrary that is the solution depends on the degree distribution of the initial graph which is the substrate of the growth previous work in the literature focuses on the asymptotic state that is when the number of nodes added to the initial graph tends to infinity rendering the effect of the initial graph negligible our contribution provides a solution for the expected degree distribution as a function of time for arbitrary initial condition previous results match our results in the asymptotic limit the results are discrete in the degree domain and continuous in the time domain where the addition of new nodes to the graph are approximated by a continuous arrival rate

range limited centrality measures in complex networks here we present a range limited approach to centrality measures in both nonweighted and weighted directed complex networks we introduce an efficient method that generates for every node and every edge its betweenness centrality based on shortest paths of lengths not longer than ensuremath ell 1 l in the case of nonweighted networks and for weighted networks the corresponding quantities based on minimum weight paths with path weights not larger than w ensuremath ell ensuremath ell ensuremath delta ensuremath ell 1 2 l r ensuremath delta these measures provide a systematic description on the positioning importance of a node edge with respect to its network neighborhoods one step out two steps out etc up to and including the whole network they are more informative than traditional centrality measures as network transport typically happens on all length scales from transport to nearest neighbors to the farthest reaches of the network we show that range limited centralities obey universal scaling laws for large nonweighted networks as the computation of traditional centrality measures is costly this scaling behavior can be exploited to efficiently estimate centralities of nodes and edges for all ranges including the traditional ones the scaling behavior can also be exploited to show that the ranking top list of nodes edges based on their range limited centralities quickly freezes as a function of the range and hence the diameter range top list can be efficiently predicted we also show how to estimate the typical largest node to node distance for a network of n nodes exploiting the afore mentioned scaling behavior these observations were made on model networks and on a large social network inferred from cell phone trace logs ensuremath sim 5 5 ifmmode times else texttimes fi 10 6 nodes and ensuremath sim 2 7 ifmmode times else texttimes fi 10 7 edges finally we apply these concepts to efficiently detect the vulnerability backbone of a network defined as the smallest percolating cluster of the highest betweenness nodes and edges and illustrate the importance of weight based centrality measures in weighted networks in detecting such backbones

percolation theory and fragmentation measures in social networks abstract we study the statistical properties of a recently proposed social networks measure of fragmentation f after removal of a fraction q of nodes or links from the network the measure f is defined as the ratio of the number of pairs of nodes that are not connected in the fragmented network to the total number of pairs in the original fully connected network we compare this measure with the one traditionally used in percolation theory p the fraction of nodes in the largest cluster relative to the total number of nodes using both analytical and numerical methods we study erd s renyi er and scale free sf networks under various node removal strategies we find that for a network obtained after removal of a fraction q of nodes above criticality p 1 f 1 2 for fixed p and close to criticality we show that 1 f better reflects the actual fragmentation for a given p 1 f has a broad distribution and thus one can improve significantly the fragmentation of the network we also study and compare the fragmentation measure f and the percolation measure p for a real national social network of workplaces linked by the households of the employees and find similar results

transcritical bifurcations in nonintegrable hamiltonian systems we report on transcritical bifurcations of periodic orbits in nonintegrable two dimensional hamiltonian systems we discuss their existence criteria and some of their properties using a recent mathematical description of transcritical bifurcations in families of symplectic maps we then present numerical examples of transcritical bifurcations in a class of generalized henon heiles hamiltonians and illustrate their stabilities and unfoldings under various perturbations of the hamiltonians we demonstrate that for hamiltonians containing straight line librating orbits the transcritical bifurcation of these orbits is the typical case which occurs also in the absence of any discrete symmetries while their isochronous pitchfork bifurcation is an exception we determine the normal forms of both types of bifurcations and derive the uniform approximation required to include transcritically bifurcating orbits in the semiclassical trace formula for the density of states of the quantum hamiltonian we compute the coarse grained density of states in a specific example both semiclassically and quantum mechanically and find excellent agreement of the results

significance of ghost orbit bifurcations in semiclassical spectra gutzwiller s trace formula for the semiclassical density of states in a chaotic system diverges near bifurcations of periodic orbits where it must be replaced with uniform approximations it is well known that when applying these approximations complex predecessors of orbits created in the bifurcation ghost orbits can produce pronounced signatures in the semiclassical spectra in the vicinity of the bifurcation it is the purpose of this paper to demonstrate that these ghost orbits can also undergo bifurcations resulting in complex nongeneric bifurcation scenarios we do so by studying an example taken from the diamagnetic kepler problem namely the period quadrupling of the balloon orbit by application of normal form theory we construct an analytic description of the complete bifurcation scenario which is then used to calculate the pertinent uniform approximation the ghost orbit bifurcation turns out to produce signatures in the semiclassical spectrum in much the same way as a bifurcation of real orbits would

bridges in complex networks a bridge in a graph is an edge whose removal disconnects the graph and increases the number of connected components we calculate the fraction of bridges in a wide range of real world networks and their randomized counterparts we find that real networks typically have more bridges than their completely randomized counterparts but they have a fraction of bridges that is very similar to their degree preserving randomizations we define an edge centrality measure called bridgeness to quantify the importance of a bridge in damaging a network we find that certain real networks have a very large average and variance of bridgeness compared to their degree preserving randomizations and other real networks finally we offer an analytical framework to calculate the bridge fraction and the average and variance of bridgeness for uncorrelated random networks with arbitrary degree distributions

decoupling process for better synchronizability on scale free networks we propose a decoupling process performed in scale free networks to enhance the synchronizability of the network together with preserving the scale free structure simulation results show that the decoupling process can effectively promote the network synchronizability which is measured in terms of eigenratio of the coupling matrix moreover we investigate the correlation between some important structural properties and the collective synchronization and find that the maximum vertex betweenness seems to be the most strongly correlated with the synchronizability among the major structural features considered we explain the effect of the decoupling process from a viewpoint of coupling information transmission our work provides some evidence that the dynamics of synchronization is related to that of information or vehicle traffic because of the low cost in modifying the coupling network the decoupling process may have potential applications

robustness of network controllability in cascading failure controllability of networks widely existing in real life systems have been a critical and attractive research subject for both network science and control systems communities research in network controllability has mostly focused on the effects of the network structure on its controllability and some studies have begun to investigate the controllability robustness of complex networks cascading failure is common phenomenon in many infrastructure networks which largely affect normal operation of networks and sometimes even lead to collapse resulting in considerable economic losses the robustness of network controllability against the cascading failure is studied by a linear load capacity model with a breakdown probability in this paper the controllability of canonical model networks under different node attack strategies is investigated random failure and malicious attack it is shown by numerical simulations that the tolerant parameter of load capacity model has an important role in the emergence of cascading failure independent to the types of network the networks with moderate average degree are more vulnerable to the cascading failure while these with high average degree are very robust in particular betweenness attack strategy is more harmful to the network controllability than degree attack one especially for the scale free networks

exploring the assortativity clustering space of a network s degree sequence nowadays there is a multitude of measures designed to capture different aspects of network structure to be able to say if a measured value is expected or not one needs to compare it with a refere

on a problem in simultaneous diophantine approximation littlewood s conjecture a consequence of hurwitz s theorem is that the right hand side of the above inequality cannot be improved by an arbitrary positive constant s more precisely for s l x 5 there exist real numbers a e i for which the inequality ilqall cq 1 has at most a finite number of solutions these a are the badly approximable numbers and we will denote by b a d the set of all such numbers that is

betweenness centrality of an edge in tree like components with finite size using the generating function formalism we propose an exact formula for estimating betweenness centrality of an edge in finite tree like components of random networks with arbitrary degree distributions as a function of the degrees of two vertices interconnected by the edge and the size of the component to which the edge belongs the formula can give an estimation of the mean value of the edge betweenness we confirm the formula by simulations for poisson and power law distributions respectively we also study the asymptotic behavior of the formula and find that there is an asymptotic power law relation with an exponent 1 5 between the edge betweenness and the size of the component to which the edge belongs

multiplicatively badly approximable numbers and generalised cantor sets abstract let p be a prime number the p adic case of the mixed littlewood conjecture states that l i m i n f q q q p q 0 for all r we show that with the additional factor of log q log log q the statement is false indeed our main result implies that the set of for which l i m i n f q q log q log log q q p q 0 is of full dimension the result is obtained as an application of a general framework for cantor sets developed in this paper

orbit bifurcations and the scarring of wave functions we extend the semiclassical theory of scarring of quantum eigenfunctions n q by classical periodic orbits to include situations where these orbits undergo generic bifurcations it is shown that n q 2 averaged locally with respect to position q and the energy spectrum en has structure around bifurcating periodic orbits with an amplitude and length scale whose dependence is determined by the bifurcation in question specifically the amplitude scales as and the length scale as and values of the scar exponents and are computed for a variety of generic bifurcations in each case the scars are semiclassically wider than those associated with isolated and unstable periodic orbits moreover their amplitude is at least as large and in most cases larger in this sense bifurcations may be said to give rise to superscars the competition between the contributions from different bifurcations to determine the moments of the averaged eigenfunction amplitude is analysed we argue that there is a resulting universal scaling in the semiclassical asymptotics of these moments for irregular states in systems with mixed phase space dynamics finally a number of these predictions are illustrated by numerical computations for a family of perturbed cat maps

weighted k shell decomposition for complex networks based on potential edge weights abstract identifying influential nodes in complex networks has attracted much attention because of its great theoretical significance and wide application existing methods consider the edges equally when designing identifying methods for the unweighted networks in this paper we propose an edge weighting method based on adding the degree of its two end nodes and for the constructed weighted networks we propose a weighted k shell decomposition method w k s further investigations on the epidemic spreading process of the susceptible infected recovered sir model and susceptible infected si model in real complex networks verify that our method is effective for detecting the node influence

vulnerability of complex networks under intentional attack with incomplete information we study the vulnerability of complex networks under intentional attack with incomplete information which means that one can only preferentially attack the most important nodes among a local region of a network the known random failure and the intentional attack are two extreme cases of our study using the generating function method we derive the exact value of the critical removal fraction fc of nodes for the disintegration of networks and the size of the giant component to validate our model and method we perform simulations of intentional attack with incomplete information in scale free networks we show that the attack information has an important effect on the vulnerability of scale free networks we also demonstrate that hiding a fraction of the nodes information is a cost efficient strategy for enhancing the robustness of complex networks