the minimum rank of symmetric matrices described by a graph a survey abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise this paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues

spectra of complex networks we propose a general approach to the description of spectra of complex networks for the spectra of networks with uncorrelated vertices and a local treelike structure exact equations are derived these equations are generalized to the case of networks with correlations between neighboring vertices the tail of the density of eigenvalues ensuremath rho ensuremath lambda at large ensuremath lambda is related to the behavior of the vertex degree distribution p k at large k in particular as p k ensuremath sim k ensuremath ensuremath gamma ensuremath rho ensuremath lambda ensuremath sim ensuremath lambda 1 ensuremath 2 ensuremath gamma we propose a simple approximation which enables us to calculate spectra of various graphs analytically we analyze spectra of various complex networks and discuss the role of vertices of low degree we show that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real world networks e g of the internet

brief paper design of highly synchronizable and robust networks in this paper the design of highly synchronizable sparse and robust dynamical networks is addressed better synchronizability means faster synchronization of the oscillators sparsity means a low ratio of links per nodes and robustness refers to the resilience of a network to the random failures or intentional removal of some of the nodes links golden spectral dynamical networks graphs are those for which the spectral spread the difference between the largest and smallest eigenvalues of the adjacency matrix is equal to the spectral gap the difference between the two largest eigenvalues of the adjacency matrix multiplied by the square of the golden ratio these networks display the property of small worldness are very homogeneous and have large isoperimetric expansion constant together with a very high synchronizability and robustness to failures of individual oscillators in particular the regular bipartite dynamical networks reported here by the first time have the best possible expansion and consequently are the most robust ones against node link failures or intentional attacks

synchronizable nodes in networked systems synchronization appears in networked systems as a phenomenon of interacting nodes the notion of synchronizability describes the property that the nodes can be forced to a synchronous behaviour by an appropriate interconnection structure whereas in earlier papers synchronizability has been investigated in terms of both the properties of the nodes and of the interconnection topology this paper shows that for linear coupled systems synchronizability is a property of the node dynamics alone if the nodes do not satisfy the necessary and sufficient synchronizability condition derived no interaction structure can ensure a synchronous behaviour of the networked system on the other hand if this condition is satisfied several interconnection structures given in this paper ensure synchronism globally in the whole state space

unified index to quantifying heterogeneity of complex networks abstract although recent studies have revealed that degree heterogeneity of a complex network has significant impact on the network performance and function a unified definition of the heterogeneity of a network with any degree distribution is absent in this paper we define a heterogeneity index 0 h 1 to quantify the degree heterogeneity of any given network we analytically show the existence of an upper bound of h 0 5 for exponential networks thus explain why exponential networks are homogeneous on the other hand we also analytically show that the heterogeneity index of an infinite power law network is between 1 and 0 5 if and only if its degree exponent is between 2 and 2 5 we further show that for any power law network with a degree exponent greater than 2 5 there always exists an exponential network such that both networks have the same heterogeneity index this may help to explain why 2 5 is a critical degree exponent for some dynamic behaviors on power law networks

spectra and eigenvectors of scale free networks we study the spectra and eigenvectors of the adjacency matrices of scale free networks when bidirectional interaction is allowed so that the adjacency matrix is real and symmetric the spectral density shows an exponential decay around the center followed by power law long tails at both spectrum edges the largest eigenvalue ensuremath lambda 1 depends on system size n as ensuremath lambda 1 ensuremath sim n 1 4 for large n and the corresponding eigenfunction is strongly localized at the hub the vertex with largest degree the component of the normalized eigenfunction at the hub is of order unity we also find that the mass gap scales as n ensuremath 0 68

synchronizability of complex networks recent interest in the study of networks associated with complex systems has led to a better understanding of the factors and parameters relating the topology of the associated networks with the dynamics of the systems and in particular their synchronization in this paper by using known and new results from spectral graph theory we characterize relevant factors which affect the synchronization of complex networks

skew adjacency matrices of graphs the spectra of the skew adjacency matrices of a graph are considered as a possible way to distinguish adjacency cospectral graphs this leads to the following topics graphs whose skew adjacency matrices are all cospectral relations between the matchings polynomial of a graph and the characteristic polynomials of its adjacency and skew adjacency matrices skew spectral radii and an analogue of the perron frobenius theorem and the number of skew adjacency matrices of a graph with distinct spectra

nonexistence of positive weak solutions of elliptic inequalities abstract in this paper we give sufficient conditions for the nonexistence of positive entire weak solutions of coercive and anticoercive elliptic inequalities both of the p laplacian and of the mean curvature type depending also on u and x inside the divergence term while a gradient factor is included on the right hand side in particular to prove our theorems we use a technique developed by mitidieri and pohozaev in e mitidieri s i pohozaev a priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities proc steklov inst math 234 2001 1 362 which relies on the method of test functions without using comparison and maximum principles their approach is essentially based first on a priori estimates and on the derivation of an asymptotics for the a priori estimate finally nonexistence of a solution is proved by contradiction

h2 norm accumulation and its impact on synchronisation of complex dynamical networks this article addresses the h 2 norm accumulation problem of linearly coupled dynamical networks an interesting outer coupling relationship is constructed under which the h 2 norm of the newly constructed network with column input and row output matrices increases exponentially fast in terms of the node number n it increases generally much faster than 2 n when n is large while the h 2 norm of each node is 1 however the h 2 norm of the network with a diffusive coupling is equal to 2 n i e increasing only linearly when the network is stable where 2 is the h 2 norm of a single node and the h 2 norm of the network with antisymmetrical coupling also increases but rather slowly with respect to the node number n other networks with block diagonal input and block diagonal output matrices behave similarly it demonstrates that the changes of h 2 norms in different networks are very complicated despite the fact that the network structures are linear finally the influence of the h 2 norm of the loca

enhancing synchronizability of diffusively coupled dynamical networks a survey in this paper we review the literature on enhancing synchronizability of diffusively coupled dynamical networks with identical nodes the last decade has witnessed intensive investigations on the collective behavior over complex networks and synchronization of dynamical systems is the most common form of collective behavior for many applications it is desired that the synchronizability the ability of networks in synchronizing activity of their individual dynamical units is enhanced there are a number of methods for improving the synchronization properties of dynamical networks through structural perturbation in this paper we survey such methods including adding removing nodes and or edges rewiring the links and graph weighting these methods often try to enhance the synchronizability through minimizing the eigenratio of the laplacian matrix of the connection graph a synchronizability measure based on the master stability function formalism we also assess the performance of the methods by numerical simulations on a number of real world networks as well as those generated through models such as preferential attachment watts strogatz and erdos renyi

generalized connection graph method for synchronization in asymmetrical networks we present a general framework for studying global complete synchronization in networks of dynamical systems with asymmetrical connections we extend the connection graph stability method originally developed for symmetrically coupled networks to the general asymmetrical case the principal new component of the method is the transformation of the directed connection graph into an undirected graph in our method for symmetrically coupled networks we have to choose a path between each pair of nodes the extension of the method to asymmetrical coupling consists in symmetrizing the graph and associating a weight to each path this weight involves the node unbalance of the two nodes this quantity is defined to be the difference between the sum of connection coefficients of the outgoing edges and the sum of the connection coefficients of the incoming edges to the node the synchronization condition for this symmetrized and weighted network then also guarantees synchronization in the original asymmetrical network

synchronization in weighted uncorrelated complex networks in a noisy environment optimization and connections with transport efficiency motivated by synchronization problems in noisy environments we study the edwards wilkinson process on weighted uncorrelated scale free networks we consider a specific form of the weights where the strength and the associated cost of a link is proportional to kikj beta with ki and kj being the degrees of the nodes connected by the link subject to the constraint that the total edge cost is fixed we find that in the mean field approximation on uncorrelated scale free graphs synchronization is optimal at beta 1 numerical results based on exact numerical diagonalization of the corresponding network laplacian confirm the mean field results with small corrections to the optimal value of beta employing our recent connections between the edwards wilkinson process and resistor networks and some well known connections between random walks and resistor networks we also pursue a naturally related problem of optimizing performance in queue limited communication networks utilizing local weighted routing schemes

nonexistence of nonnegative solutions of elliptic systems of divergence type abstract in this paper we deal with noncoercive elliptic systems of divergence type that include both the p laplacian and the mean curvature operator and whose right hand sides depend also on a gradient factor we prove that any nonnegative entire weak solution is necessarily constant the main argument of our proofs is based on previous estimates given in filippucci 2009 12 for elliptic inequalities actually the main technique for proving the central estimate has been developed by mitidieri and pohozaev 2001 23 and relies on the method of test functions no use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required

on positive radial entire solutions of second order quasilinear elliptic systems abstract we study positive radial entire solutions of second order quasilinear elliptic systems of the form p u f x u v q v g x u v x r n sufficient and or necessary conditions of f and g are obtained for to have positive radial entire solutions with prescribed asymptotic behavior at infinity we first study the upper and lower estimates for the nonlinear integral operator which is inverse of one dimensional polar form of the n dimensional m laplace operator

bicyclic oriented graphs with skew rank 6 let g be an oriented graph and s g be its skew adjacency matrix the skew rank of g denoted by sr g is the rank of s g in this paper we characterize all the bicyclic oriented graphs with skew rank 6 let g be a bicyclic oriented graph with pendant vertices but no pendant twins if s r g 6 then 6 v g 10

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

three structural properties reflecting the synchronizability of complex networks during the process of adding links we find that the synchronizability of the classical barabasi albert ba scale free or watts strogatz ws small world networks can be statistically quantified by three essentially structural quantities of these networks i e the eccentricity variance of the degree distribution and clustering coefficients the results indicate that both the eccentricity and clustering coefficient are positively linearly correlated with synchronizability while the variance is negatively linearly correlated moreover the efficiency of some particular strategies of adding links to change the synchronizability is also investigated this information can be used to guide us to design corresponding strategies of structure evolving processes to manipulate the synchronizability of a given network

spatial patterns of desynchronization bursts in networks we adapt a previous model and analysis method the master stability function extensively used for studying the stability of the synchronous state of networks of identical chaotic oscillators to the case of oscillators that are similar but not exactly identical we find that bubbling induced desynchronization bursts occur for some parameter values these bursts have spatial patterns which can be predicted from the network connectivity matrix and the unstable periodic orbits embedded in the attractor we test the analysis of bursts by comparison with numerical experiments in the case that no bursting occurs we discuss the deviations from the exactly synchronous state caused by the mismatch between oscillators

altering synchronizability by adding and deleting edges for scale free networks abstract in this paper we propose two methods for altering the synchronizability of scale free networks 1 adding edges between the max degree nodes and min degree nodes 2 deleting edges between the max degree nodes and max degree nodes after adding and deleting edges we find that the former adding process can weaken synchronizability while the latter deleting process can enhance it the two processes adding and deleting can preserve the scale free structure the study of the average clustering coefficient indicates that it is not the most closely correlated with the synchronizability among the topological features considered our work also suggests that there are some essential relations between the network synchronization and the dynamics of economic systems they can be used to deal with some problems in the real world such as relieving the economic crisis in addition the adding and deleting processes may have potential applications in modifying network structure in view of their low cost

entire solutions of completely coercive quasilinear elliptic equations ii abstract a famous theorem of sergei bernstein says that every entire solution u u x x r 2 of the minimal surface equation div d u 1 d u 2 0 is an affine function no conditions being placed on the behavior of the solution u bernstein s theorem continues to hold up to dimension n 7 while it fails to be true in higher dimensions in fact if x r n with n 8 there exist entire non affine minimal graphs bombieri de giorgi and giusti our purpose is to consider an extensive family of quasilinear elliptic type equations which has the following strong bernstein liouville property that u 0 for any entire solution u no conditions whatsoever being placed on the behavior of the solution outside of appropriate regularity assumptions in many cases moreover no conditions need be placed even on the dimension n we also study the behavior of solutions when the parameters of the problem do not allow the bernstein liouville property and give a number of counterexamples showing that the results of the paper are in many cases best possible

maximal planar networks with large clustering coefficient and power law degree distribution in this article we propose a simple rule that generates scale free networks with very large clustering coefficient and very small average distance these networks are called random apollonian networks rans as they can be considered as a variation of apollonian networks we obtain the analytic results of power law exponent ensuremath gamma 3 and clustering coefficient c frac 46 3 ensuremath 36 phantom rule 0 2em 0ex mathrm ln phantom rule 0 2em 0ex frac 3 2 ensuremath approx 0 74 which agree with the simulation results very well we prove that the increasing tendency of average distance of rans is a little slower than the logarithm of the number of nodes in rans since most real life networks are both scale free and small world networks rans may perform well in mimicking the reality the rans possess hierarchical structure as c k ensuremath sim k ensuremath 1 that are in accord with the observations of many real life networks in addition we prove that rans are maximal planar networks which are of particular practicability for layout of printed circuits and so on the percolation and epidemic spreading process are also studied and the comparisons between rans and barab asi albert ba as well as newman watts nw networks are shown we find that when the network order n the total number of nodes is relatively small as n ensuremath sim 10 4 the performance of rans under intentional attack is not sensitive to n while that of ba networks is much affected by n and the diseases spread slower in rans than ba networks in the early stage of the suseptible infected process indicating that the large clustering coefficient may slow the spreading velocity especially in the outbreaks

network synchronization diffusion and the paradox of heterogeneity many complex networks display strong heterogeneity in the degree connectivity distribution heterogeneity in the degree distribution often reduces the average distance between nodes but paradoxically may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength here we offer a solution to this apparent paradox our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states we show that for a given degree distribution the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum this enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size numerical verification of the main results is provided for representative classes of small world and scale free networks

spectral properties of supra laplacian for partially interdependent networks abstract the spectrum of the laplacian matrices of complex networks is a key factor in network functionality in this paper the spectral properties of supra laplacian for partially interdependent networks are investigated based on matrix perturbation theory refined results of the eigenvalue properties of laplacian matrices are provided which shows that the size relationship of the first order approximate solutions of the eigenvalues remains unchanged even if there is perturbation using these results the theoretical approximate formulae of the minimum non zero eigenvalue and the maximum eigenvalue of supra laplacian for partially interdependent networks are derived respectively the outcomes in this paper are more general and need fewer calculations than that in the literature 13 14 finally the simulations and applications of synchronizability and diffusion process verify the feasibility and effectiveness of the proposed solutions the findings can be instructive for networks when linking to the spectral properties of the supra laplacian

disconnected synchronized regions of complex dynamical networks this technical note addresses the synchronized region problem which is converted to a more convenient matrix stability problem for complex dynamical networks for any natural number n the existence of a network with n disconnected synchronized regions is theoretically proved and numerically demonstrated this shows the intrinsic complexity of the network synchronization problem convexity characteristic of stability for relevant matrix pencils is further discussed a smooth chua s circuit network is finally discussed as an example for illustration

better synchronizability predicted by crossed double cycle in this brief report we propose a network model named crossed double cycles which are completely symmetrical and can be considered as the extensions of nearest neighboring lattices the synchronizability measured by eigenratio r can be sharply enhanced by adjusting the only parameter the crossed length m the eigenratio r is shown very sensitive to the average distance l and the smaller average distance will lead to better synchronizability furthermore we find that in a wide interval the eigenratio r approximately obeys a power law form as r approximately l 1 5

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

projective cluster synchronization in drive response dynamical networks abstract in this paper we study the projective cluster synchronization in a drive response dynamical network with 1 n coupled partially linear chaotic systems because the scaling factors characterizing the dynamics of projective synchronization remain unpredictable pinning control ideas are adopted to direct the different scaling factors onto the desired values it is also shown that the projection cluster synchronization can be realized by controlling only one node in each cluster numerical simulations on the chaotic lorenz system are illustrated to verify the theoretical results

clustering and the synchronization of oscillator networks by manipulating the clustering coefficient of a network without changing its degree distribution we examine the effect of clustering on the synchronization of phase oscillators on networks with poisson and scale free degree distributions for both types of networks increased clustering hinders global synchronization as the network splits into dynamical clusters that oscillate at different frequencies surprisingly in scale free networks clustering promotes the synchronization of the most connected nodes hubs even though it inhibits global synchronization as a result they show an additional advanced transition instead of a single synchronization threshold this cluster enhanced synchronization of hubs may be relevant to the brain that is scale free and highly clustered

on the relationship between the skew rank of an oriented graph and the rank of its underlying graph abstract an oriented graph g is a digraph without loops and multiple arcs where g is the underlying graph of g let s g denote the skew adjacency matrix of g and a g be the adjacency matrix of g the rank resp skew rank of g resp g written as r g resp s r g refers to the rank of a g resp s g it is natural and interesting to study the relationship between s r g and r g wong ma and tian european j combin 54 2016 76 86 30 determined the sharp upper bound on s r g r g as a continuance of it in this paper a sharp lower bound on s r g r g is determined as well a sharp upper bound on s r g r g is determined all the corresponding extremal oriented graphs g are characterized respectively

global synchronization criteria of linearly coupled neural network systems with time varying coupling in this paper global synchronization of linearly coupled neural network nn systems with time varying coupling is investigated the dynamical behavior of the uncoupled system at each node is general which can be chaotic or others the coupling configuration is time varying i e the coupling matrix is not a constant matrix based on lyapunov function method and the specific property of householder transform some criteria for the global synchronization are obtained by these criteria one can verify whether the coupled system with time varying coupling is globally synchronized which is important and useful for both understanding and interpreting synchronization phenomena and designing coupling configuration finally two simulations are given to demonstrate the effectiveness of the theoretical results

non existence of entire solutions of degenerate elliptic inequalities with weights non existence results for non negative distribution entire solutions of singular quasilinear elliptic differential inequalities with weights are established such inequalities include the capillarity equation with varying gravitational field h as well as the general p poisson equation of radiative cooling with varying heat conduction coefficient g and varying radiation coefficient h since we deal with inequalities and positive weights it is not restrictive to assume h radially symmetric theorem 1 extends in several directions previous results and says that solely entire large solutions can exist while theorem 2 shows that in the p laplacian case positive entire solutions cannot exist the results are based on some qualitative properties of independent interest

failed skew zero forcing on a graph abstract given a graph g and an initial designation of each vertex in v g as filled or empty we apply the skew color change rule which states that a vertex v becomes filled if and only if it is the unique empty neighbor of some other vertex in the graph if repeated application of the color change rule results in all vertices in the graph eventually being filled then the initial set of filled vertices is a skew zero forcing set the minimum cardinality of such a set is the skew zero forcing number of g z g in this paper we introduce and investigate a new parameter the failed skew zero forcing number f g which is the maximum cardinality of a starting set that will not result in all vertices in the graph eventually being filled it turns out that f g is undefined for certain graphs we classify such graphs we provide an algorithm that determines if f g is undefined and also finds the unique perfect matching guaranteed in the case f g is undefined we provide classifications of graphs with high and low values of f g compare the parameter to other zero forcing parameters and finally establish f g for multiple graph families the skew zero forcing number was originally introduced for its application to the problem of finding the minimum rank among all skew symmetric matrices associated with a graph and it also applies to the problem among all symmetric matrices with zero diagonal we establish connections between f g and these minimum rank problems we also classify all graphs such that any set of vertices of order z g is a skew zero forcing set

on weak solutions of nonlinear weighted p laplacian elliptic inequalities abstract in this paper we give sufficient conditions for the nonexistence of nonnegative nontrivial entire weak solutions of class w loc 1 p r n c r n of p laplacian elliptic inequalities with possibly singular weights in order to get the results a new omori yau type principle is used we complement our nonexistence results by establishing existence of infinitely many positive radial solutions each of which blows up at some finite r 0 finally a criterium for the existence of positive entire large radial solutions of class w loc 1 p r n c r n is also established

on synchronizability and heterogeneity in unweighted networks abstract synchronization in classes of continuous time dynamical unweighted networks with different topologies is investigated a synchronization optimal network model based on rewiring of links is proposed compared with other networks it exhibits a stronger synchronizability we presented link density and investigated the correlation of synchronizability link density and heterogeneity in degree distribution in this work it will be shown that synchronizability of type i networks is independent of heterogeneity in the degree distribution when the link density is smaller than 0 02 synchronizability and link density are proportional when the link density is larger than 0 025 heterogeneity decides the curve slope the synchronizability of type ii networks is drastically enhanced by enhancing the link density when the link density is smaller than 0 025 however synchronizability increases weakly by enhancing the link density when the link density is larger than 0 025

synchronizability of dynamical scale free networks subject to random errors in this paper the robustness of network synchronizability against random deletion of nodes i e errors in dynamical scale free networks was studied to this end two measures of network synchronizability namely the eigenratio of the laplacian and the order parameter quantifying the degree of phase synchrony were adopted and the synchronizability robustness on preferential attachment scale free graphs was investigated the findings revealed that as the network size decreases the robustness of its synchronizability against random removal of nodes declines i e the more the number of randomly removed nodes from the network the worse its synchronizability we also showed that this dependence of the synchronizability on the network size is different with that in the growing scale free networks the profile of a number of network properties such as clustering coefficient efficiency assortativity and eccentricity as a function of the network size was investigated in these two cases growing scale free networks and those with randomly removed nodes the results showed that these processes are also different in terms of these metrics

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

stabilizing weighted complex networks real networks often consist of local units which interact with each other via asymmetric and heterogeneous connections in this paper the v stability problem is investigated for a class of asymmetric weighted coupled networks with nonidentical node dynamics which includes the unweighted network as a special case pinning control is suggested to stabilize such a coupled network the complicated stabilization problem is reduced to measuring the semi negative property of the characteristic matrix which embodies not only the network topology but also the node self dynamics and the control gains it is found that network stabilizability depends critically on the second largest eigenvalue of the characteristic matrix the smaller the second largest eigenvalue is the more the network is pinning controllable numerical simulations of two representative networks composed of non chaotic systems and chaotic systems respectively are shown for illustration and verification

classes of graphs with minimum skew rank 4 abstract the minimum skew rank of a simple graph g is the smallest possible rank among all real skew symmetric matrices whose i j entry is nonzero if and only if the edge joining i and j is in g it is known that a graph has minimum skew rank 2 if and only if it consists of a complete multipartite graph and some isolated vertices some necessary conditions for a graph to have minimum skew rank 4 are established and several families of graphs with minimum skew rank 4 are given linear algebraic techniques are developed to show that complements of trees and certain outerplanar graphs have minimum skew rank 4

the characteristic polynomial and the matchings polynomial of a weighted oriented graph abstract let g be a weighted oriented graph with skew adjacency matrix s g then g is usually referred as the weighted oriented graph associated to s g denote by g the characteristic polynomial of the weighted oriented graph g which is defined as g det i n s g i 0 n a i g n i in this paper we begin by interpreting all the coefficients of the characteristic polynomial of an arbitrary real skew symmetric matrix in terms of its associated oriented weighted graph then we establish recurrences for the characteristic polynomial and deduce a formula on the matchings polynomial of an arbitrary weighted graph in addition some miscellaneous results concerning the number of perfect matchings and the determinant of the skew adjacency matrix of an unweighted oriented graph are given

a new critical curve for a class of quasilinear elliptic systems abstract we study a class of systems of quasilinear differential inequalities associated to weakly coercive differential operators and power reaction terms the main model cases are given by the p laplacian operator as well as the mean curvature operator in nonparametric form we prove that if the exponents lie under a certain curve then the system has only the trivial solution these results hold without any restriction provided the possible solutions are more regular the underlying framework is the classical euclidean case as well as the carnot groups setting

structural and spectral properties of a family of deterministic recursive trees rigorous solutions as one of the most significant models the uniform recursive tree urt has found many applications in a variety of fields in this paper we study rigorously the structural features and spectral properties of the adjacency matrix for a family of deterministic uniform recursive trees durts that are deterministic versions of urt firstly from the perspective of complex networks we investigate analytically the main structural characteristics of durts and obtain the accurate solutions for these properties which include degree distribution average path length distribution of node betweenness and degree correlations then we determine the complete eigenvalues and their corresponding eigenvectors of the adjacency matrix for durts our research may shed light on a better understanding of the features for urt also the analytical methods used here are capable of being extended to many other deterministic networks making the precise computation of their properties especially the full spectrum characteristics possible

factors that predict better synchronizability on complex networks while shorter characteristic path length has in general been believed to enhance synchronizability of a coupled oscillator system on a complex network the suppressing tendency of the heterogeneity of the degree distribution even for shorter characteristic path length has also been reported to see this we investigate the effects of various factors such as the degree characteristic path length heterogeneity and betweenness centrality on synchronization and find a consistent trend between the synchronization and the betweenness centrality the betweenness centrality is thus proposed as a good indicator for synchronizability

a weighted local world evolving network model with aging nodes abstract in this paper the dynamical behaviors of a class of weighted local world evolving networks with aging nodes are investigated as the network grows each newly added node is connected to some existing nodes through strength age preferential attachment each time step the newly added node with randomly determined initial degree is connected to some existing nodes through a local strength age preferential attachment by using tools from stochastic analysis and continuous approximation it is theoretically proved that the strength distribution of the generated network has a power law form with exponent depending nontrivially on a decay factor 0 1 in addition the clustering coefficient correlation and small world properties of this class of networks are examined via simulations it is shown that clustering coefficient is a monotone decreasing function of the decay factor and the assortative degree correlation and small world properties appear simultaneously finally the new model is used to characterize the internet topology as a typical application

degree based attacks are not optimal for desynchronization in general networks suppressing harmful synchronization in complex networked systems is receiving increasing interest in various contexts including epileptic seizure and the internet and traffic congestion it is traditionally believed that by disrupting the most important nodes i e those nodes of high degrees synchronization can be effectively mitigated however this approach can perform poorly in more general situations such as homogeneous random or small world networks in this article we investigate how topological properties such as the heterogeneity of the network can have an impact on its desynchronization in particular we propose a topology aware scheme that chooses topologically sufficiently diversified contrarian nodes by taking into account the patterns of connectivity among them hence a maximal number of nodes in the network can be influenced leading to a more global impact and greater disruption to synchronization our scheme demonstrates significantly improved performance on various networked systems including homogeneous networks small world and even scale free networks

infection dynamics on scale free networks we discuss properties of infection processes on scale free networks relating them to the node connectivity distribution that characterizes the network considering the epidemiologically important case of a disease that confers permanent immunity upon recovery we derive analytic expressions for the final size of an epidemic in an infinite closed population and for the dependence of infection probability on an individual s degree of connectivity within the population as in an earlier study r pastor satorras and a vesipignani phys rev lett 86 3200 2001 phys rev e 63 006117 2001 for an infection that did not confer immunity upon recovery the epidemic process in contrast with many traditional epidemiological models does not exhibit threshold behavior and we demonstrate that this is a consequence of the extreme heterogeneity in the connectivity distribution of a scale free network finally we discuss effects that arise from finite population sizes showing that networks of finite size do exhibit threshold effects infections cannot spread for arbitrarily low transmission probabilities

relations between average distance heterogeneity and network synchronizability abstract by using the random interchanging algorithm we investigate the relations between average distance standard deviation of degree distribution and synchronizability of complex networks we find that both increasing the average distance and magnifying the degree deviation will make the network synchronize harder only the combination of short average distance and small standard deviation of degree distribution ensures strong synchronizability some previous studies assert that the maximal betweenness is the right quantity to estimate network synchronizability the larger the maximal betweenness the poorer the network synchronizability here we address an interesting case which strongly suggests that the single quantity maximal betweenness may not give a comprehensive description of network synchronizability

quasilinear elliptic systems in rn with multipower forcing terms depending on the gradient in this paper we deal with noncoercive elliptic multipower systems of divergence type which include p laplacian type operators as well as mean curvature operators and whose right hand sides depend on the product of both components of the solution and on a gradient factor we prove that any nonnegative nontrivial entire weak solution nonnecessarily radial is constant for nontrivial solutions we intend that both components are nontrivial the paper improves former results due to clement fleckinger mitidieri de thelin 2000 in 4 to bidaut veron and pohozaev 2001 in 3 where no gradient terms are considered on the other hand the paper contains as subcases also some recent results of filippucci 2011 12 in which systems with gradient dependence for the forcing terms are treated but not of multipower type the key ingredient of the proof technique is the method of test functions employed by mitidieri and pohozaev for studying systems in mitidieri and pohozaev 1999 17 no use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions is required

spectral reconstruction of complex networks abstract in this paper we study the reconstruction of a network topology from the eigenvalues of its laplacian matrix we introduce a simple cost function and consider the tabu search combinatorial optimization method while comparing its performance when reconstructing different categories of networks random regular small world scale free and clustered from their eigenvalues we show that this combinatorial optimization method together with the information contained in the laplacian spectrum allows an exact reconstruction of small networks and leads to good approximations in the case of networks with larger orders we also show that the method can be used to generate a quasi optimal topology for a network associated to a dynamic process like in the case of metabolic or protein protein interaction networks of organisms

mean field solution of the small world network model the small world network model is a simple model of the structure of social networks which possesses characteristics of both regular lattices and random graphs the model consists of a one dimensional lattice with a low density of shortcuts added between randomly selected pairs of points these shortcuts greatly reduce the typical path length between any two points on the lattice we present a mean field solution for the average path length and for the distribution of path lengths in the model this solution is exact in the limit of large system size and either a large or small number of shortcuts

synchronization diversity and topology of networks of integrate and fire oscillators we study synchronization dynamics of a population of pulse coupled oscillators in particular we focus our attention on the interplay between topological disorder and synchronization features of networks first we analyze synchronization time t in random networks and find a scaling law which relates t to network connectivity then we compare synchronization time for several other topological configurations characterized by a different degree of randomness the analysis shows that regular lattices perform better than a disordered network this fact can be understood by considering the variability in the number of links between two adjacent neighbors this phenomenon is equivalent to having a nonrandom topology with a distribution of interactions and it can be removed by an adequate local normalization of the couplings

synchronization transition in scale free networks clusters of synchrony we study the synchronization transition in scale free networks that display power law asymptotic behaviors in their degree distributions the critical coupling strength and the order parameter critical exponent derived by the mean field approach depend on the degree exponent ensuremath lambda which implies a close connection between structural organization and the emergence of dynamical order in complex systems we also derive the finite size scaling behavior of the order parameter finding that the giant cluster of synchronized nodes is formed in different ways between scale free networks with 2l ensuremath lambda l3 and those with ensuremath lambda g3

boolean game on scale free networks abstract inspired by the local minority game we propose a network boolean game and investigate its dynamical properties on scale free networks the system can self organize to a stable state with better performance than the random choice game although only the local information is available to each agent by introducing the heterogeneity of local interactions we find that the system will achieve the best performance when each agent s interaction frequency is linearly correlated with its information capacity generally the agents with more information gain more than those with less information while in the optimal case each agent almost has the same average profit in addition we investigate the role of irrational factor and find an interesting symmetrical behavior

spectral bounds for the betweenness of a graph in this paper we find spectral bounds laplacian matrix for the vertex and the edge betweenness of a graph we also relate the edge betweenness with the isoperimetric number and the edge forwarding and edge expansion indices of the graph allowing a new upper bound on its diameter the results are of interest as they can be used in the study of communication properties of real networks in particular for dynamical processes taking place on them broadcasting network synchronization virus spreading etc

spontaneous structure formation in a network of dynamic elements to discover the generic behaviors of dynamic networks we study a coupled map system with variable coupling strength it is found that this system spontaneously forms various types of network structure according to the parameter values depending on the synchronized or desynchronized motion of unit dynamics the network structure can be either static or dynamic the separation of units into two groups one composed of units with a large number of outgoing connections and the other units with little outgoing connections is observed in dynamic structure it is revealed that the mechanism for such separation is a positive feedback between unit and connection dynamics

the architecture of complex weighted networks networked structures arise in a wide array of different contexts such as technological and transportation infrastructures social phenomena and biological systems these highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity along with a complex topological structure real networks display a large heterogeneity in the capacity and intensity of the connections these features however have mainly not been considered in past studies where links are usually represented as binary states i e either present or absent here we study the scientific collaboration network and the world wide air transportation network which are representative examples of social and large infrastructure systems respectively in both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network we define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices this information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network these results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks

erosion of synchronization in networks of coupled oscillators we report erosion of synchronization in networks of coupled phase oscillators a phenomenon where perfect phase synchronization is unattainable in steady state even in the limit of infinite coupling an analysis reveals that the total erosion is separable into the product of terms characterizing coupling frustration and structural heterogeneity both of which amplify erosion the latter however can differ significantly from degree heterogeneity finally we show that erosion is marked by the reorganization of oscillators according to their node degrees rather than their natural frequencies

synchronization of complex dynamical networks with switching topology a switched system point of view in this paper we study the synchronization problem for complex dynamical networks with switching topology from a switched system point of view the synchronization problem is transformed into the stability problem for time varying switched systems we address two basic problems synchronization under arbitrary switching topology and synchronization via design of switching within a pre given collection of topologies when synchronization cannot be achieved by using any topology alone in this collection for the both problems we first establish synchronization criteria for general connection topology then under the condition of simultaneous triangularization of the connection matrices a common lyapunov function for the first problem and a single lyapunov and multiple lyapunov functions for the second problem are systematically constructed respectively by those of several lower dimensional dynamic systems in order to achieve synchronization using multiple lyapunov functions a stability condition and switching law design method for time varying switched systems are also presented which avoid the usual non increasing condition

statistical ensemble of scale free random graphs a thorough discussion of the statistical ensemble of scale free connected random tree graphs is presented methods borrowed from field theory are used to define the ensemble and to study analytically its properties the ensemble is characterized by two global parameters the fractal and the spectral dimensions which are explicitly calculated it is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified the stability of the scale free regime is also considered when it breaks down either a scale is spontaneously generated or else a singular node appears and the graphs become crumpled a new computer algorithm to generate these random graphs is proposed possible generalizations are also discussed in particular more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary degenerate scale free random graphs

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

exponential synchronization of kuramoto oscillators using spatially local coupling abstract we study the generalized kuramoto model of coupled phase oscillators with a finite size and discuss the asymptotic complete phase frequency synchronization the generalized kuramoto model has inherent difficulties in mathematical approaches that this model is governed by nonlinear equations and the kuramoto oscillator is arbitrarily connected with the others to overcome these mathematical barriers many researchers have adopted a linearization of homogeneous solutions and applied a perturbation method however we introduce a new method which just requires some conditions on the smallest and largest positive eigenvalues of the graph laplacian and directly compute the bounds of homogeneous solutions using this method we present analytic results for the generalized kuramoto model more specifically we give a few sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states our sufficient conditions and decay rate depend on the coupling strength the initial phase and natural frequency configurations and the graph laplacian but the conditions are independent of the system size moreover we estimate the time evolution of deviations for the phase and frequency and show that the smallest and largest positive eigenvalues for the graph laplacian affect the stability region and convergence rate for the synchronized states finally we compare our analytic results with numerical simulations using a few examples

dynamical mean field approximation to small world networks of spiking neurons from local to global and or from regular to random couplings by extending a dynamical mean field approximation previously proposed by the author h hasegawa phys rev e 67 041903 2003 we have developed a semianalytical theory which takes into account a wide range of couplings in a small world network our network consists of noisy n unit fitzhugh nagumo neurons with couplings whose average coordination number z may change from local z n to global couplings z n ensuremath 1 and or whose concentration of random couplings p is allowed to vary from regular p 0 to completely random p 1 we have taken into account three kinds of spatial correlations the on site correlation the correlation for a coupled pair and that for a pair without direct couplings the original 2n dimensional stochastic differential equations are transformed to 13 dimensional deterministic differential equations expressed in terms of means variances and covariances of state variables the synchronization ratio and the firing time precision for an applied single spike have been discussed as functions of z and p our calculations have shown that with increasing p the synchronization is worse because of increased heterogeneous couplings although the average network distance becomes shorter results calculated by our theory are in good agreement with those by direct simulations

a remark on entire solutions of quasilinear elliptic equations abstract by applying a main comparison theorem of pucci and serrin 2007 2 we cover for general equations of p laplace type the open cases of theorems b d e of farina and serrin submitted for publication 1 as described in problems 2 and 3 of section 12 of farina and serrin submitted for publication 1 moreover we provide significant improvements of theorem c and theorem 5 of farina and serrin submitted for publication 1 the latter in the context of mean curvature type operators see theorem 1 3 and theorems 5 2 5 4 below finally theorem 1 1 provides a new liouville theorem outside the context of work in farina and serrin submitted for publication 1

enhanced synchronizability by structural perturbations in this brief report we investigate the collective synchronization of a system of coupled oscillators on a barab asi albert scale free network we propose an approach of structural perturbations aiming at those nodes with maximal betweenness this method can markedly enhance the network synchronizability and is easy to realize the simulation results show that the eigenratio will sharply decrease by one half when only 0 6 of those hub nodes occur under three division processes when the network size n 2000 in addition the present study also provides numerical evidence that the maximal betweenness plays a major role in network synchronization

optimal synchronization of kuramoto oscillators a dimensional reduction approach a recently proposed dimensional reduction approach for studying synchronization in the kuramoto model is employed to build optimal network topologies to favor or to suppress synchronization the approach is based in the introduction of a collective coordinate for the time evolution of the phase locked oscillators in the spirit of the ott antonsen ansatz we show that the optimal synchronization of a kuramoto network demands the maximization of the quadratic function t l where stands for the vector of the natural frequencies of the oscillators and l for the network laplacian matrix many recently obtained numerical results can be reobtained analytically and in a simpler way from our maximization condition a computationally efficient hill climb rewiring algorithm is proposed to generate networks with optimal synchronization properties our approach can be easily adapted to the case of the kuramoto models with both attractive and repulsive interactions and again many recent numerical results can be rederived in a simpler and clearer analytical manner

synchronization of kauffman networks we analyze the synchronization transition for a pair of coupled identical kauffman networks in the chaotic phase the annealed model for kauffman networks shows that synchronization appears through a transcritical bifurcation and provides an approximate description for the whole dynamics of the coupled networks we show that these analytical predictions are in good agreement with numerical results for sufficiently large networks and study finite size effects in detail preliminary analytical and numerical results for partially disordered networks are also presented