community detection algorithms a comparative analysis uncovering the community structure exhibited by real networks is a crucial step toward an understanding of complex systems that goes beyond the local organization of their constituents many algorithms have been proposed so far but none of them has been subjected to strict tests to evaluate their performance most of the sporadic tests performed so far involved small networks with known community structure and or artificial graphs with a simplified structure which is very uncommon in real systems here we test several methods against a recently introduced class of benchmark graphs with heterogeneous distributions of degree and community size the methods are also tested against the benchmark by girvan and newman proc natl acad sci u s a 99 7821 2002 and on random graphs as a result of our analysis three recent algorithms introduced by rosvall and bergstrom proc natl acad sci u s a 104 7327 2007 proc natl acad sci u s a 105 1118 2008 blondel et al j stat mech theory exp 2008 p10008 and ronhovde and nussinov phys rev e 80 016109 2009 have an excellent performance with the additional advantage of low computational complexity which enables one to analyze large systems

distance dissimilarity index and network community structure we address the question of finding the community structure of a complex network in an earlier effort h zhou phys rev e 67 041908 2003 the concept of network random walking is introduced and a distance measure defined here we calculate based on this distance measure the dissimilarity index between nearest neighboring vertices of a network and design an algorithm to partition these vertices into communities that are hierarchically organized each community is characterized by an upper and a lower dissimilarity threshold the algorithm is applied to several artificial and real world networks and excellent results are obtained in the case of artificially generated random modular networks this method outperforms the algorithm based on the concept of edge betweenness centrality for yeast s protein protein interaction network we are able to identify many clusters that have well defined biological functions

on global behavior of solutions to an inverse problem for nonlinear parabolic equations we find conditions on data guaranteeing global nonexistence of solutions to an inverse source problem for a class of nonlinear parabolic equations we also establish a stability result on a bounded domain for a problem with the opposite sign on the power type nonlinearity

stability criteria for certain high even order delay differential equations abstract in this paper we study the asymptotic stability of the zero solution of even order linear delay differential equations of the form y 2 m t j 0 2 m 1 a j y j t j 0 2 m 1 b j y j t where a j and b j are certain constants and m 1 here 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

on the stationary solutions and inviscid limit for the generalized proudman johnson equation with o 1 forcing abstract we consider the stationary problem for the generalized proudman johnson equation x x x a x x x x x x x f where a r and 0 are parameters and and f satisfy the periodic boundary conditions on we establish the existence of a time dependent solution and the uniqueness of the stationary solution for some large then we investigate the behavior of solution sequences in the limit 0 fixing a and f where bifurcations appear and new solutions emerge in particular we analyze if there is convergence towards stationary solutions for the corresponding inviscid equation

detect overlapping and hierarchical community structure in networks abstract clustering and community structure is crucial for many network systems and the related dynamic processes it has been shown that communities are usually overlapping and hierarchical however previous methods investigate these two properties of community structure separately this paper proposes an algorithm eagle to detect both the overlapping and hierarchical properties of complex community structure together this algorithm deals with the set of maximal cliques and adopts an agglomerative framework the quality function of modularity is extended to evaluate the goodness of a cover the examples of application to real world networks give excellent results

detecting community structure using label propagation with weighted coherent neighborhood propinquity abstract community detection has become an important methodology to understand the organization and function of various real world networks the label propagation algorithm lpa is an almost linear time algorithm proved to be effective in finding a good community structure however lpa has a limitation caused by its one hop horizon specifically each node in lpa adopts the label shared by most of its one hop neighbors much network topology information is lost in this process which we believe is one of the main reasons for its instability and poor performance therefore in this paper we introduce a measure named weighted coherent neighborhood propinquity weighted cnp to represent the probability that a pair of vertices are involved in the same community in label update a node adopts the label that has the maximum weighted cnp instead of the one that is shared by most of its neighbors we propose a dynamic and adaptive weighted cnp called entropic cnp by using the principal of entropy to modulate the weights furthermore we propose a framework to integrate the weighted cnp in other algorithms in detecting community structure we test our algorithm on both computer generated networks and real world networks the experimental results show that our algorithm is more robust and effective than lpa in large scale networks

on stability of systems of delay differential equations abstract in this paper we give necessary and sufficient conditions for the asymptotic stability of the zero solution of the system of linear delay differential equations of the form x t ax t 1 ax t where a is an n n matrix 0 is constant and 0 1 we reduce this to systems of first and second order problems our stability results are given in terms of the eigenvalues of a the proof of our results are carried out by an application of pontryagin s criterion for quasi polynomials to the characteristic functions of subsystems of the delay differential equations we also provide four algorithmic stability tests and include several examples

advanced modularity specialized label propagation algorithm for detecting communities in networks abstract a modularity specialized label propagation algorithm lpam for detecting network communities was recently proposed this promising algorithm offers some desirable qualities however lpam favors community divisions where all communities are similar in total degree and thus it is prone to get stuck in poor local maxima in the modularity space to escape local maxima we employ a multistep greedy agglomerative algorithm msg that can merge multiple pairs of communities at a time combining lpam and msg we propose an advanced modularity specialized label propagation algorithm lpam experiments show that lpam successfully detects communities with higher modularity values than ever reported in two commonly used real world networks moreover lpam offers a fair compromise between accuracy and speed

stability criteria for certain third order delay differential equations in this paper we study the asymptotic stability of the zero solution of third order linear delay differential equations of the form y t p1y t p2y t q1y t q2y t v1y t v2y t where p1 p1 q1 q2 v1 and v2 are certain constants here 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

stepping community detection algorithm based on label propagation and similarity abstract community or module structure is one of the most common features in complex networks the label propagation algorithm lpa is a near linear time algorithm that is able to detect community structure effectively nevertheless when labeling a node the lpa adopts the label belonging to the majority of its neighbors which means that it treats all neighbors equally in spite of their different effects on the node another disadvantage of lpa is that the results it generates are not unique in this paper we propose a modified lpa called stepping lpa s in which labels are propagated by similarity furthermore our algorithm divides networks using a stepping framework and uses an evaluation function proposed in this paper to select the final unique partition we tested this algorithm on several artificial and real world networks the results show that stepping lpa s can obtain accurate and meaningful community structure without priori information

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

total blow up versus single point blow up traditional thermal explosion theory is used to describe reaction initiation in condensed explosives and is limited formally to nondeformable materials kassoy and poland 4 significantly extended this theory to develop an ignition model for a reactive gas in a bounded container in order to describe the induction period during this induction period there is a spatially homogeneous pressure rise in the system which causes a com pressive heating effect in the constant volume container mathematically this compressibility of the gas is expressed by means of an integral term in the induction model for the temperature perturbation 0 x t this model is given by e ak 1 j e x t dx y x 2 0 x 0 d x 2 0

stability criteria for certain second order delay differential equations with mixed coefficients in this paper we study asymptotic stability of the zero solution of the second order linear delay differential equation y t p1y t p2y t q1y t q2y t where p1p2 q1 and q2 are constants with p1p2 0 and q1q2 0 is a constant in proving our results we make use of pontryagin s theory of quasi polynomials

asymptotic stability of a mechanical robotics model with damping and delay in this paper we study the asymptotic stability of a mechanical robotics model with damping and delay this model yields a certain linear third order delay differential equation in proving our results we make use of pontryagin s theory for quasi polynomials

global existence and asymptotic behavior of solutions to a nonlocal fisher kpp type problem abstract in this work we consider a nonlocal fisher kpp reaction diffusion problem with neumann boundary condition and nonnegative initial data in a bounded domain in r n n 1 with reaction term u 1 m t where m t is the total mass at time t with the help of poho aev s identity the non existence of nontrivial stationary solutions with dirichlet boundary conditions is being shown when 1 and the initial mass is greater than or equal to one the problem has nonnegative classical solutions while if the initial mass is less than one then the problem admits global solutions for n 1 2 with any 1 2 or n 3 with any 1 1 2 n moreover the asymptotic convergence to the solution of the heat equation is proved

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

blowup of solutions for the bad boussinesq type equation abstract the paper studies the blowup of solutions to the initial boundary value problem for the bad boussinesq type equation u tt u xx bu xxxx u xx where b 0 is a real number and s is a given nonlinear function by virtue of the energy method and the fourier transform method respectively it proves that under certain assumptions on s and initial data the generalized solutions of the above mentioned problem blow up in finite time and a few examples are shown especially for the bad boussinesq equation two examples of blowup of solutions are obtained numerically

stability criteria for certain high odd order delay differential equations in this paper we study the asymptotic stability of the zero solution of odd order linear delay differential equations of the formy 2 m 1 t j 02ma jy j t j 02mb jy j t t where a j and b j are certain constants and m 1 here t 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

density shrinking algorithm for community detection with path based similarity abstract community structure is ubiquitous in real world complex networks finding the communities is the key to understand the functions of those networks a lot of works have been done in designing algorithms for community detection but it remains a challenge in the field traditional modularity optimization suffers from the resolution limit problem recent researches show that combining the density based technique with the modularity optimization can overcome the resolution limit and an efficient algorithm named denshrink was provided the main procedure of denshrink is repeatedly finding and merging micro communities broad sense into super nodes until they cannot merge analyses in this paper show that if the procedure is replaced by finding and merging only dense pairs both of the detection accuracy and runtime can be obviously improved thus an improved density based algorithm imds is provided since the time complexity path based similarity indexes are difficult to be applied in community detection for high performance in this paper the path based katz index is simplified and used in the imds algorithm

community detection in complex networks using extremal optimization we propose a method to find the community structure in complex networks based on an extremal optimization of the value of modularity the method outperforms the optimal modularity found by the existing algorithms in the literature giving a better understanding of the community structure we present the results of the algorithm for computer simulated and real networks and compare them with other approaches the efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex networks

on asymptotic forms of reactive diffusive runaway different aspects of reactive diffusive runaway are discussed in this paper it is shown that a wide range of reactive nonlinearities can be analysed in essentially the same way leading to the conclusion that there appear to be only two distinct non singular self similar or approximately self similar forms of description for non homogeneous blowup these exactly self similar and asymptotically self similar descriptions have previously been recognized but are examined here in some detail for many different types of nonlinearity using asymptotic and numerical techniques it is confirmed that exactly self similar descriptions only behave non pathologically in fairly extreme discrete cases that are very close to a reactive diffusive or steady state balance under three or more dimensions of symmetry quantitative identification of these cases indicates that they may be of dubious practical relevance it is then shown that only one class of non singular asymptotically self similar descriptions can be found using coordinate perturbation techniques resulting solutions are derived to a high asymptotic order and are found to reveal a fairly universal structure for symmetric blowup a simple analysis reveals that non symmetric blowup also fits in with this structure at least to leading order

benchmark graphs for testing community detection algorithms community structure is one of the most important features of real networks and reveals the internal organization of the nodes many algorithms have been proposed but the crucial issue of testing i e the question of how good an algorithm is with respect to others is still open standard tests include the analysis of simple artificial graphs with a built in community structure that the algorithm has to recover however the special graphs adopted in actual tests have a structure that does not reflect the real properties of nodes and communities found in real networks here we introduce a class of benchmark graphs that account for the heterogeneity in the distributions of node degrees and of community sizes we use this benchmark to test two popular methods of community detection modularity optimization and potts model clustering the results show that the benchmark poses a much more severe test to algorithms than standard benchmarks revealing limits that may not be apparent at a first analysis

on the boundedness of global solutions of abstract semilinear parabolic equations abstract in this paper we consider the question of the long time behavior of solutions of the initial value problem for evolution equations of the formdu dt au t f u t in a banach space whereais the infinitesimal generator of an analytic semigroup andfis a nonlinear function such that the initial value problem possesses nonglobal solutions for some initial data we give sufficient conditions to ensure that global solutions on 0 are necessarily bounded in a suitable norm we give several examples of problems to which the abstract result applies

a rescaling algorithm for the numerical calculation of blowing up solutions on presente un algorithme a invariance d echelle pour le calcul numerique des solutions a singularites explosives des equations d evolution non lineaires

stationary solutions of chemotaxis systems the keller segel model is a system of partial differential equations modelling a mutual attraction of amoebae caused by releasing a chemical substance chemotaxis this paper analyzes the stationary solutions of the system with general nonlinearities via bifurcation techniques and gives a criterion for bifurcation of stable nonhomogeneous aggregation patterns examples are discussed with various kinds of nonlinearities modelling the sensitivity of the chemotaxis response

further properties of a continuum of model equations with globally defined flux to develop an understanding of singularity formation in vortex sheets we consider model equations that exhibit shared characteristics with the vortex sheet equation but are slightly easier to analyze a model equation is obtained by replacing the flux term in burgers equation by alternatives that contain contributions depending globally on the solution we consider the continuum of partial differential equationsut h u u x 1 u ux uxx 0 1 0 whereh u is the hilbert transform ofu we show that when 1 2 for 0 the solution of the equation exists for all time and is unique we also show with a combination of analytical and numerical means that the solution when 1 2 and 0 is analytic using a pseudo spectral method in space and the adams moulton fourth order predictor corrector in time we compute the numerical solution of the equation with 1 2 for various viscosities the results confirm that for 0 the solution is well behaved and analytic the numerical results also confirm that for 0 and 1 2 the solution becomes singular in finite time and finite viscosity prevents singularity formation we also present for a certain class of initial conditions solutions of the equation with 0 1 3 and 1 that become infinite for 0 in finite time

a local multiresolution algorithm for detecting communities of unbalanced structures abstract in complex networks such as computer and information networks social networks or biological networks a community structure is a common and important property community detection in complex networks has attracted a lot of attention in recent years community detection is the problem of finding closely related groups within a network modularity optimisation is a widely accepted method for community detection it has been shown that the modularity optimisation has a resolution limit because it is unable to detect communities with sizes smaller than a certain number of vertices defined with network size in this paper we propose a metric for describing community structures that enables community detection better than other metrics we present a fast local expansion algorithm for community detection the proposed algorithm provides online multiresolution community detection from a source vertex experimental results show that the proposed algorithm is efficient in both real world and synthetic networks

phase transition model for community detection abstract motivated by social and biological interactions a novel type of phase transition model is provided in order to investigate the emergence of the clustering phenomenon in networks the model has two types of interactions one is attractive and the other is repulsive in each iteration the phase of a node or an agent moves toward the average phase of its neighbors and moves away from the average phase of its non neighbors the velocities of the two types of phase transition are controlled by two parameters respectively it is found that the phase transition phenomenon is closely related to the topological structure of the underlying network and thus can be applied to identify its communities and overlapping groups by giving each node of the network a randomly generated initial phase and updating these phases by the phase transition model until they reach stability one or two communities will be detected according to the nodes stable phases confusable nodes are moved into a set named o f by removing the detected communities and the nodes in o f another one or two communities will be detected by an iteration of the algorithm in this way all communities and the overlapping nodes are detected simulations on both real world networks and the lfr benchmark graphs have verified the efficiency of the proposed scheme

blow up in a system of partial differential equations with conserved first integral part ii problems with convection a reaction diffusion convection equation with a nonlocal term is studied the nonlocal operator acts to conserve the spatial integral of the unknown function as time evolves the equations are parameterised by and for 1 the equation arises as a similarity solution of the navier stokes equations and the nonlocal term plays the role of pressure for 0 the equation is a nonlocal reaction diffusion problem the aim of the paper is to determine for which values of the parameter blow up occurs and to study its form in particular interest is focused on the three cases 1 2 and 1 it is observed that for any 0 1 2 nonuniform global blow up occurs if 1 2 1 then the blow up is global and uniform while for 1 the navier stokes equations there are exact solutions with initial data of arbitrarily large l l 2 and h 1 norms that decay to zero furthermore one of these exact solutions is proved to be nonlinearly stable in l 2 for arbitrarily large supremum norm an understanding of this transition from blow up behaviour to decay behaviour is achieved by a combination of analysis asymptotics and numerical techniques

blowup in a partial differential equation with conserved first integral a reaction diffusion equation with a nonlocal term is studied the nonlocal term acts to conserve the spatial integral of the unknown function as time evolves such equations give insight into biological and chemical problems where conservation properties predominate the aim of the paper is to understand how the conservation property affects the nature of blowup the equation studied has a trivial steady solution that is proved to be stable existence of nontrivial steady solutions is proved and their instability established numerically blowup is proved for sufficiently large initial data by using a comparison principle in fourier space the nature of the blowup is investigated by a combination of asymptotic and numerical calculations

detection of community structure in networks based on community coefficients abstract determining community structure in networks is fundamental to the analysis of the structural and functional properties of those networks including social networks computer networks and biological networks modularity function q which was proposed by newman and girvan was once the most widely used criterion for evaluating the partition of a network into communities however modularity q is subject to a serious resolution limit in this paper we propose a new function for evaluating the partition of a network into communities this is called community coefficient c using community coefficient c we can automatically identify the ideal number of communities in the network without any prior knowledge we demonstrate that community coefficient c is superior to the modularity q and does not have a resolution limit we also compared the two widely used community structure partitioning methods the hierarchical partitioning algorithm and the normalized cuts ncut spectral partitioning algorithm we tested these methods on computer generated networks and real world networks whose community structures were already known the ncut algorithm and community coefficient c were found to produce better results than hierarchical algorithms unlike several other community detection methods the proposed method effectively partitioned the networks into different community structures and indicated the correct number of communities

symmetric singularity formation in lubrication type equations for interface motion fourth order degenerate diffusion equations arise in a lubrication approximation of a thin film or neck driven by surface tension numerical studies of the lubrication equation le h t h n h xxx x 0 with various boundary conditions indicate that singularity formation in which h x t t to 0 occurs for small enough n with anomalous or second type scaling inconsistent with usual dimensional analysis this paper considers locally symmetric or even singularities in the le and in the modified lubrication equation mle h t h n h xxxx 0 both equations have the property that entropy bounds forbid finite time singularities when n is sufficiently large power series expansions for local symmetric similarity solutions are proposed for equation le with n 1 and mle for all n in mathbb r in the latter case special boundary conditions that force singularity formation are required to produce singularities when n is large matching conditions at higher order terms in the

adaptive multi resolution modularity for detecting communities in networks abstract community structure is a common topological property of complex networks which attracted much attention from various fields optimizing quality functions for community structures is a kind of popular strategy for community detection such as modularity optimization here we introduce a general definition of modularity by which several classical multi resolution modularity can be derived and then propose a kind of adaptive multi resolution modularity that can combine the advantages of different modularity by applying the modularity to various synthetic and real world networks we study the behaviors of the methods showing the validity and advantages of the multi resolution modularity in community detection the adaptive modularity as a kind of multi resolution method can naturally solve the first type limit of modularity and detect communities at different scales it can quicken the disconnecting of communities and delay the breakup of communities in heterogeneous networks and thus it is expected to generate the stable community structures in networks more effectively and have stronger tolerance against the second type limit of modularity

a nonlocal sturm liouville eigenvalue problem a nonlocal eigenvalue problem of the form u a x u bu u with homogeneous dirichlet boundary conditions is considered where b is a rank one bounded linear operator and x belongs to some bounded interval on the real line the behaviour of the eigenvalues is studied using methods of linear perturbation theory in particular some results are given which ensure that the spectrum remains real a sturm type comparison result is obtained finally these results are applied to the study of some nonlocal reaction diffusion equations

global blow up of a nonlinear heat equation solutions to semilinear parabolic equations of the form u t u f u x in which blow up at some finite time t are investigated for slowly growing functions f for nonlinearities such as f s 2 s ln 2 s 1 b with 0 u becomes infinite throughout as t t it is alsofound that for marginally more quickly growing functions e g f s 2 s ln 2 s 2 u is unbounded on some subset of which has positive measure and is unbounded throughout if is a small enough region

on blow up and degeneracy for the semilinear heat equation with source the asymptotic behaviour of the solution of the semilinear parabolic equation u t u xx 1 u ln 2 l u for t 0 x u x t 0 for t 0 and u 0 x u 0 x 0 in which blows up at a finite time t 0 is investigated it is proved that for some two parametric set of initial functions u 0 the behaviour of u t x near t t 0 is described by the approximate self similar solution v a t x exp t 0 t 1 cos 2 x 2 1 satisfying the first order nonlinear hamilton jacobi equation v t v x 2 1 v 1 v ln 2 1 v some open problems of degeneracy near a finite blow up time for other semilinear or quasilinear parabolic equations with source u t u 1 u ln 1 u 1 u t u u l u t u e u u t ln 1 u u 1 u ln 1 u 0 1 are discussed

numerical solution of fast diffusion or slow diffusion equations in this paper the author proposes a semi discretization in time of a nonlinear reaction diffusion equation fast or slow diffusion the solution of which may vanish or blow up in a finite time the approximate value at each time step is solution of a nonlinear equation which is solved by using an iterative method it is proved that if the exact solution vanishes or blows up in a finite time it is the same for the numerical solution

ordered community structure in networks abstract community structure in networks is often a consequence of homophily or assortative mixing based on some attribute of the vertices for example researchers may be grouped into communities corresponding to their research topic this is possible if vertex attributes have unordered discrete values but many networks exhibit assortative mixing by some ordered discrete or continuous attribute such as age or geographical location in such cases the identification of discrete communities may be difficult or impossible we consider how the notion of community structure can be generalized to networks that have assortative mixing by ordered attributes we propose a method of generating synthetic networks with ordered communities and investigate the effect of ordered community structure on the spread of infectious diseases we also show that current community detection algorithms fail to recover community structure in ordered networks and evaluate an alternative method using a layout algorithm to recover the ordering

detecting communities by the core vertex and intimate degree in complex networks abstract in this paper we present a new approach to extract communities in the complex networks with considerable accuracy we introduce the core vertex and the intimate degree between the community and its neighboring vertices first we find the core vertices as the initial community these core vertices are then expanded using intimate degree function during extracting community structure from the given network in addition our algorithm successfully finds common nodes between communities experimental results using some real world networks data shows that the performance of our algorithm is satisfactory

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

blowup for nonlinear equations using a comparison principle in fourier space on decrit une nouvelle methode pour demontrer l eclatement en temps fini pour une classe d equations paraboliques semi lineaires

blow up versus extinction in a nonlocal p laplace equation with neumann boundary conditions abstract this paper studies a fast diffusive p laplace equation with the nonlocal source u q u q d x in a bounded domain subject to homogeneous neumann boundary value condition a critical criterion is determined that the changing sign solutions blow up in finite time with q 1 and non positive initial energy associated and must be global for any initial energy if q 1 in particular the conditions are obtained under which the changing sign solutions vanish in finite time

a new prediction scheme for input delay compensation in restricted feedback linearizable systems the input output inversion of a system under the effect of input delays typically relies on the ability to predict the future of the system s state indeed if the latter is known ahead of time one can cope with the input delay by using a prediction of the state instead of the state itself such methods are efficient when the plant is stable but become numerically unstable otherwise we present a new method to compensate input delays our approach relies on imposing a desired error dynamics which is designed to be linear and asymptotically stable at the origin then the state prediction is computed from the state reference trajectory and the predicted error dynamics in this technical note we concentrate on the case study of systems in strict feedback form and present a simple backstepping procedure

detecting network communities by propagating labels under constraints we investigate the recently proposed label propagation algorithm lpa for identifying network communities we reformulate the lpa as an equivalent optimization problem giving an objective function whose maxima correspond to community solutions by considering properties of the objective function we identify conceptual and practical drawbacks of the label propagation approach most importantly the disparity between increasing the value of the objective function and improving the quality of communities found to address the drawbacks we modify the objective function in the optimization problem producing a variety of algorithms that propagate labels subject to constraints of particular interest is a variant that maximizes the modularity measure of community quality performance properties and implementation details of the proposed algorithms are discussed bipartite as well as unipartite networks are considered

singularities in a modified kuramoto sivashinsky equation describing interface motion for phase transition abstract phase transitions can be modeled by the motion of an interface between two locally stable phases a modified kuramoto sivashinsky equation h t 2 h 4 h 1 h 2 2 h 2 h xx h yy h xy 2 describes near planar interfaces which are marginally long wave unstable we study the question of finite time singularity formation in this equation in one and two space dimensions on a periodic domain such singularity formation does not occur in the kuramoto sivashinsky equation 0 for all 1 0 we provide sufficient conditions on the initial data and size of the domain to guarantee a finite time blow up in which a second derivative of h becomes unbounded using a bifurcation theory analysis we show a parallel between the stability of steady periodic solutions and the question of finite time blow up in one dimension finally we consider the local structure of the blow up in the one dimensional case via similarity solutions and numerical simulations that employ a dynamically adaptive self similar grid the simulations resolve the singularity to over 25 decades in h xx l and indicate that the singularities are all locally described by a unique self similar profile in h xx we discuss the relevance of these observations to the full intrinsic equations of motion and the associated physics

community mining with new node similarity by incorporating both global and local topological knowledge in a constrained random walk abstract detection of community is a crucial step to understand the structure and dynamics of complex networks most of conventional community detection methods focus on optimizing a certain objective function or on clustering nodes based on their similarities which leads to a phenomenon that they have preference for specific types of networks but are not general using constrained random walk we exploit global and local topology structures of network to propose a modified transition matrix and further to define a new similarity metric named isim between two nodes in contrast to the existing similarities isim does not work directly on the observed data but in a convergent stable space this feature makes isim robust to the observed noisy data in real world networks isim not only measures node s distance but also captures node s topology structure in network experiments on synthetic and real world networks demonstrate that isim can be successfully applied to community detection in broader types of networks and outperforms other community detection methods

modularity from fluctuations in random graphs and complex networks the mechanisms by which modularity emerges in complex networks are not well understood but recent reports have suggested that modularity may arise from evolutionary selection we show that finding the modularity of a network is analogous to finding the ground state energy of a spin system moreover we demonstrate that due to fluctuations stochastic network models give rise to modular networks specifically we show both numerically and analytically that random graphs and scale free networks have modularity we argue that this fact must be taken into consideration to define statistically significant modularity in complex networks

blow up and extinction for a thin film equation with initial boundary value conditions abstract in this paper we consider an initial boundary value problem for a thin film equation with the nonlocal source u q u q d x in a bounded domain due to the comparison principle being invalid we use the potential well method to give a threshold result of global existence and non existence for the sign changing weak solutions further we obtain the conditions under which the global solutions extinct in finite time and study the asymptotic behavior of the non extinct solutions

numerical simulation of blowup in nonlocal reaction diffusion equations using a moving mesh method in this paper we implement the moving mesh pde method for simulating the blowup in reaction diffusion equations with temporal and spacial nonlinear nonlocal terms by a time dependent transformation the physical equation is written into a lagrangian form with respect to the computational variables the time dependent transformation function satisfies a parabolic partial differential equation usually called moving mesh pde mmpde the transformed physical equation and mmpde are solved alternately by central finite difference method combined with a backward time stepping scheme the integration time steps are chosen to be adaptive to the blowup solution by employing a simple and efficient approach the monitor function in mmpdes plays a key role in the performance of the moving mesh pde method the dominance of equidistribution is utilized to select the monitor functions and a formal analysis is performed to check the principle a variety of numerical examples show that the blowup profiles can be expressed correctly in the computational coordinates and the blowup rates are determined by the tests

limits of modularity maximization in community detection modularity maximization is the most popular technique for the detection of community structure in graphs the resolution limit of the method is supposedly solvable with the introduction of modified versions of the measure with tunable resolution parameters we show that multiresolution modularity suffers from two opposite coexisting problems the tendency to merge small subgraphs which dominates when the resolution is low the tendency to split large subgraphs which dominates when the resolution is high in benchmark networks with heterogeneous distributions of cluster sizes the simultaneous elimination of both biases is not possible and multiresolution modularity is not capable to recover the planted community structure not even when it is pronounced and easily detectable by other methods for any value of the resolution parameter this holds for other multiresolution techniques and it is likely to be a general problem of methods based on global optimization

loops and multiple edges in modularity maximization of networks the modularity maximization model proposed by newman and girvan for the identification of communities in networks works for general graphs possibly with loops and multiple edges however the applications usually correspond to simple graphs these graphs are compared to a null model where the degree distribution is maintained but edges are placed at random therefore in this null model there will be loops and possibly multiple edges sharp bounds on the expected number of loops and their impact on the modularity are derived then building upon the work of massen and doye but using algebra rather than simulation we propose modified null models associated with graphs without loops but with multiple edges graphs with loops but without multiple edges and graphs without loops nor multiple edges we validate our models by using the exact algorithm for clique partitioning of grotschel and wakabayashi

blowup in diffusion equations a survey this paper deals with quasilinear reaction diffusion equations for which a solution local in time exists if the solution ceases to exist for some finite time we say that it blows up in contrast to linear equations blowup can occur even if the data are smooth and well defined for all times depending on the equation either the solution or some of its derivatives become singular we shall concentrate on those cases where the solution becomes unbounded in finite time this can occur in quasilinear equations if the heat source is strong enough there exist many theoretical studies on the question on the occurrence of blowup in this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup the asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power a better knowledge would be useful also for their numerical treatment thus not surprisingly the numerical analysis of this type of problems is still at a rather early stage the goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems

global existence and non existence problem for a reaction diffusion equation with a conserved first integral in this paper we prove the global existence and non existence of solutions of the following problem rdc cases u t u xx u 2 int u 2 x text d x quad x in 0 1 quad t 0 u x 0 t u x 1 t 0 quad t 0 u x 0 u 0 x quad x in 0 1 int 0 1 u x t text d x 0 quad t 0 endcases moreover let u m x be a stationary solution of problem rdc with m zeros in the interval 0 1 for m in n and if we take u 0 x epsilon u m x then we have that the solution exists globally if 0 1 this result verifies the numerical results of budd et al 1993 siam jl appl math 53 718 742 that the non zero stationary solutions are unstable

locally optimal heuristic for modularity maximization of networks community detection in networks based on modularity maximization is currently done with hierarchical divisive or agglomerative as well as partitioning heuristics hybrids and in a few papers exact algorithms we consider here the case of hierarchical networks in which communities should be detected and propose a divisive heuristic which is locally optimal in the sense that each of the successive bipartitions is done in a provably optimal way this heuristic is compared with the spectral based hierarchical divisive heuristic of newman proc natl acad sci usa 103 8577 2006 and with the hierarchical agglomerative heuristic of clauset newman and moore phys rev e 70 066111 2004 computational results are given for a series of problems of the literature with up to 4941 vertices and 6594 edges they show that the proposed divisive heuristic gives better results than the divisive heuristic of newman and than the agglomerative heuristic of clauset et al