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

solving set optimization problems based on the concept of null set abstract we introduce the concept of null set in the hyperspace that is consisting of all nonempty subsets of a given normed space based on the concepts of null set and convex cone we can define two partial orderings according to the algebraic difference and hukuhara difference between any two elements in the hyperspace which will be used to define the solution concepts of set optimization problems on the other hand we transform set optimization problems into a conventional vector optimization problem under these settings we can apply the technique of scalarization to solve this transformed vector optimization problem finally we show that the optimal solution of scalarized problem is also the optimal solution of original set optimization problem

statistical significance of communities in networks nodes in real world networks are usually organized in local modules these groups called communities are intuitively defined as subgraphs with a larger density of internal connections than of external links in this work we define a measure aimed at quantifying the statistical significance of single communities extreme and order statistics are used to predict the statistics associated with individual clusters in random graphs these distributions allows us to define one community significance as the probability that a generic clustering algorithm finds such a group in a random graph the method is successfully applied in the case of real world networks for the evaluation of the significance of their communities

four types of nonlinear scalarizations and some applications in set optimization abstract there are two types of criteria of solutions for the set valued optimization problem the vectorial criterion and set optimization criterion the first criterion consists of looking for efficient points of set valued map and is called set valued vector optimization problem on the other hand kuroiwa tanaka ha started developing a new approach to set valued optimization which is based on comparison among values of the set valued map in this paper we treat the second type criterion and call set optimization problem the aim of this paper is to investigate four types of nonlinear scalarizing functions for set valued maps and their relationships these scalarizing functions are generalization of tammer weidner s scalarizing functions for vectors as applications of the scalarizing functions for sets we present nonconvex separation type theorems gordan s type alternative theorems for set valued map optimality conditions for set optimization problem and takahashi s minimization theorems for set valued map

recursive markov chains stochastic grammars and monotone systems of nonlinear equations we define recursive markov chains rmcs a class of finitely presented denumerable markov chains and we study algorithms for their analysis informally an rmc consists of a collection of finite state markov chains with the ability to invoke each other in a potentially recursive manner rmcs offer a natural abstract model for probabilistic programs with procedures they generalize in a precise sense a number of well studied stochastic models including stochastic context free grammars scfg and multi type branching processes mt bp we focus on algorithms for reachability and termination analysis for rmcs what is the probability that an rmc started from a given state reaches another target state or that it terminatesq these probabilities are in general irrational and they arise as least fixed point solutions to certain monotone systems of nonlinear equations associated with rmcs we address both the qualitative problem of determining whether the probabilities are 0 1 or in between and the quantitative problems of comparing the probabilities with a given bound or approximating them to desired precision we show that all these problems can be solved in pspace using a decision procedure for the existential theory of reals we provide a more practical algorithm based on a decomposed version of multi variate newton s method and prove that it always converges monotonically to the desired probabilities we show this method applies more generally to any monotone polynomial system we obtain polynomial time algorithms for various special subclasses of rmcs among these for scfgs and mt bps equivalently for 1 exit rmcs the qualitative problem can be solved in p time for linearly recursive rmcs the probabilities are rational and can be computed exactly in p time we show that our pspace upper bounds cannot be substantially improved without a breakthrough on long standing open problems the square root sum problem and an arithmetic circuit decision problem that captures p time on the unit cost rational arithmetic ram model we show that these problems reduce to the qualitative problem and to the approximation problem to within any nontrivial error for termination probabilities of general rmcs and to the quantitative decision problem for termination extinction of scfgs mt bps

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

pointwise well posedness in set optimization with cone proper sets abstract this paper deals with the well posedness property in the setting of set optimization problems by using a notion of well posed set optimization problem due to zhang et al 2009 18 and a scalarization process we characterize this property through the well posedness in the tykhonov sense of a family of scalar optimization problems and we show that certain quasiconvex set optimization problems are well posed our approach is based just on a weak boundedness assumption called cone properness that is unavoidable to obtain a meaningful set optimization problem

set relations and optimality conditions in set valued maps abstract problems in set valued optimization can be solved via set optimization in this paper properties of set optimization are investigated conditions for existence of solutions are established directional derivatives are studied for set valued mappings necessary and sufficient conditions in the existence of solutions are showed with directional derivatives

about hahn banach extension theorems and applications to set valued optimization in this paper we revise several generalizations of the algebraic hanh banach extension theorem for set valued maps and give new versions in terms of set relations and selections we apply our results by establishing optimality conditions for a set valued constrained optimization problem considering several types of solutions finally we give necessary conditions for the existence of affine selections via affinelike maps in the framework of linear spaces

existence theorems for set optimization problems abstract in this paper we consider set optimization problems by using a partial ordering based on a set relation we establish several existence results of minimal elements more general than those existing in the literature as an application we present sufficient conditions for the existence of solutions of a set valued optimization problem under some semi continuity assumption

nonconvex vector optimization of set valued mappings in this paper we discuss properties such as monotonicity and continuity of the gerstewitz s nonconvex separation functional with the aid of this functional necessary and sufficient optimality conditions for nonconvex optimization problems of set valued mappings are obtained in topological vector spaces

spectral methods and cluster structure in correlation based networks abstract we investigate how in complex systems the eigenpairs of the matrices derived from the correlations of multichannel observations reflect the cluster structure of the underlying networks for this we use daily return data from the nyse and focus specifically on the spectral properties of weight w i j c i j i j and diffusion matrices d i j w i j s j i j where c i j is the correlation matrix and s i j w i j the strength of node j the eigenvalues and corresponding eigenvectors of the weight matrix are ranked in descending order as in the earlier observations the first eigenvector stands for a measure of the market correlations its components are to first approximation equal to the strengths of the nodes and there is a second order roughly linear correction the high ranking eigenvectors excluding the highest ranking one are usually assigned to market sectors and industrial branches our study shows that both for weight and diffusion matrices the eigenpair analysis is not capable of easily deducing the cluster structure of the network without a priori knowledge in addition we have studied the clustering of stocks using the asset graph approach with and without spectrum based noise filtering it turns out that asset graphs are quite insensitive to noise and there is no sharp percolation transition as a function of the ratio of bonds included thus no natural threshold value for that ratio seems to exist we suggest that these observations can be of use for other correlation based networks as well

scaling invariance in spectra of complex networks a diffusion factorial moment approach a new method called diffusion factorial moment is used to obtain scaling features embedded in the spectra of complex networks for an erdos renyi network with connecting probability p mathit er l1 n the scaling parameter is ensuremath delta 0 51 while for p mathit er ensuremath geqslant 1 n the scaling parameter deviates from it significantly for ws small world networks in the special region p r 0 05 0 2 typical scale invariance is found for growing random networks in the range of ensuremath theta 0 33 049 we have ensuremath delta 0 6 ifmmode pm else textpm fi 0 1 and the value of ensuremath delta oscillates around ensuremath delta 0 6 abruptly in the range of ensuremath theta 0 54 1 we have basically ensuremath delta g0 7 scale invariance is one of the common features of the three kinds of networks which can be employed as a global measurement of complex networks in a unified way

well posedness for set optimization problems abstract in this paper three kinds of well posedness for set optimization are first introduced by virtue of a generalized gerstewitz s function the equivalent relations between the three kinds of well posedness and the well posedness of three kinds of scalar optimization problems are established respectively then sufficient and necessary conditions of well posedness for set optimization problems are obtained by using a generalized forcing function respectively finally various criteria and characterizations of well posedness are given for set optimization problems

on approximate solutions in set valued optimization problems in this paper we introduce several concepts of approximate solutions of set valued optimization problems with vector and set optimization we prove existence results and necessary and sufficient conditions by using limit sets

on solutions of set valued optimization problems we consider two criteria of a solution associated with a set valued optimization problem a vector criterion and a set criterion we show how solutions of a vector type can help to find solutions of a set type and reciprocally as an application we obtain a sufficient condition for the existence of solutions of a set type via vector optimization

overlapping modularity at the critical point of k clique percolation one of the most remarkable social phenomena is the formation of communities in social networks corresponding to families friendship circles work teams etc since people usually belong to several different communities at the same time the induced overlaps result in an extremely complicated web of the communities themselves thus uncovering the intricate community structure of social networks is a non trivial task with great potential for practical applications gaining a notable interest in the recent years the clique percolation method cpm is one of the earliest overlapping community finding methods which was already used in the analysis of several different social networks in this approach the communities correspond to k clique percolation clusters and the general heuristic for setting the parameters of the method is to tune the system just below the critical point of k clique percolation however this rule is based on simple physical principles and its validity was never subject to quantitative analysis here we examine the quality of the partitioning in the vicinity of the critical point using recently introduced overlapping modularity measures according to our results on real social and other networks the overlapping modularities show a maximum close to the critical point justifying the original criteria for the optimal parameter settings

λ c contingent derivatives of set valued maps abstract in this paper a new notion of differentiability of set valued maps is introduced based on ordered spaces techniques c lower and c upper contingent derivatives are defined conditions for c differentiability are given furthermore we show that these concepts are suitable for the formulation of optimality conditions for set valued optimization problems with set optimization criteria

discrete synchronization of massively connected systems using hierarchical couplings abstract we study the synchronization of massively connected dynamical systems for which the interactions come from the succession of couplings forming a global hierarchical coupling process motivations of this work come from the growing necessity of understanding properties of complex systems that often exhibit a hierarchical structure starting with a set of 2 n systems the couplings we consider represent a two by two matching process that gather them in larger and larger groups of systems providing to the whole set a structure in n stages corresponding to n scales of hierarchy this leads us naturally to the synchronization of a cantor set of systems indexed by 0 1 n using the closed open sets defined by n tuples of 0 and 1 that permit us to make the link with the finite previous situation of 2 n systems we obtain a global synchronization result generalizing this case in the same context we deal with this question when some defects appear in the hierarchy that is to say when some couplings among certain systems do not happen at a given stage of the hierarchy we prove we can accept an infinite number of broken links inside the hierarchy while keeping a local synchronization under the condition that these defects are present at the n smallest scales of the hierarchy for a fixed integer n and they be enough spaced out in those scales

quantitative function for community detection we propose a quantitative function for community partition i e modularity density or d value we demonstrate that this quantitative function is superior to the widely used modularity q and also prove its equivalence with the objective function of the kernel k means both theoretical and numerical results show that optimizing the new criterion not only can resolve detailed modules that existing approaches cannot achieve but also can correctly identify the number of communities

efficient sampling of complex network with modified random walk strategies abstract we present two novel random walk strategies choosing seed node csn random walk and no retracing nr random walk different from the classical random walk sampling the csn and nr strategies focus on the influences of the seed node choice and path overlap respectively three random walk samplings are applied in the erdos renyi er barabasi albert ba watts strogatz ws and the weighted usair networks respectively then the major properties of sampled subnets such as sampling efficiency degree distributions average degree and average clustering coefficient are studied the similar conclusions can be reached with these three random walk strategies firstly the networks with small scales and simple structures are conducive to the sampling secondly the average degree and the average clustering coefficient of the sampled subnet tend to the corresponding values of original networks with limited steps and thirdly all the degree distributions of the subnets are slightly biased to the high degree side however the nr strategy performs better for the average clustering coefficient of the subnet in the real weighted usair networks some obvious characters like the larger clustering coefficient and the fluctuation of degree distribution are reproduced well by these random walk strategies

random matrix theory models of electric grid topology abstract the random matrix theory is useful in the study of large systems such as electric grids these transmission systems can be modeled as complex networks with high voltage lines the edges that connect nodes representing power plants and substations we draw upon established literature of complex systems theory and introduce methods from nuclear and statistical physics to identify new characteristics of these networks we show that most grids can be characterized by the gaussian orthogonal ensemble an indicator of chaos in many complex systems under certain circumstances however grids may be described by poisson statistics an indicator of regularity we use the random matrix formalism to describe the interconnection of multiple grids and construct a simple model of a distributed grid

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

nonconvex scalarization in set optimization with set valued maps abstract the aim of this work is to obtain scalar representations of set valued optimization problems without any convexity assumption using a criterion of solution introduced by kuroiwa d kuroiwa some duality theorems of set valued optimization with natural criteria in proceedings of the international conference on nonlinear analysis and convex analysis world scientific river edge nj 1999 pp 221 228 which is based on ordered relations between sets we characterize this type of solutions by means of nonlinear scalarization the scalarizing function is a generalization of the gerstewitz s nonconvex separation function as applications of our results we give two existence theorems for set valued optimization problems

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

edge ratio and community structure in networks a hierarchical divisive algorithm is proposed for identifying communities in complex networks to that effect the definition of community in the weak sense of radicchi et al proc natl acad sci u s a 101 2658 2004 is extended into a criterion for a bipartition to be optimal one seeks to maximize the minimum for both classes of the bipartition of the ratio of inner edges to cut edges a mathematical program is used within a dichotomous search to do this in an optimal way for each bipartition this includes an exact solution of the problem of detecting indivisible communities the resulting hierarchical divisive algorithm is compared with exact modularity maximization on both artificial and real world data sets for two problems of the former kind optimal solutions are found for five problems of the latter kind the edge ratio algorithm always appears to be competitive moreover it provides additional information in several cases notably through the use of the dendrogram summarizing the resolution finally both algorithms are compared on reduced versions of the data sets of girvan and newman proc natl acad sci u s a 99 7821 2002 and of lancichinetti et al phys rev e 78 046110 2008 results for these instances appear to be comparable

some minimax theorems of set valued maps and their applications abstract in the paper we consider two types of scalarization functions of sets and investigate their properties moreover based on two set relations we propose two kinds of minimax and maximin values of set valued maps respectively and show some minimax theorems of set valued maps with respect to those minimax and maximin values by using several properties of the above two functions as an application of these results we give common saddle point theorems of vector valued functions

epidifferentiability and hypodifferentiability of pseudoconvex maps in set optimization problems abstract in this paper we study set valued optimization problems with set optimization criteria with this aim we introduce two classes of pseudoconvex maps whose elements are epidifferentiable or hypodifferentiable we establish optimality conditions in terms of the epiderivatives and hypoderivatives of associated maps of infima or suprema under pseudoconvexity conditions finally by connecting set optimization criteria with vector criterion we apply these results in order to solve a vector optimization problem with constraints given by a set valued map

diffusion on complex networks a way to probe their large scale topological structures a diffusion process on complex networks is introduced in order to uncover their large scale topological structures this is achieved by focusing on the slowest decaying diffusive modes of the network the proposed procedure is applied to real world networks like a friendship network of known modular structure and an internet routing network for the friendship network its known structure is well reproduced in case of the internet where the structure is far less well known one indeed finds a modular structure and modules can roughly be associated with individual countries quantitatively the modular structure of the internet manifests itself in an approximately 10 times larger participation ratio of its slowest decaying modes as compared to the null model a random scale free network the extreme edges of the internet are found to correspond to russian and us military sites c 2004 elsevier b v all rights reserved less

subspaces generated by translations in rearrangement invariant spaces in the setting of rearrangement invariant r i banach function spaces e on 0 we study the complementability of subspaces qa generated by sequences of translations of functions a e 0 1 an r i function space e is said to be nice in short e n if every subspace of type qa is complemented we give necessary and sufficient conditions for an r i function space to be nice we determinate the orlicz lorentz and marcinkiewicz spaces belonging to the class n as an application we obtain a new characterization of the lp spaces 1 p among the class of r i function spaces

sampling properties of random graphs the degree distribution we discuss two sampling schemes for selecting random subnets from a network random sampling and connectivity dependent sampling and investigate how the degree distribution of a node in the network is affected by the two types of sampling here we derive a necessary and sufficient condition that guarantees that the degree distributions of the subnet and the true network belong to the same family of probability distributions for completely random sampling of nodes we find that this condition is satisfied by classical random graphs for the vast majority of networks this condition will however not be met we furthermore discuss the case where the probability of sampling a node depends on the degree of a node and we find that even classical random graphs are no longer closed under this sampling regime we conclude by relating the results to real eschericia coli protein interaction network data

exploring complex networks through random walks most real complex networks such as protein interactions social contacts and the internet are only partially known and available to us while the process of exploring such networks in many cases resembles a random walk it becomes a key issue to investigate and characterize how effectively the nodes and edges of such networks can be covered by different strategies at the same time it is critically important to infer how well can topological measurements such as the average node degree and average clustering coefficient be estimated during such network explorations the present article addresses these problems by considering random barabasi albert ba and geographical network models with varying connectivity explored by three types of random walks traditional preferential to untracked edges and preferential to unvisited nodes a series of relevant results are obtained including the fact that networks of the three studied models with the same size and average node degree allow similar node and edge coverage efficiency the identification of linear scaling with the size of the network of the random walk step at which a given percentage of the nodes edges is covered and the critical result that the estimation of the averaged node degree and clustering coefficient by random walks on ba networks often leads to heavily biased results many are the theoretical and practical implications of such results

modelling hierarchical and modular complex networks division and independence abstract we introduce a growing network model which generates both modular and hierarchical structure in a self organized way to this end we modify the barabasi albert model into the one evolving under the principles of division and independence as well as growth and preferential attachment pa a newly added vertex chooses one of the modules composed of existing vertices and attaches edges to vertices belonging to that module following the pa rule when the module size reaches a proper size the module is divided into two and a new module is created the karate club network studied by zachary is a simple version of the current model we find that the model can reproduce both modular and hierarchical properties characterized by the hierarchical clustering function of a vertex with degree k c k being in good agreement with empirical measurements for real world networks

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

detecting community structure via the maximal sub graphs and belonging degrees in complex networks abstract community structure is a common phenomenon in complex networks and it has been shown that some communities in complex networks often overlap each other so in this paper we propose a new algorithm to detect overlapping community structure in complex networks to identify the overlapping community structure our algorithm firstly extracts fully connected sub graphs which are maximal sub graphs from original networks then two maximal sub graphs having the key pair vertices can be merged into a new larger sub graph using some belonging degree functions furthermore we extend the modularity function to evaluate the proposed algorithm in addition overlapping nodes between communities are founded successfully finally we report the comparison between the modularity and the computational complexity of the proposed algorithm with some other existing algorithms the experimental results show that the proposed algorithm gives satisfactory results

explicit determination of mean first passage time for random walks on deterministic uniform recursive trees the determination of mean first passage time mfpt for random walks in networks is a theoretical challenge and is a topic of considerable recent interest within the physics community in this paper according to the known connections between mfpt effective resistance and the eigenvalues of graph laplacian we first study analytically the mfpt between all node pairs of a class of growing treelike networks which we term deterministic uniform recursive trees durts since one of its particular cases is a deterministic version of the famous uniform recursive tree the interesting quantity is determined exactly through the recursive relation of the laplacian spectra obtained from the special construction of durts the analytical result shows that the mfpt between all couples of nodes in durts varies as n text text ln text n for large networks with node number n second we study trapping on a particular network of durts focusing on a special case with the immobile trap positioned at a node having largest degree we determine exactly the average trapping time att that is defined as the average of fpt from all nodes to the trap in contrast to the scaling of the mfpt the leading behavior of att is a linear function of n interestingly we show that the behavior for att of the trapping problem is related to the trapping location which is in comparison with the phenomenon of trapping on fractal t graph although both networks exhibit tree structure finally we believe that the methods could open the way to exactly calculate the mfpt and att in a wide range of deterministic media

multiplicator space and complemented subspaces of rearrangement invariant space abstract we show that the multiplicator space m x of an rearrangement invariant r i space x on 0 1 and the nice part n 0 x of x that is the set of all a x for which the subspaces generated by sequences of dilations and translations of a are uniformly complemented coincide when the space x is separable in the general case the nice part is larger than the multiplicator space several examples of descriptions of m x and n 0 x for concrete x are presented

the natural rearrangement invariant structure on tensor products we prove that the only rearrangement invariant r i spaces for which there exists a crossnorm verifying that the tensor product of these spaces preserves the natural r i space structure in the sense that it makes the multiplication operator b a topological isomorphism are the lp spaces

generalized hukuhara gâteaux and frechet derivatives of interval valued functions and their application in optimization with interval valued functions abstract in this article the notions of gh directional derivative gh g teaux derivative and gh frechet derivative for interval valued functions are proposed the existence of gh frechet derivative is shown to imply the existence of gh g teaux derivative and the existence of gh g teaux derivative is observed to indicate the presence of gh directional derivative for an interval valued gh lipschitz function it is proved that the existence of gh g teaux derivative implies the existence of gh frechet derivative it is observed that for an interval valued convex function on a linear space the gh directional derivative exists at any point for every direction concepts of linear and monotonic interval valued functions are studied in the sequel further it is shown that the proposed derivatives are useful to check the convexity of an interval valued function and to characterize efficient points of an optimization problem with interval valued objective function it is observed that at an efficient point of an interval valued function none of its gh directional derivatives dominates zero and the gh g teaux derivative must contain zero the entire study is supported by suitable illustrative examples

optimization of synchronization in gradient clustered networks we consider complex clustered networks with a gradient structure where the sizes of the clusters are distributed unevenly such networks describe actual networks in biophysical systems and in technological applications more closely than the previous models theoretical analysis predicts that the network synchronizability can be optimized by the strength of the gradient field but only when the gradient field points from large to small clusters a remarkable finding is that if the gradient field is sufficiently strong synchronizability of the network is mainly determined by the properties of the subnetworks in the two largest clusters these results are verified by numerical eigenvalue analysis and by direct simulation of synchronization dynamics on coupled oscillator networks

strict efficiency in set valued optimization in this paper we develop the notion of a phi minimizer or strict efficiency for a set valued map its properties and relations with other similar notions are studied in special circumstances under suitable conditions we prove that a point is a phi minimizer of a vector function if and only if it is a phi minimizer of a certain family of linear scalarizations we also establish a characterization of strict efficiency through a nonlinear scalarization which is a generalization of the gerstewitz function defined on the power set of the image space the final part is focused on minimizers of order one and we provide several necessary or sufficient conditions without convexity assumptions through different kinds of derivatives as contingent radial among others various illustrative examples showing the applicability of our results are also presented

epidifferentiability of the map of infima in hilbert spaces this article deals with derivatives for set valued maps that take values in ordered vector spaces in particular it concerns about the relationship between the epiderivatives of a set valued map and its associated map of infima when the image space is a real separable hilbert space ordered by an orthonormal basis by using a variational technique based on a decoupling of the ordering cone into half spaces we show that both epiderivatives coincide under certain hypothesis of compactness and stability furthermore we obtain some computation formulas for these derivatives in terms of associated scalar set valued maps

markov chain methods for analyzing urban networks complex transport networks abstracted as graphs undirected directed or multi component can be effectively analyzed by random walks or diffusions we have unified many concepts into one framework and studied in details the structural and spectral properties of spatial graphs for five compact urban patterns

strict approximate solutions in set valued optimization with applications to the approximate ekeland variational principle this work deals with strict solutions of set valued optimization problems under the set optimality criterion in this context we introduce a new approximate solution concept and we obtain several properties of these solutions when the error is fixed and also for their limit behavior when the error tends to zero then we prove a general existence result which is applied to obtain approximate ekeland variational principles

diffusion on hierarchical systems of weakly coupled networks abstract we analyzed diffusion dynamics on weakly coupled networks interconnected networks by means of separation of time scales using an adiabatic approximation we reduced the system dynamics to a markov chain with aggregated variables and derived a transport equation that is analogous to fick s first law and includes a driving force entropy production is a sum of microscopic entropy transport which results from the particle s migration between networks of different topologies and macroscopic entropy production of the markov chain equilibrium particles partition between different sub networks depends only on internal sub network parameters by changing structure of networks one can not only modify diffusion constants but can also induce or reverse the direction of the particles flow between different networks our framework confirmed by numerical simulations is also useful for considering diffusion in nested systems corresponding to hierarchical networks with several different time scales thus it can serve to uncover hidden hierarchy levels from observations of diffusion processes

optimality conditions for set valued maps with set optimization abstract problems in set valued optimization can be solved via set optimization in this paper optimality conditions are studied for set valued maps with set optimization optimality requirements are established for continuous selections using directional derivatives necessary and sufficient conditions for the existence of solutions are shown for set valued maps under generalized convexity assumptions and with the notion of the contingent derivative

detecting overlapping communities in networks using the maximal sub graph and the clustering coefficient abstract in this paper we present an alternate algorithm for detecting overlapping community structures in the complex network two concepts named the maximal sub graph and the clustering coefficient between two neighboring communities are introduced first all the maximal sub graphs are extracted from the original networks and then merge them by considering the clustering coefficient of two neighboring maximal sub graphs and a new extended modularity is proposed to quantify this algorithm the other advantage of this algorithm is that the overlapping vertex can be detected the effectiveness of our algorithm is tested on some real networks finally we compare the computational complexity of this algorithm with selected close related algorithms the results show that this algorithm gives satisfactory results

degree dependent intervertex separation in complex networks we study the mean length l k of the shortest paths between a vertex of degree k and other vertices in growing networks where correlations are essential in a number of deterministic scale free networks we observe a power law correction to a logarithmic dependence l k a ln n k gamma 1 2 ck gamma 1 n in a wide range of network sizes here n is the number of vertices in the network gamma is the degree distribution exponent and the coefficients a and c depend on a network we compare this law with a corresponding l k dependence obtained for random scale free networks growing through the preferential attachment mechanism in stochastic and deterministic growing trees with an exponential degree distribution we observe a linear dependence on degree l k approximately a ln n ck we compare our findings for growing networks with those for uncorrelated graphs

on the equivalence of the label propagation method of community detection and a potts model approach abstract we show that the recently introduced label propagation method for detecting communities in complex networks is equivalent to finding the local minima of a simple potts model applying to empirical data the number of such local minima was found to be very high much larger than the number of nodes in the graph the aggregation method for combining information from more local minima shows a tendency to fragment the communities into very small pieces

localization of eigenvector centrality in networks with a cut vertex we show that eigenvector centrality exhibits localization phenomena on networks that can be easily partitioned by the removal of a vertex cut set the most extreme example being networks with a cut vertex three distinct types of localization are identified in these structures one is related to the well established hub node localization phenomenon and the other two are introduced and characterized here we gain insights into these problems by deriving the relationship between eigenvector centrality and katz centrality this leads to an interpretation of the principal eigenvector as an approximation to more robust centrality measures which exist in the full span of an eigenbasis of the adjacency matrix

modularity and extreme edges of the internet we study the spectral properties of a diffusion process taking place on the internet network focusing on the slowest decaying modes these modes identify an underlying modular structure roughly corresponding to individual countries for instance in the slowest decaying mode the diffusion current flows from russia to u s military sites quantitatively the modular structure manifests itself in a 10 times larger participation ratio of its slow decaying modes compared to a random scale free network we propose to use the fraction of nodes participating in slow decaying modes as a general measure of the modularity of a network for the 100 slowest decaying modes of the internet this fraction is 30 finally we suggest that the degree of isolation of an individual module can be assessed by comparing its participation in different diffusion modes

why social networks are different from other types of networks we argue that social networks differ from most other types of networks including technological and biological networks in two important ways first they have nontrivial clustering or network transitivity and second they show positive correlations also called assortative mixing between the degrees of adjacent vertices social networks are often divided into groups or communities and it has recently been suggested that this division could account for the observed clustering we demonstrate that group structure in networks can also account for degree correlations we show using a simple model that we should expect assortative mixing in such networks whenever there is variation in the sizes of the groups and that the predicted level of assortative mixing compares well with that observed in real world networks

social significance of community structure statistical view community structure analysis is a powerful tool for social networks that can simplify their topological and functional analysis considerably however since community detection methods have random factors and real social networks obtained from complex systems always contain error edges evaluating the significance of a partitioned community structure is an urgent and important question in this paper integrating the specific characteristics of real society we present a framework to analyze the significance of a social community the dynamics of social interactions are modeled by identifying social leaders and corresponding hierarchical structures instead of a direct comparison with the average outcome of a random model we compute the similarity of a given node with the leader by the number of common neighbors to determine the membership vector an efficient community detection algorithm is proposed based on the position of the nodes and their corresponding leaders then using a log likelihood score the tightness of the community can be derived based on the distribution of community tightness we establish a connection between p value theory and network analysis and then we obtain a significance measure of statistical form finally the framework is applied to both benchmark networks and real social networks experimental results show that our work can be used in many fields such as determining the optimal number of communities analyzing the social significance of a given community comparing the performance among various algorithms etc

covariance correlation matrix and the multiscale community structure of networks empirical studies show that real world networks often exhibit multiple scales of topological descriptions however it is still an open problem how to identify the intrinsic multiple scales of networks in this paper we consider detecting the multiscale community structure of network from the perspective of dimension reduction according to this perspective a covariance matrix of network is defined to uncover the multiscale community structure through the translation and rotation transformations it is proved that the covariance matrix is the unbiased version of the well known modularity matrix we then point out that the translation and rotation transformations fail to deal with the heterogeneous network which is very common in nature and society to address this problem a correlation matrix is proposed through introducing the rescaling transformation into the covariance matrix extensive tests on real world and artificial networks demonstrate that the correlation matrix significantly outperforms the covariance matrix identically the modularity matrix as regards identifying the multiscale community structure of network this work provides a novel perspective to the identification of community structure and thus various dimension reduction methods might be used for the identification of community structure through introducing the correlation matrix we further conclude that the rescaling transformation is crucial to identify the multiscale community structure of network as well as the translation and rotation transformations

localization and centrality in networks eigenvector centrality is a common measure of the importance of nodes in a network here we show that under common conditions the eigenvector centrality displays a localization transition that causes most of the weight of the centrality to concentrate on a small number of nodes in the network in this regime the measure is no longer useful for distinguishing among the remaining nodes and its efficacy as a network metric is impaired as a remedy we propose an alternative centrality measure based on the nonbacktracking matrix which gives results closely similar to the standard eigenvector centrality in dense networks where the latter is well behaved but avoids localization and gives useful results in regimes where the standard centrality fails