a maximum entropy framework for nonexponential distributions probability distributions having power law tails are observed in a broad range of social economic and biological systems we describe here a potentially useful common framework we derive distribution functions for situations in which a joiner particle k pays some form of price to enter a community of size where costs are subject to economies of scale maximizing the boltzmann gibbs shannon entropy subject to this energy like constraint predicts a distribution having a power law tail it reduces to the boltzmann distribution in the absence of economies of scale we show that the predicted function gives excellent fits to 13 different distribution functions ranging from friendship links in social networks to protein protein interactions to the severity of terrorist attacks this approach may give useful insights into when to expect power law distributions in the natural and social sciences

optimal network configuration for maximal coherence resonance in excitable systems we analyze the coherence resonance phenomenon in an ensemble of noise driven excitable neurons giving special attention to the role of the interaction topology the neural architecture is modeled using a spatially embedded network in which we can tune the network organization between scale free like with dominating long range connections and a network with mostly adjacent neurons connected we found that besides an optimal noise intensity also an optimal network configuration exists at which the largest average coherence of noise induced spikes is achieved furthermore we show that long as well as short range interactions between neurons should exist in order to achieve the optimal response of the neuronal network

models of social networks based on social distance attachment we propose a class of models of social network formation based on a mathematical abstraction of the concept of social distance social distance attachment is represented by the tendency of peers to establish acquaintances via a decreasing function of the relative distance in a representative social space we derive analytical results corroborated by extensive numerical simulations showing that the model reproduces the main statistical characteristics of real social networks large clustering coefficient positive degree correlations and the emergence of a hierarchy of communities the model is confronted with the social network formed by people that shares confidential information using the pretty good privacy pgp encryption algorithm the so called web of trust of pgp

a vector partitioning approach to detecting community structure in complex networks in recent years the problem of community structure detection has attracted more and more attention and many approaches have been proposed recently newman pointed out that this issue can be transformed into the problem of constrained maximization of the assignment matrix over possible divisions of a network he presents further that this maximization process can be written in terms of the eigenspectrum of the modularity matrix on the basis of this work and the vector partition approach in computer science we propose a kind of multiway division approach for detecting community structure of complex networks experimental results indicate that the algorithm works well and is effective at finding both good communities and the appropriate number of communities

fuzzy communities and the concept of bridgeness in complex networks we consider the problem of fuzzy community detection in networks which complements and expands the concept of overlapping community structure our approach allows each vertex of the graph to belong to multiple communities at the same time determined by exact numerical membership degrees even in the presence of uncertainty in the data being analyzed we create an algorithm for determining the optimal membership degrees with respect to a given goal function based on the membership degrees we introduce a measure that is able to identify outlier vertices that do not belong to any of the communities bridge vertices that have significant membership in more than one single community and regular vertices that fundamentally restrict their interactions within their own community while also being able to quantify the centrality of a vertex with respect to its dominant community the method can also be used for prediction in case of uncertainty in the data set analyzed the number of communities can be given in advance or determined by the algorithm itself using a fuzzified variant of the modularity function the technique is able to discover the fuzzy community structure of different real world networks including but not limited to social networks scientific collaboration networks and cortical networks with high confidence

weighted scale free networks with variable power law exponents abstract we present a weighted scale free network model in which the power law exponents can be controlled by the model parameters the network is generated through the weight driven preferential attachment of new nodes to existing nodes and the growth of the weights of existing links the simplicity of the model enables us to derive analytically the various statistical properties such as the distributions of degree strength and weight the degree strength and degree weight relationship and the dependencies of these power law exponents on the model parameters finally we demonstrate that networks of words coauthorship of researchers and collaboration of actor actresses are quantitatively well described by this model

statistical mechanics of the international trade network analyzing real data on international trade covering the time interval 1950 2000 we show that in each year over the analyzed period the network is a typical representative of the ensemble of maximally random weighted networks whose directed connections bilateral trade volumes are only characterized by the product of the trading countries gdps it means that time evolution of this network may be considered as a continuous sequence of equilibrium states i e a quasistatic process this in turn allows one to apply the linear response theory to make and also verify simple predictions about the network in particular we show that bilateral trade fulfills a fluctuation response theorem which states that the average relative change in imports exports between two countries is a sum of the relative changes in their gdps yearly changes in trade volumes prove that the theorem is valid

classifying thick subcategories of the stable category of cohen macaulay modules abstract various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics in this paper as a higher dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p group due to benson carlson and rickard we consider classifying thick subcategories of the stable category of cohen macaulay modules over a gorenstein local ring the main result of this paper yields a complete classification of the thick subcategories of the stable category of cohen macaulay modules over a local hypersurface in terms of specialization closed subsets of the prime ideal spectrum of the ring which are contained in its singular locus

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

scale free property of local world networks and their community structures abstract the scale free property and community structures of complex networks formed by local events have been studied theoretically numerically and empirically we showed analytically and numerically that the degree distribution function p k of the local world evolving network exhibits a crossover from an exponential to power law form by increasing the local world size m for m much larger than the crossover local world size m co the distribution function p k has a power law form for any degree k 1 below m co however p k obeys a power law for 1 k k co and decays exponentially for k k co the crossover size m co and the crossover degree k co have been also elucidated in addition we constructed the drug prescription network dpn as a real local world network in which the local world subsets are definitely specified to reveal how the local world nature affects properties of real world networks we found that the community structure of the dpn strongly correlates with the local worlds

hamiltonian dynamics of preferential attachment prediction and control of network dynamics are grand challenge problems in network science the lack of understanding of fundamental laws driving the dynamics of networks is among the reasons why many practical problems of great significance remain unsolved for decades here we study the dynamics of networks evolving according to preferential attachment pa known to approximate well the large scale growth dynamics of a variety of real networks we show that this dynamics is hamiltonian thus casting the study of complex networks dynamics to the powerful canonical formalism in which the time evolution of a dynamical system is described by hamilton s equations we derive the explicit form of the hamiltonian that governs network growth in pa this hamiltonian turns out to be nearly identical to graph energy in the configuration model which shows that the ensemble of random graphs generated by pa is nearly identical to the ensemble of random graphs with scale free degree distributions in other words pa generates nothing but random graphs with power law degree distribution the extension of the developed canonical formalism for network analysis to richer geometric network models with non degenerate groups of symmetries may eventually lead to a system of equations describing network dynamics at small scales

the analysis and dissimilarity comparison of community structure abstract based on a database of collaboration recording in econophysics scientists and other networks hierarchical clustering method and the algorithm of girvan and newman are applied to detect their community structure the interesting results for community structure of econophysicists collaboration network are shown a dissimilarity function d is proposed to quantitatively measure the difference between community structures obtained by different methods using this measurement the differences between the process and community results obtained by aforementioned algorithms are given the effectiveness of hierarchical clustering method and gn algorithm for detecting community structure in various networks is discussed

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

analysis of weighted networks the connections in many networks are not merely binary entities either present or not but have associated weights that record their strengths relative to one another recent studies of networks have by and large steered clear of such weighted networks which are often perceived as being harder to analyze than their unweighted counterparts here we point out that weighted networks can in many cases be analyzed using a simple mapping from a weighted network to an unweighted multigraph allowing us to apply standard techniques for unweighted graphs to weighted ones as well we give a number of examples of the method including an algorithm for detecting community structure in weighted networks and a simple proof of the maximum flow minimum cut theorem

compatibility as underlying mechanism behind the evolution of networks abstract in this work we propose a dynamic network model based on an evolutive rule called compatibility the presence of common topological properties observed in two real complex networks are well reproduced by the proposed model these results suggest that the compatibility between nodes is sufficient to generate the complexity observed in real systems even considering that the new added nodes do not have the ability to know global information of the system

generating multi scaling networks with two types of vertices abstract a variety of scale free networks have been created since the pioneer work by barabasi and albert science 286 1999 509 most of these models are homogeneous since they are composed of the same kind of nodes in the realistic world however elements nodes or vertices in the network may play different roles or have different functions in this work we develop an alternative way of vertex classification other than the ordinary modularity method by introducing two types of vertices the interaction between two neighbor vertices is dependent on their types it is found that the vertex degree exhibits a multi scaling law distribution with the scaling exponent of each types of vertex adjustable this network model may exhibit some interesting properties concerning the dynamical processes on it

laplacian spectra as a diagnostic tool for network structure and dynamics we examine numerically the three way relationships among structure laplacian spectra and frequency synchronization dynamics on complex networks we study the effects of clustering degree distribution and a particular type of coupling asymmetry input normalization all of which are known to have effects on the synchronizability of oscillator networks we find that these topological factors produce marked signatures in the laplacian eigenvalue distribution and in the localization properties of individual eigenvectors using a set of coordinates based on the laplacian eigenvectors as a diagnostic tool for synchronization dynamics we find that the process of frequency synchronization can be visualized as a series of quasi independent transitions involving different normal modes particular features of the partially synchronized state can be understood in terms of the behavior of particular modes or groups of modes for example there are important partially synchronized states in which a set of low lying modes remain unlocked while those in the main spectral peak are locked we find therefore that spectra are correlated with dynamics in ways that go beyond results relating a single threshold to a single extremal eigenvalue

pointwise estimates for a class of non homogeneous kolmogorov equations we consider a class of ultraparabolic differential equations that satisfy the hormander s hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions the method extends the moser s iteration procedure and has previously been employed in the case of operators verifying a further homogeneity assumption here we remove that assumption by proving some potential estimates and some ad hoc sobolev type inequalities for solutions

a class of improved algorithms for detecting communities in complex networks abstract detecting communities in complex networks is of considerable importance for understanding both the structure and function of the networks here we propose a class of improved algorithms for community detection by combining the betweenness algorithm of girvan and newman with the edge weight defined by the edge clustering coefficient the improved algorithms are tested on some artificial and real world networks and the results show that they can detect communities of networks more effectively in both unweighted and weighted cases in addition the technique for improving the betweenness algorithm in this paper thanks to its compatibility can directly be applied to various detection algorithms

time cumulative scale free networks without both growth and preferential attachment abstract a model of nongrowing networks with time cumulative scale free sf property based on the apply reply process is proposed in this paper this model is not only without growth but also without a direct expression of preferential attachment the network structure is evolved by the change of nodes active probability which is mathematically equivalent to a special diffusion process this model only allows zero or one edge connecting on a node at any moment and the power law degree distributions can be exhibited from the statistic of the network after a long time accumulation our results imply that both growth and direct preferential attachments are not necessary in the generation of the sf property

communication and correlation among communities given a network and a partition in communities we consider the issues how communities influence each other and when two given communities do communicate specifically we address these questions in the context of small world networks where an arbitrary quenched graph is given and long range connections are randomly added we prove that among the communities a superposition principle applies and gives rise to a natural generalization of the effective field theory already presented by m ostilli and j f f mendes phys rev e 78 031102 2008 n 1 which here n 1 consists in a sort of effective tap thouless anderson and palmer equations in which each community plays the role of a microscopic spin the relative susceptibilities derived from these equations calculated at finite or zero temperature where the method provides an effective percolation theory give us the answers to the above issues unlike the case n 1 asymmetries among the communities may lead via the tap like structure of the equations to many metastable states whose number in the case of negative shortcuts among the communities may grow exponentially fast with n as examples we consider the n viana bray communities model and the n one dimensional small world communities model despite being the simplest ones the relevance of these models in network theory as e g in social networks is crucial and no analytic solution were known until now connections between percolation and the fractal dimension of a network are also discussed finally as an inverse problem we show how from the relative susceptibilities a natural and efficient method to detect the community structure of a generic network arises

accuracy and precision of methods for community identification in weighted networks based on brief review of approaches for community identification and measurement for sensitivity characterization the accuracy and precision of several approaches for detecting communities in weighted networks are investigated in weighted networks the community structure should take both links and link weights into account and the partition of networks should be evaluated by weighted modularity q w the results reveal that link weight has important effects on communities especially in dense networks potts model and weighted extremal optimization weo algorithm work well on weighted networks then potts model and weo algorithms are used to detect communities in rhesus monkey network the results gives nice understanding for real community structure

evolution of a modified binomial random graph by agglomeration in the classical erd s renyi random graph g n p there are n vertices and each of the possible edges is independently present with probability p the random graph g n p is homogeneous in the sense that all vertices have the same characteristics on the other hand numerous real world networks are inhomogeneous in this respect such an inhomogeneity of vertices may influence the connection probability between pairs of vertices the purpose of this paper is to propose a new inhomogeneous random graph model which is obtained in a constructive way from the erd s renyi random graph g n p given a configuration of n vertices arranged in n subsets of vertices we call each subset a super vertex we define a random graph with n super vertices by letting two super vertices be connected if and only if there is at least one edge between them in g n p our main result concerns the threshold for connectedness we also analyze the phase transition for the emergence of the giant component and the degree distribution even though our model begins with g n p it assumes the existence of some community structure encoded in the configuration furthermore under certain conditions it exhibits a power law degree distribution both properties are important for real world applications

strong correlations between text quality and complex networks features abstract concepts of complex networks have been used to obtain metrics that were correlated to text quality established by scores assigned by human judges texts produced by high school students in portuguese were represented as scale free networks word adjacency model from which typical network features such as the in outdegree clustering coefficient and shortest path were obtained another metric was derived from the dynamics of the network growth based on the variation of the number of connected components the scores assigned by the human judges according to three text quality criteria coherence and cohesion adherence to standard writing conventions and theme adequacy development were correlated with the network measurements text quality for all three criteria was found to decrease with increasing average values of outdegrees clustering coefficient and deviation from the dynamics of network growth among the criteria employed cohesion and coherence showed the strongest correlation which probably indicates that the network measurements are able to capture how the text is developed in terms of the concepts represented by the nodes in the networks though based on a particular set of texts and specific language the results presented here point to potential applications in other instances of text analysis

quantitative description and modeling of real networks we present data analysis and modeling of two particular cases of study in the field of growing networks we analyze world wide web data set and authorship collaboration networks in order to check the presence of correlation in the data the results are reproduced with good agreement through a suitable modification of the standard albert barabasi model of network growth in particular intrinsic relevance of sites plays a role in determining the future degree of the vertex

riemannian geometric entropy for measuring network complexity a central issue in the science of complex systems is the quantitative characterization of complexity in the present work we address this issue by resorting to information geometry actually we propose a constructive way to associate with a in principle any network a differentiable object a riemannian manifold whose volume is used to define the entropy the effectiveness of the latter in measuring network complexity is successfully proved through its capability of detecting a classical phase transition occurring in both random graphs and scale free networks as well as of characterizing small exponential random graphs configuration models and real networks

lp compactness of inhomogeneous kinetic equation with rough force abstract we are concerned with the inhomogeneous kinetic equations with rough force l p compactness of the inhomogeneous kinetic equation with rough force is constructed by using phase space analysis and the fourier multiplier method

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

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

large scale topological and dynamical properties of the internet we study the large scale topological and dynamical properties of real internet maps at the autonomous system level collected in a 3 yr time interval we find that the connectivity structure of the internet presents statistical distributions settled in a well defined stationary state the large scale properties are characterized by a scale free topology consistent with previous observations correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the internet the study of the internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models we compare the properties of growing network models with the present real internet data analysis

social aggregation as a cooperative game abstract a new approach for the description of phenomena of social aggregation is suggested on the basis of psychological concepts as for instance social norms and cultural coordinates we deduce a general mechanism for social aggregation in which different clusters of individuals can merge according to cooperation among the agents in their turn the agents can cooperate or defect according to the clusters distribution inside the system the fitness of an individual increases with the size of its cluster but decreases with the work the individual had to do in order to join it in order to test the reliability of such a new approach we introduce a couple of simple toy models with the features illustrated above we see from this preliminary study how cooperation is the most convenient strategy only in the presence of very large clusters while on the other hand it is not necessary to have one hundred percent of cooperators for reaching a totally ordered configuration with only one megacluster filling the whole system

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

partition of networks into basins of attraction we study partition of networks into basins of attraction based on a steepest ascent search for the node of highest degree each node is associated with or attracted to its neighbor of maximal degree as long as the degree is increasing a node that has no neighbors of higher degree is a peak attracting all the nodes in its basin maximally random scale free networks exhibit different behavior based on their degree distribution exponent gamma for small gamma broad distribution networks are dominated by a giant basin whereas for large gamma narrow distribution there are numerous basins with peaks attracting mainly their nearest neighbors we derive expressions for the first two moments of the number of basins we also obtain the complete distribution of basin sizes for a class of hierarchical deterministic scale free networks that resemble random nets finally we generalize the problem to regular networks and lattices where all degrees are equal and thus the attractiveness of a node must be determined by an assigned weight rather than the degree we derive the complete distribution of basins of attraction resulting from randomly assigned weights in one dimensional chains

generation of scale free networks using a simple preferential rewiring dynamics abstract we propose a simple dynamical process for non growing networks where steady states in the long time limit exhibit power law degree distributions with the exponent 2 at each time step two nodes i and j are randomly selected and one incoming link to i is redirected to j with the rewiring probability r determined only by degrees of two nodes k i and k j while higher degree nodes are preferred to get another link this is an application of the general model introduced earlier s ree phys rev e 73 2006 026115 to take the structure of networks into account we also consider three possible distinctions for the model i how we choose a rewiring link out of all incoming links to i three cases ii whether links are directed or not two cases iii types of networks considering the existence of self loops and multiple links two cases as a result we specify the total of 12 different cases of the model we then observe numerically that networks will evolve to steady states with power law degree distributions when parameters of the model satisfy certain conditions this work is from an effort to find a simple model of the network dynamics generating scale free networks and has a potential to become an underlying mechanism for wide range of scale free non growing networks

orlov spectra bounds and gaps the orlov spectrum is a new invariant of a triangulated category it was introduced by d orlov building on work of a bondal m van den bergh and r rouquier the supremum of the orlov spectrum of a triangulated category is called the ultimate dimension in this work we study orlov spectra of triangulated categories arising in mirror symmetry we introduce the notion of gaps and outline their geometric significance we provide the first large class of examples where the ultimate dimension is finite categories of singularities associated to isolated hypersurface singularities similarly given any nonzero object in the bounded derived category of coherent sheaves on a smooth calabi yau hypersurface we produce a generator by closing the object under a certain monodromy action and uniformly bound this generator s generation time in addition we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived fukaya category of a symplectic surface of genus greater than one

random networks with tunable degree distribution and clustering we present an algorithm for generating random networks with arbitrary degree distribution and clustering frequency of triadic closure we use this algorithm to generate networks with exponential power law and poisson degree distributions with variable levels of clustering such networks may be used as models of social networks and as a testable null hypothesis about network structure finally we explore the effects of clustering on the point of the phase transition where a giant component forms in a random network and on the size of the giant component some analysis of these effects is presented

gibbs entropy of network ensembles by cavity methods the gibbs entropy of a microcanonical network ensemble is the logarithm of the number of network configurations compatible with a set of hard constraints this quantity characterizes the level of order and randomness encoded in features of a given real network here we show how to relate this entropy to large deviations of conjugated canonical ensembles we derive exact expression for this correspondence using the cavity methods for the configuration model for the ensembles with constraint degree sequence and community structure and for the ensemble with constraint degree sequence and number of links at a given distance

scale free graph with preferential attachment and evolving internal vertex structure we extend the classical barabasi albert preferential attachment procedure to graphs with internal vertex structure given by weights of vertices in our model weight dynamics depends on the current vertex degree distribution and the preferential attachment procedure takes into account both weights and degrees of vertices we prove that such a coupled dynamics leads to scale free graphs with exponents depending on parameters of the weight dynamics

spin models on random graphs with controlled topologies beyond degree constraints we study ising spin models on finitely connected random interaction graphs which are drawn from an ensemble in which not only the degree distribution p k can be chosen arbitrarily but which allows for further fine tuning of the topology via preferential attachment of edges on the basis of an arbitrary function q k k of the degrees of the vertices involved we solve these models using finite connectivity equilibrium replica theory within the replica symmetric ansatz in our ensemble of graphs phase diagrams of the spin system are found to depend no longer only on the chosen degree distribution but also on the choice made for q k k the increased ability to control interaction topology in solvable models beyond prescribing only the degree distribution of the interaction graph enables a more accurate modeling of real world interacting particle systems by spin systems on suitably defined random graphs

detecting communities in large networks we develop an algorithm to detect community structure in complex networks the algorithm is based on spectral methods and takes into account weights and link orientation since the method detects efficiently clustered nodes in large networks even when these are not sharply partitioned it turns to be specially suitable for the analysis of social and information networks we test the algorithm on a large scale data set from a psychological experiment of word association in this case it proves to be successful both in clustering words and in uncovering mental association patterns

interacting individuals leading to zipf s law institut de physique th eorique universit e de fribourg ch 1700 february 1 2008 we present a general approach to explain the zipf s law of city distribution if the simplestinteraction pairwise is assumed individuals tend to form cities in agreement with the well knownstatisticspacs numbers 89 50 r 05 20 y 05 40 j

recursive filtration method for detecting community structure in networks abstract community detection is a topic of considerable recent interest within complex networks but most methods proposed so far are divisive and agglomerative methods which delete only one edge each time to split the network or agglomerating only one node each time until no individual node remains unlike those we propose a method to split networks in parallel by deleting many edges in each filtration operation and propose a community recursive coefficient crc denoted by m instead of q modularity to quantify the effect of the splitting results in this paper we proved that recursive optimizing of the local m is equivalent to acquiring the maximal global q value corresponding to good divisions for a network with m edges c communities and arbitrary topology the method split the network at most c 1 times and detected the community structure in time o m 2 c 1 m we give several example applications and show that the method can detect local communities according to the densities of external links to them in increasing order especially in large networks

community structure discovery method based on the gaussian kernel similarity matrix abstract community structure discovery in complex networks is a popular issue and overlapping community structure discovery in academic research has become one of the hot spots based on the gaussian kernel similarity matrix and spectral bisection this paper proposes a new community structure discovery method first by adjusting the gaussian kernel parameter to change the scale of similarity we can find the corresponding non overlapping community structure when the value of the modularity is the largest relatively second the changes of the gaussian kernel parameter would lead to the unstable nodes jumping off so with a slight change in method of non overlapping community discovery we can find the overlapping community nodes finally synthetic data karate club and political books datasets are used to test the proposed method comparing with some other community discovery methods to demonstrate the feasibility and effectiveness of this method

analysis of nonlinear synchronization dynamics of oscillator networks by laplacian spectral methods we analyze the synchronization dynamics of phase oscillators far from the synchronization manifold including the onset of synchronization on scale free networks with low and high clustering coefficients we use normal coordinates and corresponding time averaged velocities derived from the laplacian matrix which reflects the network s topology in terms of these coordinates synchronization manifests itself as a contraction of the dynamics onto progressively lower dimensional submanifolds of phase space spanned by laplacian eigenvectors with lower eigenvalues differences between high and low clustering networks can be correlated with features of the laplacian spectrum for example the inhibition of full synchoronization at high clustering is associated with a group of low lying modes that fail to lock even at strong coupling while the advanced partial synchronization at low coupling noted elsewhere is associated with high eigenvalue modes

chinese character structure analysis based on complex networks abstract in this paper chinese character networks are modelled using complex networks theory we analyze statistical properties of the networks and find that character networks also display two important features as other real networks i e small world feature and the non poisson distribution these results indicate that the discovered features of chinese character structure reflect the combinatorial nature of chinese characters we also simulate the formation of chinese phono semantic characters using bipartite graph theory the bipartite graph model generates non poisson distributions and disassortative mixing as the empirical networks which effectively explain the origin and formation of phono semantic characters

scale free networks from varying vertex intrinsic fitness a new mechanism leading to scale free networks is proposed in this letter it is shown that in many cases of interest the connectivity power law behavior is neither related to dynamical properties nor to preferential attachment assigning a quenched fitness value x i to every vertex and drawing links among vertices with a probability depending on the fitnesses of the two involved sites gives rise to what we call a good get richer mechanism in which sites with larger fitness are more likely to become hubs i e to be highly connected

compactness in kinetic transport equations and hypoellipticity abstract we establish improved hypoelliptic estimates on the solutions of kinetic transport equations using a suitable decomposition of the phase space our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions f x v l 1 satisfying some appropriate transport relation v x f 1 x 2 1 v 2 g may be inferred solely from additional integrability and compactness with respect to v in a forthcoming work the authors make a crucial application of this new approach to the study of the hydrodynamic limit of the boltzmann equation with a rough force field arsenio and saint raymond in preparation 4

chinese lexical networks the structure function and formation abstract in this paper chinese phrases are modeled using complex networks theory we analyze statistical properties of the networks and find that phrase networks display some important features not only small world and the power law distribution but also hierarchical structure and disassortative mixing these statistical traits display the global organization of chinese phrases the origin and formation of such traits are analyzed from a macroscopic chinese culture and philosophy perspective it is interesting to find that chinese culture and philosophy may shape the formation and structure of chinese phrases to uncover the structural design principles of networks network motif patterns are studied it is shown that they serve as basic building blocks to form the whole phrase networks especially triad 38 feed forward loop plays a more important role in forming most of the phrases and other motifs the distinct structure may not only keep the networks stable and robust but also be helpful for information processing the results of the paper can give some insight into chinese language learning and language acquisition it strengthens the idea that learning the phrases helps to understand chinese culture on the other side understanding chinese culture and philosophy does help to learn chinese phrases the hub nodes in the networks show the close relationship with chinese culture and philosophy learning or teaching the hub characters hub linking phrases and phrases which are meaning related based on motif feature should be very useful and important for chinese learning and acquisition

detecting the community structure in complex networks based on quantum mechanics abstract in this paper we develop a novel method to detect the community structure in complex networks this approach is based on the combination of kernel based clustering using quantum mechanics the spectral clustering technique and the concept of the bayesian information criterion we test the proposed algorithm on zachary s karate club network and the world of american college football experimental results indicate that our algorithm is efficient and effective at finding both the optimal number of clusters and the best clustering of community structures

scale free property of directed networks with two intrinsic node weights abstract this paper proposes a network model to understand the scale free property of directed networks the proposed model assigns two intrinsic variables incoming and outgoing weights to every node a directed link is established from node i to node j if the sum of the outgoing weight of node i and the incoming weight of node j exceeds a predetermined threshold the proposed model allows us to know the exact analytical expressions for degree distributions and clustering we analytically find that the in degree and out degree distributions have power law tails and their scaling exponents are controllable within the range 1 the average clustering coefficient of nodes with out degree or in degree n also has a power law tail as a function of n we also find that the scaling exponent of the clustering coefficient depends on the correlation between incoming and outgoing weights

number of cliques in random scale free network ensembles abstract in this paper we calculate the average number of cliques in random scale free networks in the limit of large network size n 1 we consider first the hidden variable ensemble and subsequently the molloy reed ensemble in both cases we find that cliques i e fully connected subgraphs appear also when the average degree is finite this is in contrast to what happens in erdos and renyi graphs in which diverging average degree is required to observe cliques of size c 3 moreover we show that in random scale free networks the clique number i e the size of the largest clique present in the network diverges with the system size

some extensions of theorems of knorrer and herzog popescu abstract a construction due to knorrer shows that if n is a maximal cohen macaulay module over a hypersurface defined by f y 2 then the first syzygy of n y n decomposes as the direct sum of n and its own first syzygy this was extended by herzog popescu to hypersurfaces f y n replacing n y n by n y n 1 n we show in the same setting as herzog popescu that the first syzygy of n y k n is always an extension of n by its first syzygy and moreover that this extension has useful approximation properties we give two applications first we construct a ring over which every finitely generated module has an eventually 2 periodic projective resolution prompting us to call it a non commutative hypersurface ring second we give upper bounds on the dimension of the stable module category a k a the singularity category of a hypersurface defined by a polynomial of the form x 1 a 1 x d a d

communicability graph and community structures in complex networks we use the concept of the network communicability e estrada n hatano communicability in complex networks phys rev e 77 2008 036111 to define communities in a complex network the communities are defined as the cliques of a communicability graph which has the same set of nodes as the complex network and links determined by the communicability function then the problem of finding the network communities is transformed to an all clique problem of the communicability graph we discuss the efficiency of this algorithm of community detection in addition we extend here the concept of the communicability to account for the strength of the interactions between the nodes by using the concept of inverse temperature of the network finally we develop an algorithm to manage the different degrees of overlapping between the communities in a complex network we then analyze the usa airport network for which we successfully detect two big communities of the eastern airports and of the western central airports as well as two bridging central communities in striking contrast a well known algorithm groups all but two of the continental airports into one community

topological structural classes of complex networks we use theoretical principles to study how complex networks are topologically organized at large scale using spectral graph theory we predict the existence of four different topological structural classes of networks these classes correspond respectively to highly homogenous networks lacking structural bottlenecks networks organized into highly interconnected modules with low inter community connectivity networks with a highly connected central core surrounded by a sparser periphery and networks displaying a combination of highly connected groups quasicliques and groups of nodes partitioned into disjoint subsets quasibipartites here we show by means of the spectral scaling method that these classes really exist in real world ecological biological informational technological and social networks we show that neither of three network growth mechanisms random with uniform distribution preferential attachment and random with the same degree sequence as real network is able to reproduce the four structural classes of complex networks these models reproduce two of the network classes as a function of the average degree but completely fail in reproducing the other two classes of networks

spectral tripartitioning of networks we formulate a spectral graph partitioning algorithm that uses the two leading eigenvectors of the matrix corresponding to a selected quality function to split a network into three communities in a single step in so doing we extend the recursive bipartitioning methods developed by newman m e j newman proc natl acad sci u s a 103 8577 2006 phys rev e 74 036104 2006 to allow one to consider the best available two way and three way divisions at each recursive step we illustrate the method using simple bucket brigade examples and then apply the algorithm to examine the community structures of the coauthorship graph of network scientists and of u s congressional networks inferred from roll call voting similarities

fluctuation driven dynamics of the internet topology we study the dynamics of the internet topology based on empirical data on the level of the autonomous systems it is found that the fluctuations occurring in the stochastic process of connecting and disconnecting edges are important features of the internet dynamics the network s overall growth can be described approximately by a single characteristic degree growth rate g eff approximately 0 016 and the fluctuation strength sigma eff approximately 0 14 together with the vertex growth rate alpha approximately 0 029 a stochastic model which incorporates these values and an adaptation rule newly introduced reproduces several features of the real internet topology such as the correlations between the degrees of different vertices

quantifying network heterogeneity despite degree distributions give some insights about how heterogeneous a network is they fail in giving a unique quantitative characterization of network heterogeneity this is particularly the case when several different distributions fit for the same network when the number of data points is very scarce due to network size or when we have to compare two networks with completely different degree distributions here we propose a unique characterization of network heterogeneity based on the difference of functions of node degrees for all pairs of linked nodes we show that this heterogeneity index can be expressed as a quadratic form of the laplacian matrix of the network which allows a spectral representation of network heterogeneity we give bounds for this index which is equal to zero for any regular network and equal to one only for star graphs using it we study random networks showing that those generated by the erdos renyi algorithm have zero heterogeneity and those generated by the preferential attachment method of barabasi and albert display only 11 of the heterogeneity of a star graph we finally study 52 real world networks and we found that they display a large variety of heterogeneities we also show that a classification system based on degree distributions does not reflect the heterogeneity properties of real world networks

what big data tells sampling the social network by communication channels big data has become the primary source of understanding the structure and dynamics of the society at large scale the network of social interactions can be considered as a multiplex where each layer corresponds to one communication channel and the aggregate of all of them constitutes the entire social network however usually one has information only about one of the channels or even a part of it which should be considered as a subset or sample of the whole here we introduce a model based on a natural bilateral communication channel selection mechanism which for one channel leads to consistent changes in the network properties for example while it is expected that the degree distribution of the whole social network has a maximum at a value larger than one we get a monotonically decreasing distribution as observed in empirical studies of single channel data we also find that assortativity may occur or get strengthened due to the sampling method we analyze the far reaching consequences of our findings

mixing properties of growing networks and simpson s paradox the mixing properties of networks are usually inferred by comparing the degree of a node with the average degree of its neighbors this kind of analysis often leads to incorrect conclusions assortative patterns may appear reversed by a mechanism known as simpson s paradox we prove this fact by analytical calculations and simulations on three classes of growing networks based on preferential attachment and fitness where the disassortative behavior observed is a spurious effect our results give a crucial contribution to the debate about the origin of disassortative mixing since networks previously classified as disassortative reveal instead assortative behavior to a careful analysis

synchronization processes in complex networks we present an extended analysis based on the dynamics towards synchronization of a system of coupled oscillators of the hierarchy of communities in complex networks in the synchronization process different structures corresponding to well defined communities of nodes appear in a hierarchical way the analysis also provides a useful connection between synchronization dynamics complex networks topology and spectral graph analysis

multifractal network generator we introduce a new approach to constructing networks with realistic features our method in spite of its conceptual simplicity it has only two parameters is capable of generating a wide variety of network types with prescribed statistical properties e g with degree or clustering coefficient distributions of various very different forms in turn these graphs can be used to test hypotheses or as models of actual data the method is based on a mapping between suitably chosen singular measures defined on the unit square and sparse infinite networks such a mapping has the great potential of allowing for graph theoretical results for a variety of network topologies the main idea of our approach is to go to the infinite limit of the singular measure and the size of the corresponding graph simultaneously a very unique feature of this construction is that with the increasing system size the generated graphs become topologically more structured we present analytic expressions derived from the parameters of the to be iterated initial generating measure for such major characteristics of graphs as their degree clustering coefficient and assortativity coefficient distributions the optimal parameters of the generating measure are determined from a simple simulated annealing process thus the present work provides a tool for researchers from a variety of fields such as biology computer science biology or complex systems enabling them to create a versatile model of their network data

scale free networks from a hamiltonian dynamics contrary to many recent models of growing networks we present a model with fixed number of nodes and links where a dynamics favoring the formation of links between nodes with degree of connectivity as different as possible is introduced by applying a local rewiring move the network reaches equilibrium states assuming broad degree distributions which have a power law form in an intermediate range of the parameters used interestingly in the same range we find nontrivial hierarchical clustering

community detection in complex networks by density based clustering abstract we proposed a method to find the community structure in a complex network by density based clustering physical topological distance is introduced in density based clustering for determining a distance function of specific influence functions according to the distribution of the data the community structures are uncovered the method keeps a better connection mode of the community structure than the existing algorithms in terms of modularity which can be viewed as a basic characteristic of community detection in the future moreover experimental results indicate that the proposed method is efficient and effective to be used for community detection of medium and large networks

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