kernel spectral clustering with memory effect evolving graphs describe many natural phenomena changing over time such as social relationships trade markets methabolic networks etc in this framework performing community detection and analyzing the cluster evolution represents a criti cal task here we propose a new model for this purpose where the smoothness of the clustering results over time can be considered as a valid prior knowledge it is based on a constrained optimization formulation typical of least squares support vector machines ls svm where the objective function is designed to explicitly incorporate temporal smoothness the latter allo ws the model to cluster the current data well and to be consistent with the recent history we also propose new model selection criteria in order to carefully choose the hyper parameters of our model which is a crucial issue to achieve good performances we successfully test the model on four toy problems and on a real world network we also compare our model with evolutionary spectral clustering which is a stateof the art algorithm for community detection of evolving networks illustrating that the kernel spectral clustering with memory effect can achieve better or equal performances

milp formulations for the modularity density maximization problem abstract cluster analysis refers to finding subsets of vertices of a graph called clusters which are more likely to be joined pairwise than vertices in different clusters in the last years this topic has been studied by many researchers and several methods have been proposed one of the most popular is to maximize the modularity which represents the fraction of edges within clusters minus the expected fraction of such edges in a random graph with the same degree distribution however this criterion presents some issues for example the resolution limit i e the difficulty to detect clusters having small sizes in this paper we focus on a recent measure called modularity density which improves the resolution limit issue of modularity the problem of maximizing the modularity density can be described by means of a 0 1 nlp formulation we derive some properties of the optimal solution which will be used to tighten the formulation and we propose some milp reformulations which yield an improvement of the resolution time

performance of modularity maximization in practical contexts although widely used in practice the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood in practical contexts here we present a broad characterization of its performance in such situations first we revisit and clarify the resolution limit phenomenon for modularity maximization second we show that the modularity function q exhibits extreme degeneracies it typically admits an exponential number of distinct high scoring solutions and typically lacks a clear global maximum third we derive the limiting behavior of the maximum modularity q max for one model of infinitely modular networks showing that it depends strongly both on the size of the network and on the number of modules it contains finally using three real world metabolic networks as examples we show that the degenerate solutions can fundamentally disagree on many but not all partition properties such as the composition of the largest modules and the distribution of module sizes these results imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts they also explain why many heuristics are often successful at finding high scoring partitions in practice and why different heuristics can disagree on the modular structure of the same network we conclude by discussing avenues for mitigating some of these behaviors such as combining information from many degenerate solutions or using generative models

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

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

degrees of freedom of full duplex cellular networks effect of self interference it was recently shown that full duplex fd operation at a base station bs can provide up to twice as many degrees of freedom dof as a conventional half duplex hd cellular network in the absence of self interference in practice however self interference in an fd bs may not be eliminated completely due to imperfect cancellation and it is not yet known whether fd operation can improve the sum dof of cellular networks in the presence of residual self interference in this paper we provide a complete characterization of the sum dof in an fd bs cellular network with self interference where the rank of the self interference matrix is arbitrary specifically we propose a self interference cancellation scheme that maximizes the sum dof and we prove its optimality by deriving the matching upper bound our results show that even in the presence of residual self interference an fd bs network can still outperform a conventional hd bs network and furthermore the sum dof coincides with that of an fd bs network with no self interference under certain conditions we also derive an achievable sum rate under ergodic phase fading showing that not only a sum dof gain but also a sum rate gain can be obtained over the entire signal to noise ratio range even if residual self interference exists

the similarity of weights on edges and discovering of community structure abstract in this paper we propose a weighted modularity q w based on the similarity of weights on edges and a threshold coefficient to evaluate the equivalence of edge weights simulations on benchmark networks and real networks show that optimization on the modularity enable us to obtain groups of nodes within which the edge weights are distributed uniformly but at random between them the communities can reveal the uniform connections stable relationships measured by the similarity of weights on edges between nodes or some similarity between nodes functions furthermore with the dynamical moving of we observe that optimization on the q w allows for the discovering of a special hierarchical organization which reveals different levels of uniform connections between nodes in networks the substructures revealed by the hierarchical organization enable us to obtain more information of networks and give a potential way for partly remedying the intrinsic resolution problem of modularity

efficient modularity density heuristics for large graphs abstract modularity density maximization is a community detection optimization problem which improves the resolution limit degeneracy of modularity maximization this paper presents seven scalable heuristics for modularity density and compares them with literature results from exact mixed integer linear programming and gaod imeme net hain and bmd heuristics the results are also compared with cnm and louvain which are scalable heuristics for modularity maximization the results suggest that our seven heuristics are faster than gaod imeme net hain and bmd modularity density heuristics our experiments also show that some of our heuristics surpassed the objective function value reported by imeme net hain and bmd for some real graphs our seven heuristics were tested with real graphs from the stanford large network dataset collection and the experiments show that they are scalable this feature was confirmed by an amortized complexity analysis which reveals average linear time for three of our heuristics hypothesis tests suggest that four proposed heuristics are state of the art since they are scalable for hundreds of thousands of nodes for the modularity density problem and they find the high objective value partitions for the largest instances ground truth experiments in artificial random graphs were performed and suggest that our heuristics lead to better cluster detection than both cnm and louvain

detecting community structure using biased random merging abstract decomposing a network into small modules or communities is beneficial for understanding the structure and dynamics of the network one of the most prominent approaches is to repeatedly join communities together in pairs with the greatest increase in modularity so that a dendrogram that shows the order of joins is obtained then the community structure is acquired by cutting the dendrogram at the levels corresponding to the maximum modularity however there tends to be multiple pairs of communities that share the maximum modularity increment and the greedy agglomerative procedure may only merge one of them although the modularity function typically admits a lot of high scoring solutions the greedy strategy may fail to reach any of them in this paper we propose an enhanced data structure in order to enable diverse choices of merging operations in community finding procedure this data structure is actually a max heap equipped with an extra array that stores the maximum modularity increments and the corresponding community pairs is merged in the next move by randomly sampling elements in this head array additional diverse community structures can be efficiently extracted the head array is designed to host the community pairs corresponding to the most significant increments in modularity so that the modularity structures obtained out of the sampling exhibit high modularity scores that are on the average even greater than what the cnm algorithm produces our method is tested on both real world and computer generated networks

statistical mechanics of the minimum vertex cover problem in stochastic block models the minimum vertex cover min vc problem is a well known np hard problem earlier studies illustrate that the problem defined over the erdos renyi random graph with a mean degree c exhibits computational difficulty in searching the min vc set above a critical point c e 2 718 here we address how this difficulty is influenced by the mesoscopic structures of graphs for this we evaluate the critical condition of difficulty for the stochastic block model we perform a detailed examination of the specific cases of two equal size communities characterized by in and out degrees which are denoted by c in and c out respectively our analysis based on the cavity method indicates that the solution search once becomes difficult when c in c out exceeds e from below but becomes easy again when c out is sufficiently larger than c in in the region c out e experiments based on various search algorithms support the theoretical prediction

detecting communities in clustered networks based on group action on set abstract in this paper we propose a well targeted algorithm gas algorithm for detecting communities in high clustered networks by presenting group action technology on community division during the processing of this algorithm the underlying community structure of a clustered network emerges simultaneously as the corresponding partition of orbits by the permutation groups acting on the node set are achieved as the derivation of the orbit partition an algebraic structure r cycle can be considered as the origin of the community to be a priori estimation for the community structure of the algorithm the community separability is introduced to indicate whether a network has distinct community structure by executing the algorithm on several typical networks and the lfr benchmark it shows that this gas algorithm can detect communities accurately and effectively in high clustered networks furthermore we compare the gas algorithm and the clique percolation algorithm on the lfr benchmark it is shown that the gas algorithm is more accurate at detecting non overlapping communities in clustered networks it is suggested that algebraic techniques can uncover fresh light on detecting communities in complex networks

statistical mechanics of community detection starting from a general ansatz we show how community detection can be interpreted as finding the ground state of an infinite range spin glass our approach applies to weighted and directed networks alike it contains the ad hoc introduced quality function from j reichardt and s bornholdt phys rev lett 93 218701 2004 and the modularity q as defined by newman and girvan phys rev e 69 026113 2004 as special cases the community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices we elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study further we show how hierarchies and overlap in the community structure can be detected computationally efficient local update rules for optimization procedures to find the ground state are given we show how the ansatz may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension finally we give expectation values for the modularity of random graphs which can be used in the assessment of statistical significance of community structure

fluctuations between high and low modularity topology in time resolved functional connectivity modularity is an important topological attribute for functional brain networks recent human fmri studies have reported that modularity of functional networks varies not only across individuals being related to demographics and cognitive performance but also within individuals co occurring with fluctuations in network properties of functional connectivity estimated over short time intervals however characteristics of these time resolved functional networks during periods of high and low modularity have remained largely unexplored in this study we investigate basic spatiotemporal properties of time resolved networks in the high and low modularity periods during rest with a particular focus on their spatial connectivity patterns temporal homogeneity and test retest reliability we show that spatial connectivity patterns of time resolved networks in the high and low modularity periods are represented by increased and decreased dissociation of the default mode network module from task positive network modules respectively we also find that the instances of time resolved functional connectivity sampled from within the high respectively low modularity period are relatively homogeneous respectively heterogeneous over time indicating that during the low modularity period the default mode network interacts with other networks in a variable manner we confirmed that the occurrence of the high and low modularity periods varies across individuals with moderate inter session test retest reliability and that it is correlated with previously reported individual differences in the modularity of functional connectivity estimated over longer timescales our findings illustrate how time resolved functional networks are spatiotemporally organized during periods of high and low modularity allowing one to trace individual differences in long timescale modularity to the variable occurrence of network configurations at shorter timescales

combinatorial approach to modularity communities are clusters of nodes with a higher than average density of internal connections their detection is of great relevance to better understand the structure and hierarchies present in a network modularity has become a standard tool in the area of community detection providing at the same time a way to evaluate partitions and by maximizing it a method to find communities in this work we study the modularity from a combinatorial point of view our analysis as the modularity definition relies on the use of the configurational model a technique that given a graph produces a series of randomized copies keeping the degree sequence invariant we develop an approach that enumerates the null model partitions and can be used to calculate the probability distribution function of the modularity our theory allows for a deep inquiry of several interesting features characterizing modularity such as its resolution limit and the statistics of the partitions that maximize it additionally the study of the probability of extremes of the modularity in the random graph partitions opens the way for a definition of the statistical significance of network partitions

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

asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications in this paper we extend our previous work on the stochastic block model a commonly used generative model for social and biological networks and the problem of inferring functional groups or communities from the topology of the network we use the cavity method of statistical physics to obtain an asymptotically exact analysis of the phase diagram we describe in detail properties of the detectability undetectability phase transition and the easy hard phase transition for the community detection problem our analysis translates naturally into a belief propagation algorithm for inferring the group memberships of the nodes in an optimal way i e that maximizes the overlap with the underlying group memberships and learning the underlying parameters of the block model finally we apply the algorithm to two examples of real world networks and discuss its performance

scalable detection of statistically significant communities and hierarchies using message passing for modularity modularity is a popular measure of community structure however maximizing the modularity can lead to many competing partitions with almost the same modularity that are poorly correlated with each other it can also produce illusory communities in random graphs where none exist we address this problem by using the modularity as a hamiltonian at finite temperature and using an efficient belief propagation algorithm to obtain the consensus of many partitions with high modularity rather than looking for a single partition that maximizes it we show analytically and numerically that the proposed algorithm works all of the way down to the detectability transition in networks generated by the stochastic block model it also performs well on real world networks revealing large communities in some networks where previous work has claimed no communities exist finally we show that by applying our algorithm recursively subdividing communities until no statistically significant subcommunities can be found we can detect hierarchical structure in real world networks more efficiently than previous methods

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

minimum spanning trees for community detection abstract a simple deterministic algorithm for community detection is provided by using two rounds of minimum spanning trees by comparing the first round minimum spanning tree 1st mst with the second round spanning tree 2nd mst of the network communities are detected and their overlapping nodes are also identified to generate the two msts a distance matrix is defined and computed from the adjacent matrix of the network compared with the resistance matrix or the communicability matrix used in community detection in the literature the proposed distance matrix is very simple in computation the proposed algorithm is tested on real world social networks graphs which are failed by the modularity maximization and the lfr benchmark graphs for community detection

finding local community structure in networks although the inference of global community structure in networks has recently become a topic of great interest in the physics community all such algorithms require that the graph be completely known here we define both a measure of local community structure and an algorithm that infers the hierarchy of communities that enclose a given vertex by exploring the graph one vertex at a time this algorithm runs in time o k2d for general graphs when d is the mean degree and k is the number of vertices to be explored for graphs where exploring a new vertex is time consuming the running time is linear o k we show that on computer generated graphs the average behavior of this technique approximates that of algorithms that require global knowledge as an application we use this algorithm to extract meaningful local clustering information in the large recommender network of an online retailer

multiway spectral community detection in networks one of the most widely used methods for community detection in networks is the maximization of the quality function known as modularity of the many maximization techniques that have been used in this context some of the most conceptually attractive are the spectral methods which are based on the eigenvectors of the modularity matrix spectral algorithms have however been limited by and large to the division of networks into only two or three communities with divisions into more than three being achieved by repeated two way division here we present a spectral algorithm that can directly divide a network into any number of communities the algorithm makes use of a mapping from modularity maximization to a vector partitioning problem combined with a fast heuristic for vector partitioning we compare the performance of this spectral algorithm with previous approaches and find it to give superior results particularly in cases where community sizes are unbalanced we also give demonstrative applications of the algorithm to two real world networks and find that it produces results in good agreement with expectations for the networks studied

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

network community detection enhancement by proper weighting in this paper we show how proper assignment of weights to the edges of a complex network can enhance the detection of communities and how it can circumvent the resolution limit and the extreme degeneracy problems associated with modularity our general weighting scheme takes advantage of graph theoretic measures and it introduces two heuristics for tuning its parameters we use this weighting as a preprocessing step for the greedy modularity optimization algorithm of newman to improve its performance the result of the experiments of our approach on computer generated and real world data networks confirm that the proposed approach not only mitigates the problems of modularity but also improves the modularity optimization

modularity optimization by conformational space annealing we propose a modularity optimization method mod csa based on stochastic global optimization algorithm conformational space annealing csa our method outperforms simulated annealing in terms of both efficiency and accuracy finding higher modularity partitions with less computational resources required the high modularity values found by our method are higher than or equal to the largest values previously reported in addition the method can be combined with other heuristic methods and implemented in parallel fashion allowing it to be applicable to large graphs with more than 10 000 nodes

when are networks truly modular abstract the study of cluster or community structure of complex networks contributes to the understanding of networks at a functional level in many networks latent classes of nodes are suspected which manifest themselves as communities i e groups of nodes with a high link density among the nodes of the same class and low link density between nodes of different classes community detection algorithms are used to infer these classes e g by finding a partition of the network which maximizes a quality function such as the network modularity q m newman m girvan finding and evaluating community structure in networks phys rev e 69 2004 026113 however it is known from numerical experiments that even purely random networks display intrinsic modularity and may be partitioned yielding high values of q extending on our earlier results j reichardt s bornholdt statistical mechanics of community detection phys rev e 74 2006 016110 the mapping of the community detection problem onto finding the ground state of a spin glass is exploited in order to derive analytical expressions for the expected modularity in random graphs and assess the theoretical limits to community detection the results are independent of any specific community detection algorithm and allow for differentiation between modularity arising purely due to the search process in the large configuration space of possible partitionings on the one hand or due to the actual presence of different classes of nodes on the other hand

normalized modularity optimization method for community identification with degree adjustment as a fundamental problem in network study community identification has attracted much attention from different fields representing a seminal work in this area the modularity optimization method has been widely applied and studied however this method has issues in resolution limit and extreme degeneracy and may not perform well for networks with unbalanced structures although several methods have been proposed to overcome these limitations they are all based on the original idea of defining modularity through comparing the total number of edges within the putative communities in the observed network with that in an equivalent randomly generated network in this paper we show that this modularity definition is not suitable to analyze some networks such as those with unbalanced structures instead we propose to define modularity through the average degree within the communities and formulate modularity as comparing the sum of average degree within communities of the observed network to that of an equivalent randomly generated network in addition we also propose a degree adjusted approach for further improvement when there are unbalanced structures we analyze the theoretical properties of our degree adjusted method numerical experiments for both artificial networks and real networks demonstrate that average degree plays an important role in network community identification and our proposed methods have better performance than existing ones

local resolution limit free potts model for community detection we report on an exceptionally accurate spin glass type potts model for community detection with a simple algorithm we find that our approach is at least as accurate as the best currently available algorithms and robust to the effects of noise it is also competitive with the best currently available algorithms in terms of speed and size of solvable systems we find that the computational demand often exhibits superlinear scaling o l 1 3 where l is the number of edges in the system and we have applied the algorithm to synthetic systems as large as 40 ifmmode times else texttimes fi 10 6 nodes and over 1 ifmmode times else texttimes fi 10 9 edges a previous stumbling block encountered by popular community detection methods is the so called resolution limit being a local measure of community structure our potts model is free from this resolution limit effect and it further remains a local measure on weighted and directed graphs we also address the mitigation of resolution limit effects for two other popular potts models

ganc greedy agglomerative normalized cut for graph clustering this paper describes a graph clustering algorithm that aims to minimize the normalized cut criterion and has a model order selection procedure the performance of the proposed algorithm is comparable to spectral approaches in terms of minimizing normalized cut however unlike spectral approaches the proposed algorithm scales to graphs with millions of nodes and edges the algorithm consists of three components that are processed sequentially a greedy agglomerative hierarchical clustering procedure model order selection and a local refinement for a graph of n nodes and o n edges the computational complexity of the algorithm is o nlog 2n a major improvement over the o n 3 complexity of spectral methods experiments are performed on real and synthetic networks to demonstrate the scalability of the proposed approach the effectiveness of the model order selection procedure and the performance of the proposed algorithm in terms of minimizing the normalized cut metric

hiv dynamics modeling data analysis and optimal treatment protocols we present an overview of some concepts and methodologies we believe useful in modeling hiv pathogenesis after a brief discussion of motivation for and previous efforts in the development of mathematical models for progression of hiv infection and treatment we discuss mathematical and statistical ideas relevant to structured treatment interruptions sti among these are model development and validation procedures including parameter estimation data reduction and representation and optimal control relative to sti results from initial attempts in each of these areas by an interdisciplinary team of applied mathematicians statisticians and clinicians are presented

overlapping community detection in complex networks using symmetric binary matrix factorization discovering overlapping community structures is a crucial step to understanding the structure and dynamics of many networks in this paper we develop a symmetric binary matrix factorization model to identify overlapping communities our model allows us not only to assign community memberships explicitly to nodes but also to distinguish outliers from overlapping nodes in addition we propose a modified partition density to evaluate the quality of community structures we use this to determine the most appropriate number of communities we evaluate our methods using both synthetic benchmarks and real world networks demonstrating the effectiveness of our approach

weight conserving characterization of complex functional brain networks complex functional brain networks are large networks of brain regions and functional brain connections statistical characterizations of these networks aim to quantify global and local properties of brain activity with a small number of network measures important functional network measures include measures of modularity measures of the goodness with which a network is optimally partitioned into functional subgroups and measures of centrality measures of the functional influence of individual brain regions characterizations of functional networks are increasing in popularity but are associated with several important methodological problems these problems include the inability to characterize densely connected and weighted functional networks the neglect of degenerate topologically distinct high modularity partitions of these networks and the absence of a network null model for testing hypotheses of association between observed nontrivial network properties and simple weighted connectivity properties in this study we describe a set of methods to overcome these problems specifically we generalize measures of modularity and centrality to fully connected and weighted complex networks describe the detection of degenerate high modularity partitions of these networks and introduce a weighted connectivity null model of these networks we illustrate our methods by demonstrating degenerate high modularity partitions and strong correlations between two complementary measures of centrality in resting state functional magnetic resonance imaging mri networks from the 1000 functional connectomes project an open access repository of resting state functional mri datasets our methods may allow more sound and reliable characterizations and comparisons of functional brain networks across conditions and subjects

finding community structure in very large networks the discovery and analysis of community structure in networks is a topic of considerable recent interest within the physics community but most methods proposed so far are unsuitable for very large networks because of their computational cost here we present a hierarchical agglomeration algorithm for detecting community structure which is faster than many competing algorithms its running time on a network with n vertices and m edges is o md phantom rule 0 2em 0ex mathrm log phantom rule 0 2em 0ex n where d is the depth of the dendrogram describing the community structure many real world networks are sparse and hierarchical with m ensuremath sim n and d ensuremath sim mathrm log phantom rule 0 2em 0ex n in which case our algorithm runs in essentially linear time o n phantom rule 0 2em 0ex mathrm log 2 phantom rule 0 2em 0ex n as an example of the application of this algorithm we use it to analyze a network of items for sale on the web site of a large on line retailer items in the network being linked if they are frequently purchased by the same buyer the network has more than 400 000 vertices and 2 ifmmode times else texttimes fi 10 6 edges we show that our algorithm can extract meaningful communities from this network revealing large scale patterns present in the purchasing habits of customers

fast super paramagnetic clustering abstract we map stock market interactions to spin models to recover their hierarchical structure using a simulated annealing based super paramagnetic clustering spc algorithm this is directly compared to a modified implementation of a maximum likelihood approach we call fast super paramagnetic clustering f spc the methods are first applied to standard toy test case problems and then to a data set of 447 stocks traded on the new york stock exchange nyse over 1249 days the signal to noise ratio of stock market correlation matrices is briefly considered our result recover approximately clusters representative of standard economic sectors and mixed ones whose dynamics shine light on the adaptive nature of financial markets and raise concerns relating to the effectiveness of industry based static financial market classification in the world of real time data analytics a key result is that we show that f spc maximum likelihood solutions converge to ones found within the super paramagnetic phase where the entropy is maximum and those solutions are qualitatively better for high dimensionality data sets

modularity density of network community divisions the problem of dividing a network into communities is extremely complex and grows very rapidly with the number of nodes and edges that are involved in order to develop good algorithms to identify optimal community divisions it is extremely beneficial to identify properties that are similar for most networks we introduce the concept of modularity density the distribution of modularity values as a function of the number of communities and find strong indications that the general features of this modularity density are quite similar for different networks the region of high modularity generally has very low probability density and occurs where the number of communities is small the properties and shape of the modularity density may give valuable information and aid in the search for efficient algorithms to find community divisions with high modularities

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

robustness of community structure in networks the discovery of community structure is a common challenge in the analysis of network data many methods have been proposed for finding community structure but few have been proposed for determining whether the structure found is statistically significant or whether conversely it could have arisen purely as a result of chance in this paper we show that the significance of community structure can be effectively quantified by measuring its robustness to small perturbations in network structure we propose a suitable method for perturbing networks and a measure of the resulting change in community structure and use them to assess the significance of community structure in a variety of networks both real and computer generated

axioms for graph clustering quality functions we investigate properties that intuitively ought to be satisfied by graph clustering quality functions that is functions that assign a score to a clustering of a graph graph clustering also known as network community detection is often performed by optimizing such a function two axioms tailored for graph clustering quality functions are introduced and the four axioms introduced in previous work on distance based clustering are reformulated and generalized for the graph setting we show that modularity a standard quality function for graph clustering does not satisfy all of these six properties this motivates the derivation of a new family of quality functions adaptive scale modularity which does satisfy the proposed axioms adaptive scale modularity has two parameters which give greater flexibility in the kinds of clusterings that can be found standard graph clustering quality functions such as normalized cut and unnormalized cut are obtained as special cases of adaptive scale modularity in general the results of our investigation indicate that the considered axiomatic framework covers existing good quality functions for graph clustering and can be used to derive an interesting new family of quality functions

asymptotic properties of receding horizon optimal control problems the asymptotic behavior of receding horizon optimal control problems with terminal cost chosen as a control liapunov function is analyzed for regulator as well as disturbance attenuation problems both the continuous as well as the discrete time cases are treated further the approximation of the continuous time optimal control problem by the discrete time receding horizon problems is studied

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

efficient and principled method for detecting communities in networks a fundamental problem in the analysis of network data is the detection of network communities groups of densely interconnected nodes which may be overlapping or disjoint here we describe a method for finding overlapping communities based on a principled statistical approach using generative network models we show how the method can be implemented using a fast closed form expectation maximization algorithm that allows us to analyze networks of millions of nodes in reasonable running times we test the method both on real world networks and on synthetic benchmarks and find that it gives results competitive with previous methods we also show that the same approach can be used to extract nonoverlapping community divisions via a relaxation method and demonstrate that the algorithm is competitively fast and accurate for the nonoverlapping problem

adaptive modularity maximization via edge weighting scheme modularity maximization is one of the state of the art methods for community detection that has gained popularity in the last decade yet it suffers from the resolution limit problem by preferring under certain conditions large communities over small ones to solve this problem we propose to expand the meaning of the edges that are currently used to indicate propensity of nodes for sharing the same community in our approach this is the role of edges with positive weights while edges with negative weights indicate aversion for putting their end nodes into one community we also present a novel regression model which assigns weights to the edges of a graph according to their local topological features to enhance the accuracy of modularity maximization algorithms we construct artificial graphs based on the parameters sampled from a given unweighted network and train the regression model on ground truth communities of these artificial graphs in a supervised fashion the extraction of local topological edge features can be done in linear time making this process efficient experimental results on real and synthetic networks show that the state of the art community detection algorithms improve their performance significantly by finding communities in the weighted graphs produced by our model

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

dynamic fluctuations coincide with periods of high and low modularity in resting state functional brain networks we investigate the relationship of resting state fmri functional connectivity estimated over long periods of time with time varying functional connectivity estimated over shorter time intervals we show that using pearson s correlation to estimate functional connectivity implies that the range of fluctuations of functional connections over short time scales is subject to statistical constraints imposed by their connectivity strength over longer scales we present a method for estimating time varying functional connectivity that is designed to mitigate this issue and allows us to identify episodes where functional connections are unexpectedly strong or weak we apply this method to data recorded from n 80 participants and show that the number of unexpectedly strong weak connections fluctuates over time and that these variations coincide with intermittent periods of high and low modularity in time varying functional connectivity we also find that during periods of relative quiescence regions associated with default mode network tend to join communities with attentional control and primary sensory systems in contrast during periods where many connections are unexpectedly strong weak default mode regions dissociate and form distinct modules finally we go on to show that while all functional connections can at times manifest stronger more positively correlated or weaker more negatively correlated than expected a small number of connections mostly within the visual and somatomotor networks do so a disproportional number of times our statistical approach allows the detection of functional connections that fluctuate more or less than expected based on their long time averages and may be of use in future studies characterizing the spatio temporal patterns of time varying functional connectivity c 2015 the authors published by elsevier inc

information theoretic approach to network modularity exploiting recent developments in information theory we propose illustrate and validate a principled information theoretic algorithm for module discovery and the resulting measure of network modularity this measure is an order parameter a dimensionless number between 0 and 1 comparison is made with other approaches to module discovery and to quantifying network modularity using monte carlo generated erd os like modular networks finally the network information bottleneck nib algorithm is applied to a number of real world networks including the social network of coauthors at the 2004 aps march meeting

feedback control of the immune response of renal transplant recipients with inequality constraints this paper describes a model of the immunologic response of latent viruses and a donor kidney in a renal transplant recipient an optimal control problem with state variable inequality constraints is considered to maintain the balance between over suppression where latent viruses are reactivated and under suppression where the transplanted kidney is rejected a feedback methodology based on the model predictive control mpc method is proposed to design sub optimal treatment regimes in addition the problem of implementing the mpc methodology and nonlinear kalman filter with inaccurate or incomplete observation data and long measurement periods is addressed the results of numerical simulations show that a sub optimal dynamic immunosuppression therapy method can help strike a balance between the over suppression and under suppression of the immune system

learning simultaneous adaptive clustering and classification via moea clustering learning and classification learning are two major tasks in pattern recognition the traditional hybrid clustering and classification algorithms handle them in a sequential way rather than a simultaneous way fortunately multiobjective optimization provides a way to solve this problem in this paper an algorithm that learns simultaneous clustering and classification adaptively via multiobjective evolutionary algorithm is proposed the main idea of this paper is to optimize two objective functions which represent fuzzy cluster connectedness and classification error rate to achieve the goal of simultaneous learning firstly we adopt a graph based representation scheme to encode so that it can generate a set of solutions with different number of clusters in a single run then the relationship between clustering and classification is built via the bayesian theory during the optimization process the quality of clustering and classification is measured by the objective functions and the feedback drawn from both aspects is used to guide the mutation at last a set of nondominated solutions are generated from which the final pareto optimal solution is selected by using adjusted rand index the results on synthetic datasets and real life datasets demonstrate the rationality and effectiveness of the proposed algorithm furthermore we apply the proposed algorithm to image segmentation including texture images and synthetic aperture radar images the experimental results show the superiority of the proposed algorithm compared with other five algorithms highlightsa simultaneous adaptive clustering and classification learning via moea is proposed new clustering objective function is designed and two objective functions complement with each other the number of clusters can be determined adaptively during the learning process the drawback from clustering classification learning is used to guide the search we extend it to texture image and sar image segmentation to show its effectiveness

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

modular structure of functional networks in olfactory memory graph theory enables the study of systems by describing those systems as a set of nodes and edges graph theory has been widely applied to characterize the overall structure of data sets in the social technological and biological sciences including neuroscience modular structure decomposition enables the definition of sub networks whose components are gathered in the same module and work together closely while working weakly with components from other modules this processing is of interest for studying memory a cognitive process that is widely distributed we propose a new method to identify modular structure in task related functional magnetic resonance imaging fmri networks the modular structure was obtained directly from correlation coefficients and thus retained information about both signs and weights the method was applied to functional data acquired during a yes no odor recognition memory task performed by young and elderly adults four response categories were explored correct hit and incorrect false alarm fa recognition and correct and incorrect rejection we extracted time series data for 36 areas as a function of response categories and age groups and calculated condition based weighted correlation matrices overall condition based modular partitions were more homogeneous in young than elderly subjects using partition similarity based statistics and a posteriori statistical analyses we demonstrated that several areas including the hippocampus caudate nucleus and anterior cingulate gyrus belonged to the same module more frequently during hit than during all other conditions modularity values were negatively correlated with memory scores in the hit condition and positively correlated with bias scores liberal conservative attitude in the hit and fa conditions we further demonstrated that the proportion of positive and negative links between areas of different modules i e the proportion of correlated and anti correlated areas accounted for most of the observed differences in signed modularity taken together our results provided some evidence that the neural networks involved in odor recognition memory are organized into modules and that these modular partitions are linked to behavioral performance and individual strategies

a simple min cut algorithm we present an algorithm for finding the minimum cut of an undirected edge weighted graph it is simple in every respect it has a short and compact description is easy to implement and has a surprisingly simple proof of correctness its runtime matches that of the fastest algorithm known the runtime analysis is straightforward in contrast to nearly all approaches so far the algorithm uses no flow techniques roughly speaking the algorithm consists of about v nearly identical phases each of which is a maximum adjacency search

surveying network community structure in the hidden metric space abstract most real world networks from various fields share a universal topological property as community structure in this paper we propose a node similarity based mechanism to explore the formation of modular networks by applying the concept of hidden metric spaces of complex networks it is demonstrated that network community structure could be formed according to node similarity in the underlying hidden metric space to clarify this we generate a set of observed networks using a typical kind of hidden metric space model by detecting and analyzing corresponding communities both in the observed network and the hidden space we show that the values of the fitness are rather close and the assignments of nodes for these two kinds of community structures detected based on the fitness parameter are extremely matching ones furthermore our research also shows that networks with strong clustering tend to display prominent community structures with large values of network modularity and fitness

nonparametric bayesian inference of the microcanonical stochastic block model a principled approach to characterize the hidden structure of networks is to formulate generative models and then infer their parameters from data when the desired structure is composed of modules or communities a suitable choice for this task is the stochastic block model sbm where nodes are divided into groups and the placement of edges is conditioned on the group memberships here we present a nonparametric bayesian method to infer the modular structure of empirical networks including the number of modules and their hierarchical organization we focus on a microcanonical variant of the sbm where the structure is imposed via hard constraints i e the generated networks are not allowed to violate the patterns imposed by the model we show how this simple model variation allows simultaneously for two important improvements over more traditional inference approaches 1 deeper bayesian hierarchies with noninformative priors replaced by sequences of priors and hyperpriors which not only remove limitations that seriously degrade the inference on large networks but also reveal structures at multiple scales 2 a very efficient inference algorithm that scales well not only for networks with a large number of nodes and edges but also with an unlimited number of modules we show also how this approach can be used to sample modular hierarchies from the posterior distribution as well as to perform model selection we discuss and analyze the differences between sampling from the posterior and simply finding the single parameter estimate that maximizes it furthermore we expose a direct equivalence between our microcanonical approach and alternative derivations based on the canonical sbm