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

oscillations of first order neutral delay differential equations abstract consider the neutral delay differential equation d dt y t py t qy t 0 t t 0 where q and are positive constants while p 1 0 for the case p 1 0 see ladas and sficas oscillations of neutral delay differential equations to appear the following results are then proved theorem 1 assume p and q 1 p 1 e then every solution of eq oscillates theorems 3 assume p 0 then every nonoscillatory solution y t of eq tends to zero as t theorem 4 assume p 0 then a necessary condition for all solutions of eq to oscillate is that theorem 5 assume p 0 and q 1 p 1 e then every solution of eq oscillates extensions of these results to equations with variable coefficients are also obtained

step by step community detection in volume regular graphs abstract spectral techniques have proved amongst the most effective approaches to graph clustering however in general they require explicit computation of the main eigenvectors of a suitable matrix usually the laplacian matrix of the graph recent work e g becchetti et al soda 2017 suggests that observing the temporal evolution of the power method applied to an initial random vector may at least in some cases provide enough information on the space spanned by the first two eigenvectors so as to allow recovery of a hidden partition without explicit eigenvector computations while the results of becchetti et al apply to perfectly balanced partitions and or graphs that exhibit very strong forms of regularity we extend their approach to graphs containing a hidden k partition and characterized by a milder form of volume regularity we show that the class of k volume regular graphs is the largest class of undirected possibly weighted graphs whose transition matrix admits k stepwise eigenvectors i e vectors that have constant entries over the components corresponding to the same set of the hidden partition to obtain this result we highlight a connection between volume regularity and lumpability of markov chains moreover we prove that if the stepwise eigenvectors are those associated to the first k largest eigenvalues of the transition matrix of a random walk on the graph and the gap between the k th and the k 1 th eigenvalues is sufficiently large the averaging dynamics of becchetti et al recovers the underlying community structure of the graph in logarithmic time with high probability

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

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

an efficient community detection algorithm using greedy surprise maximization community detection is an important and crucial problem in complex network analysis although classical modularity function optimization approaches are widely used for identifying communities the modularity function q suffers from its resolution limit recently the surprise function s was experimentally proved to be better than the q function however up until now there has been no algorithm available to perform searches to directly determine the maximal surprise values in this paper considering the superiority of the s function over the q function we propose an efficient community detection algorithm called agso algorithm based on greedy surprise optimization and its improved version fagso fast agso which are based on greedy surprise optimization and do not suffer from the resolution limit in addition f agso does not need the number of communities k to be specified in advance tests on experimental networks show that f agso is able to detect optimal partitions in both simple and even more complex networks moreover algorithms based on surprise maximization perform better than those algorithms based on modularity maximization including blondel guillaume lambiotte lefebvre bgll clauset newman moore cnm and the other state of the art algorithms such as infomap order statistics local optimization method oslom and label propagation algorithm lpa

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

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

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

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

a semigroup associated to a linear control system on a lie group abstract let us consider a linear control system on a connected lie group g it is known that the accessibility set a from the identity e is in general not a semigroup in this article we associate a new algebraic object s to which turns out to be a semigroup allowing the use of the semigroup machinery to approach in particular we obtain some controllability results

detection of community structures in networks via global optimization abstract we present an analysis of communality structure in networks based on the application of simulated annealing techniques in this case we use as cost function the already introduced modularity q 1 which is based on the relative number of links within a commune against the number of links that would correspond in case the links were distributed randomly we compare the results of our approach against other methodologies based on betweenness analysis and show that in all cases a better community structure can be attained

oscillation and nonoscillation in first order neutral differential equations abstract verifiable sufficient conditions are obtained respectively for the oscillation and nonoscillation of the neutral differential equation x t cx t px t 0

spectral bounds and oscillations of a differential difference system of neutral type abstract this paper is concerned with the oscillatory behavior of a differential difference system of neutral type spectral bounds are used in order to obtain sufficient conditions such that all their solutions are oscillatory

controllability of linear systems on solvable lie groups linear systems on lie groups are a natural generalization of linear systems on euclidian spaces for such systems this paper studies controllability by taking into consideration the eigenvalues of an associated derivation mathcal d when the state space is a solvable connected lie group controllability of the system is guaranteed if the reachable set of the neutral element is open and the derivation mathcal d has only pure imaginary eigenvalues for bounded systems on nilpotent lie groups such conditions are also necessary

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

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

oscillation of first order neutral delay differential equations new necessary and sufficient conditions and verifiable sufficient conditions are obtained for the oscillation of the neutral delay differential equations these sufficient conditions include and are in many cases weaker than those known

fast community detection based on sector edge aggregation metric model in hyperbolic space by studying the edge aggregation characteristic of nodes in hyperbolic space sector edge aggregation metric seam model is proposed and theoretically proved in this paper in hyperbolic disk seam model determines the minimum angular range of a sector which possesses the maximal edge aggregation of nodes the set of nodes within such sector has dense internal links which corresponds with the characteristic of community structure based on seam model we propose a fast community detection algorithm called greedy optimization modularity algorithm goma which employs greedy optimization strategy and hyperbolic coordinates goma firstly divides initial communities according to the quantitative results of sector edge aggregation given by seam and the nodes hyperbolic coordinates then based on greedy optimization strategy only merges the two angular neighboring communities in hyperbolic disk to optimize the network modularity function and consequently obtains high quality community detection the strategies of initial community partition and merger in hyperbolic space greatly improve the speed of searching the most optimal modularity experimental results indicate that goma is able to detect out high quality community structure in synthetic and real networks and performs better when applied to the large scale and dense networks with strong clustering

oscillation of first order neutral functional differential equations abstract some new sufficient conditions are obtained for oscillation of a kind of neutral differential equations these conditions improve some recent results

efficient modularity optimization by multistep greedy algorithm and vertex mover refinement identifying strongly connected substructures in large networks provides insight into their coarse grained organization several approaches based on the optimization of a quality function e g the modularity have been proposed we present here a multistep extension of the greedy algorithm msg that allows the merging of more than one pair of communities at each iteration step the essential idea is to prevent the premature condensation into few large communities upon convergence of the msg a simple refinement procedure called vertex mover vm is used for reassigning vertices to neighboring communities to improve the final modularity value with an appropriate choice of the step width the combined msg vm algorithm is able to find solutions of higher modularity than those reported previously the multistep extension does not alter the scaling of computational cost of the greedy algorithm

a review on some classes of algebraic systems in this paper we review some algebraic control system precisely linear and bilinear systems on euclidean spaces and invariant and linear systems on lie groups the fourth classes of systems have

comment on evolutionary method for finding communities in bipartite networks in a recent paper zhan zhang guan and zhou phys rev e 83 066120 2011 presented a modified adaptive genetic algorithm maga tailored to the discovery of maximum modularity partitions of the node set into communities in unipartite bipartite and directed networks the authors claim that detection of communities in unipartite networks or in directed networks can be transformed into the same task in bipartite networks actually some tests show that it is not the case for the proposed transformations and why experimental results of maga for modularity maximization of untransformed unipartite or bipartite networks are also discussed