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

bounding option prices by semidefinite programming a cutting plane algorithm in a recent article bertsimas and popescu showed that a tight upper bound on a europeantype call option price given the first n moments of the distribution of the underlying security price can be obtained by solving an associated semidefinite programming problem sdp the purpose of this paper is to improve and extend their results we will show that a tight lower bound can be calculated by solving another sdp also we will show that these problems can be solved very quickly by a newly developed cutting plane algorithm when n is less than six or seven

on the existence of supervisory policies that enforce liveness in discrete event dynamic systems modeled by controlled petri nets we consider discrete state plants represented by controlled petri nets ctlpns where a subset of transitions can be prevented from firing by a supervisor a transition in a ctlpn can fire at a marking if there are sufficient tokens in its input places and it is permitted to fire by the supervisor a ctlpn is live if it is possible to fire any transition from every marking that is reachable under supervision in this paper we derive a necessary and sufficient condition for the existence of a supervisory policy that enforces liveness in ctlpns we show this condition cannot be tested for an arbitrary ctlpn however for bounded ctlpns or ctlpns where each transition is individually controllable we show the existence of a supervisory policy which enforces that liveness is decidable we also show the existence of a supervisory policy that enforces liveness is necessary and sufficient for the existence of a minimally restrictive supervisory policy

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

multiresolution community detection for megascale networks by information based replica correlations we use a potts model community detection algorithm to accurately and quantitatively evaluate the hierarchical or multiresolution structure of a graph our multiresolution algorithm calculates correlations among multiple copies replicas of the same graph over a range of resolutions significant multiresolution structures are identified by strongly correlated replicas the average normalized mutual information the variation in information and other measures in principle give a quantitative estimate of the best resolutions and indicate the relative strength of the structures in the graph because the method is based on information comparisons it can in principle be used with any community detection model that can examine multiple resolutions our approach may be extended to other optimization problems as a local measure our potts model avoids the resolution limit that affects other popular models with this model our community detection algorithm has an accuracy that ranks among the best of currently available methods using it we can examine graphs over 40 ifmmode times else texttimes fi 10 6 nodes and more than 1 ifmmode times else texttimes fi 10 9 edges we further report that the multiresolution variant of our algorithm can solve systems of at least 200 text 000 nodes and 10 ifmmode times else texttimes fi 10 6 edges on a single processor with exceptionally high accuracy for typical cases we find a superlinear scaling o l 1 3 for community detection and o l 1 3 text text log text n for the multiresolution algorithm where l is the number of edges and n is the number of nodes in the system

sequential algorithm for fast clique percolation in complex network research clique percolation introduced by palla der enyi and vicsek nature london 435 814 2005 is a deterministic community detection method which allows for overlapping communities and is purely based on local topological properties of a network here we present a sequential clique percolation algorithm scp to do fast community detection in weighted and unweighted networks for cliques of a chosen size this method is based on sequentially inserting the constituent links to the network and simultaneously keeping track of the emerging community structure unlike existing algorithms the scp method allows for detecting k clique communities at multiple weight thresholds in a single run and can simultaneously produce a dendrogram representation of hierarchical community structure in sparse weighted networks the scp algorithm can also be used for implementing the weighted clique percolation method recently introduced by farkas et al new j phys 9 180 2007 the computational time of the scp algorithm scales linearly with the number of k cliques in the network as an example the method is applied to a product association network revealing its nested community structure

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

overlapping decompositions expansions contractions and stability of hybrid systems inclusion and extension principles are defined and studied for hybrid systems the controller design problem for hybrid systems is then considered within the framework of the extension principle it is shown that if the extension principle is used then any controller designed in the expanded spaces is contractible to the original spaces for implementation the stability of hybrid systems is also considered a number of definitions of stability for hybrid systems are given and it is shown that if a system which includes another system is stable then the included system is also stable furthermore it is shown that if a controller designed for the expanded system achieves stability then the contracted controller also achieves stability for the original system decentralized controller design with overlapping decompositions is also discussed within the framework of extension an example is provided to illustrate the proposed approach

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

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

local modularity measure for network clusterizations many complex networks have an underlying modular structure i e structural subunits communities or clusters characterized by highly interconnected nodes the modularity q has been introduced as a measure to assess the quality of clusterizations q has a global view while in many real world networks clusters are linked mainly locally among each other local cluster connectivity here we introduce a measure of localized modularity lq which reflects local cluster structure optimization of q and lq on the clusterization of two biological networks shows that the localized modularity identifies more cohesive clusters yielding a complementary view of higher granularity

community detection in complex networks by dynamical simplex evolution we benchmark the dynamical simplex evolution dse method with several of the currently available algorithms to detect communities in complex networks by comparing correctly identified nodes for different levels of fuzziness of random networks composed of well defined communities the potential benefits of the dse method to detect hierarchical substructures in complex networks are discussed

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

pair wise decomposition and coordinated control of complex systems pair wise decomposition and coordinated control for complex interconnected systems with any information structure constraints are proposed in this paper based on the inclusion principle a permuted inclusion principle is presented to expand the system into an expanded space from which a recurrent reverse order of pair wise subsystems is extracted by a standard lq control pair wise controllers can be designed to stabilize and coordinate the pair wise subsystems according to these controllers a coordinated compensator of subsystem interconnections is introduced in order to obtain a coordination based controller for the expanded system therefore a coordinated controller of the system can be contracted from the expanded space under the conditions of the inclusion principle moreover a sufficient condition of connective stability for the system is also proved by the lyapunov function and m matrix a main advantage of the proposed decomposition and control is that it allows any variation of information structure constraints of the system especially the increase and decrease of subsystem numbers finally the effectiveness of the proposed methodology is demonstrated by an application of automatic generation control to a four area electric power system

optimal inequalities in probability theory a convex optimization approach we propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for p x in s for a set s defined by polynomial inequalities and a random vector x defined on omega subseteq cal r n that has a given collection of up to kth order moments in the univariate case we provide optimal bounds on p x in s when the first k moments of x are given as the solution of a semidefinite optimization problem in k 1 dimensions in the multivariate case if the sets s and omega are given by polynomial inequalities we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n for fixed k we characterize the complexity of the problem of deriving tight moment inequalities we show that it is np hard to find tight bounds for k geq 4 and omega cal r n and for k geq 2 and omega cal r n when the data in the problem is rational for k 1 and omega cal r n we show that we can find tight upper bounds by solving n convex optimization problems when the set s is convex and we provide a polynomial time algorithm when s and omega are unions of convex sets over which linear functions can be optimized efficiently for the case k 2 and omega cal r n we present an efficient algorithm for finding tight bounds when s is a union of convex sets over which convex quadratic functions can be optimized efficiently

superreplication of financial derivatives via convex programming we give a method based on convex programming to calculate the optimal superreplicating and subreplicating prices and corresponding hedging portfolios of a financial derivative in terms of other financial derivatives in a discrete time setting our method produces a model that matches the superreplicating or subreplicating price within an arbitrary precision and is consistent with the other financial derivatives prices applications include robust replication in terms of call prices with various strikes and maturities of forward start options volatility and variance swaps and derivatives cliquet calls barrier options and lookback and asian options numerical examples show that in some cases the best superreplicating and or subreplicating prices are within 10 of the price obtained by a standard model but considerably differ from it in other cases our method can incorporate bid ask spreads interest rates and dividends and various limitations to the diffusion model the supplementary material is a

a cutting plane algorithm for semi definite programming problems with applications to failure discriminant analysis we will propose a new cutting plane algorithm for solving a class of semi definite programming problems sdp with a small number of variables and a large number of constraints problems of this type appear when we try to classify a large number of multi dimensional data into two groups by a hyper ellipsoidal surface among such examples are cancer diagnosis failure discrimination of enterprises also a certain class of option pricing problems can be formulated as this type of problem we will show that the cutting plane algorithm is much more efficient than the standard interior point algorithms for solving sdp

community detection as an inference problem we express community detection as an inference problem of determining the most likely arrangement of communities we then apply belief propagation and mean field theory to this problem and show that this leads to fast accurate algorithms for community detection

overlapping decompositions and expansions of petri nets overlapping decompositions and expansions are considered to design decentralized controllers for discrete event systems dess modeled by petri nets the inclusion principle for petri nets is first defined it is shown that properties like boundedness reversibility and liveness with a mild additional condition carry over from the including net to the included net moreover a new property called obstruction is introduced for the including net and it is shown that if obstruction does not occur in the including net then deadlock does not occur in the included net an expansion procedure which guarantees inclusion for an overlappingly decomposed petri net is then introduced

detecting complex network modularity by dynamical clustering based on cluster desynchronization properties of phase oscillators we introduce an efficient method for the detection and identification of modules in complex networks the performance of the algorithm is tested on computer generated and real world networks whose modular structure is already known or has been studied by means of other methods the algorithm attains a high level of precision especially when the modular units are very mixed and hardly detectable by the other methods with a computational effort o kn on a generic graph with n nodes and k links