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

enhanced group classification of nonlinear diffusion reaction equations with gradient dependent diffusivity abstract we carry out the enhanced group classification of a class of 1 1 dimensional nonlinear diffusion reaction equations with gradient dependent diffusivity using the two step version of the method of furcate splitting for simultaneously finding the equivalence groups of an unnormalized class of differential equations and a collection of its subclasses we suggest an optimized version of the direct method the optimization includes the preliminary study of admissible transformations within the entire class and the successive splitting of the corresponding determining equations with respect to arbitrary elements and their derivatives depending on auxiliary constraints associated with each of required subclasses in the course of applying the suggested technique to subclasses of the class under consideration we construct for the first time a nontrivial example of finite dimensional effective generalized equivalence group using the method of lie reduction and the generalized separation of variables exact solutions of some equations under consideration are found

mutations of the cluster algebra of type a _ 1 1 and the periodic discrete toda lattice a direct connection between two sequences of points one of which is generated by seed mutations of the cluster algebra of type and the other by time evolutions of the periodic discrete toda lattice is explicitly given in this construction each of them is realized as an orbit of a qrt map and specialization of the parameters in the maps and appropriate choices of the initial points relate them the connection with the periodic discrete toda lattice enables us a geometric interpretation of the seed mutations of the cluster algebra of type as an addition of points on an elliptic curve

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

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

extended symmetry analysis of an isothermal no slip drift flux model abstract we perform extended group analysis for a system of differential equations modeling an isothermal no slip drift flux the maximal lie invariance algebra of this system is proved to be infinite dimensional we also find the complete point symmetry group of this system including discrete symmetries using the megaideal based version of the algebraic method optimal lists of one and two dimensional subalgebras of the maximal lie invariance algebra in question are constructed and employed for obtaining reductions of the system under study since this system contains a subsystem of two equations that involves only two of three dependent variables we also perform group analysis of this subsystem the latter can be linearized by a composition of a fiber preserving point transformation with a two dimensional hodograph transformation to the klein gordon equation we also employ both the linearization and the generalized hodograph method for constructing the general solution of the entire system under study we find inter alia genuinely generalized symmetries for this system and present the connection between them and the lie symmetries of the subsystem we mentioned earlier hydrodynamic conservation laws and their generalizations are also constructed

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

differential algebraic and bi hamiltonian integrability analysis of the riemann hierarchy revisited a differential algebraic approach to studying the lax integrability of the generalized riemann type hydrodynamic hierarchy is revisited and its new lax representation is constructed in exact form the bi hamiltonian integrability of the generalized riemann type hierarchy is discussed by means of the gradient holonomic and symplectic methods and the related compatible poissonian structures for n 3 and n 4 are constructed

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

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

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

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

integrable mappings via rational elliptic surfaces we present a geometric description of the qrt map which is an integrable mapping introduced by quispel roberts and thompson in terms of the addition formula of a rational elliptic surface by this formulation we classify all the cases when the qrt map is periodic and show that its period is 2 3 4 5 or 6 a generalization of the qrt map which acts birationally on a pencil of k3 surfaces or calabi yau manifolds is also presented

hierarchy of conservation laws of diffusion convection equations we introduce notions of equivalence of conservation laws with respect to lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems we also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials to construct conservation laws we develop and apply the most direct method which is effective to use in the case of two independent variables admitting possibility of dependence of conserved vectors on a number of potentials we generalize the iteration procedure proposed by bluman and doran wu for finding nonlocal potential conservation laws as an example we completely classify potential conservation laws including arbitrary order local ones of diffusion convection equations with respect to the equivalence group and construct an exhaustive list of locally inequivalent potential systems corresponding to these equations

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

solutions of the generalized periodic discrete toda equation a box ball system with more than one kind of ball is obtained by the generalized periodic discrete toda equation pd toda equation we present an algebraic geometric study of the periodic toda equation the time evolution of the pd toda equation is linearized on the picard group of an algebraic variety and theta function solutions are obtained

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

inverse problem on conservation laws abstract the explicit formulation of the general inverse problem on conservation laws is presented for the first time within this problem one aims to derive the general form of systems of differential equations that admit a prescribed set of conservation laws the particular cases of the inverse problem on first integrals of ordinary differential equations and on conservation laws for evolution equations are studied we also solve the inverse problem on conservation laws for differential equations admitting an infinite dimensional space of zeroth order conservation law characteristics this particular case is further studied in the context of conservative first order parameterization schemes for the two dimensional incompressible euler equations we exhaustively classify conservative first order parameterization schemes for the eddy vorticity flux that lead to a class of closed averaged euler equations possessing conservation of generalized circulation generalized momentum and energy

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

a geometric realization of the periodic discrete toda lattice and its tropicalization an explicit formula concerning curve intersections equivalent to the time evolution of the periodic discrete toda lattice pdtl is presented first the time evolution is realized as a point addition on a hyperelliptic curve which is the spectral curve of the pdtl then the point addition is translated into curve intersections next it is shown that the curves which appear in the curve intersections are explicitly given by using the conserved quantities of the pdtl finally the formulation is lifted to the framework of tropical geometry and a tropical geometric realization of the periodic box ball system is constructed via tropical curve intersections

a new conservation theorem a general theorem on conservation laws for arbitrary differential equations is proved the theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables the new theorem does not require existence of a lagrangian and is based on a concept of an adjoint equation for non linear equations suggested recently by the author it is proved that the adjoint equation inherits all symmetries of the original equation accordingly one can associate a conservation law with any group of lie lie backlund or non local symmetries and find conservation laws for differential equations without classical lagrangians

on symmetry analysis of differential systems on functional manifolds abstract in this short letter there is successively sketched a classical e noether based analytical scheme of constructing the complete set of global conservation laws to a smooth vector field on a functional manifold there is also described an analytical algorithm of constructing the complete set of differential conservation laws to a general smooth differential functional system on a functional manifold with values in an arbitrary vector bundle it is demonstrated that the algorithm reduces as a special case to that presented recently by c s anco

dirac constraints in field theory lifts of hamiltonian systems to the cotangent bundle to every hamiltonian system is associated a degenerate lagrangian formulation following dirac s theory of constraints one finds a family of lifts of the system to the cotangent bundle of the underlying manifold the lifted system admits a reduction to the original equation such that the original hamiltonian formulation is the pullback of the canonical symplectic form invariants of the original equation can be lifted to invariants of the extended system this procedure is applied to integrable systems such as the korteweg de vries equation

partitioning and modularity of graphs with arbitrary degree distribution we solve the graph bipartitioning problem in dense graphs with arbitrary degree distribution using the replica method we find the cut size to scale universally with k in contrast earlier results studying the problem in graphs with a poissonian degree distribution had found a scaling with kfu and anderson j phys a 19 1605 1986 our results also generalize to the problem of q partitioning they can be used to find the expected modularity q newman and girvan phys rev e 69 026113 2004 of random graphs and allow for the assessment of the statistical significance of the output of community detection algorithms given a graph or network gn m of n nodes and m edges the problem of graph partitioning is finding a partition of the nodes into q equal sized parts such that the number of edges connecting different parts the cut size is minimal the solution of this problem has many important practical applications in multiprocessor scheduling for parallel computing very large scale integrated chip design vlsi and data mining 1 due to the problem being np complete 2 i e no algorithm is known which is guaranteed to find an optimal solution in a number of steps that grows only polynomially with the size of the graph only heuristics exist to approximate a solution furthermore one cannot determine in polynomial time whether a solution found is indeed optimal the only way to assess the quality of a heuristic algorithm or a particular result is to compare with rigorous bounds on the cut size

local conservation laws of second order evolution equations generalizing results by bryant and griffiths 1995 duke math j 78 531 we completely describe local conservation laws of second order 1 1 dimensional evolution equations up to contact equivalence the possible dimensions of spaces of conservation laws prove to be 0 1 2 and infinity the canonical forms of equations with respect to contact equivalence are found for all nonzero dimensions of spaces of conservation laws