graph optimisation problems and the potts glass the np complete problems of partitioning and colouring of random graphs with p partitions and colours respectively are mapped onto the statistical mechanical problem of p state potts glasses an estimate of the cost functions of these optimisation problems has been derived using the potts glass mean field theory this estimate applies to dense graphs in the thermodynamic limit an exact expression for the cost function in the large p limit is given

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

variational formula for the time constant of first passage percolation consider first passage percolation with positive stationary ergodic weights on the square lattice in d dimensions let t x be the first passage time from the origin to x in z d the convergence of t nx n to the time constant as n tends to infinity is a consequence of the subadditive ergodic theorem this convergence can be viewed as a problem of homogenization for a discrete hamilton jacobi bellman hjb equation by borrowing several tools from the continuum theory of stochastic homogenization for hjb equations we derive an exact variational formula duality principle for the timeconstant under a symmetry assumption we will use the variational formula to construct an explicit iteration that produces the limit shape

stochastic homogenization of certain nonconvex hamilton jacobi equations abstract in this paper we prove the stochastic homogenization of certain nonconvex hamilton jacobi equations the nonconvex hamiltonians which are generally uneven and inseparable are generated by a sequence of quasiconvex hamiltonians and a sequence of quasiconcave hamiltonians through the min max formula we provide a monotonicity assumption on the contact values between those stably paired hamiltonians so as to guarantee the stochastic homogenization

on the stability of the replica symmetric theory of the matching problem the longitudinal sector a replica symmetric theory of the matching problem was given by orland 1985 and by mezard and paris 1985 here the authors investigate the stability of this solution in the zero temperature limit they find that the longitudinal spectrum is continuous and lies in the interval 0 lambda 2 they conclude that the solution of mezard and parisi is stable to longitudinal fluctuation they give the explicit form of the longitudinal eigenvectors the method used lends itself to dealing with replica symmetry broken solutions where necessary

exit times of n dimensional random walks we study the exit time t of a sum of independent identically distributed random vectors x1 x2 from a subset r of n dimensional euclidean space when n 2 we assume that r is invariant under positive dilations and that the boundary of r satisfies certain regularity conditions the random vector x1 is to have mean zero and a nonsingular covariance matrix we show that there is a critical exponent e independent of x1 x2 such that 0 e implies etp 2

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

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

data clustering using a model granular magnet we present a new approach to clustering based on the physical properties of an inhomogeneous ferromagnet no assumption is made regarding the underlying distribution of the data we assign a potts spin to each data point and introduce an interaction between neighboring points whose strength is a decreasing function of the distance between the neighbors this magnetic system exhibits three phases at very low temperatures it is completely ordered all spins are aligned at very high temperatures the system does not exhibit any ordering and in an intermediate regime clusters of relatively strongly coupled spins become ordered whereas different clusters remain uncorrelated this intermediate phase is identified by a jump in the order parameters the spin spin correlation function is used to partition the spins and the corresponding data points into clusters we demonstrate on three synthetic and three real data sets how the method works detailed comparison to the performance of other techniques clearly indicates the relative success of our method

stochastic optimization on continuous domains with finite time guarantees by markov chain monte carlo methods we introduce bounds on the finite time performance of markov chain monte carlo mcmc algorithms in solving global stochastic optimization problems defined over continuous domains it is shown that mcmc algorithms with finite time guarantees can be developed with a proper choice of the target distribution and by studying their convergence in total variation norm this work is inspired by the concept of finite time learning with known accuracy and confidence developed in statistical learning theory

graph bipartitioning problem replica symmetric solutions of the graph bipartitioning problem with finite connectivity are presented with the constraint sum mathrm i 1 mathrm n mathrm s mathrm i 0 strictly enforced another solution can be obtained which gives a lower cost function than that given by the spin glass solution it is believed that this new solution is currently the best candidate for the true solution of the graph bipartitioning problem

random walks in cones we study the asymptotic behavior of a multidimensional random walk in a general cone we find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone the main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments for the proof of tail asymptotics and integral limit theorems we use a strong approximation of random walks by brownian motion for the proof of local limit theorems we suggest a rather simple approach which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks we also discuss some possible applications of our results to ordered random walks and lattice path enumeration

stochastic homogenization of a front propagation problem with unbounded velocity we study the homogenization of hamilton jacobi equations which arise in front propagation problems in stationary ergodic media our results are obtained for fronts moving with possible unbounded velocity we show by an example that the homogenized hamiltonian which always exists may be unbounded in this context we show convergence results if we start with a compact initial front on the other hand if the media satisfies a finite range of dependence condition we prove that the effective hamiltonian is bounded and obtain classical homogenization in this context

fluctuations in the random link matching problem using the replica approach and the cavity method we study the fluctuations of the optimal cost in the random link matching problem by means of replica arguments we derive the exact expression of its variance moreover we study the large deviation function deriving its expression in two different ways namely using both the replica method and the cavity method

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

galerkin finite element approximations for stochastic space time fractional wave equations the traditional wave equation models wave propagation in an ideal medium to characterize wave propagation in inhomogeneous media with frequency dependent power law attenuation the space time fractional wave equation is needed further incorporating the additive white gaussian noise external noise coming from many natural sources leads to the stochastic space time fractional wave equation this paper discusses the galerkin finite element approximations for the stochastic space time fractional wave equation forced by an additive space time white noise we first discretize the space time additive noise which introduces a modeling error and results in a regularized stochastic space time fractional wave equation then the regularity of the regularized equation is analyzed for the discretization in space the finite element approximation is used and the definition of the discrete fractional laplacian is introduced we derive the mean squared l 2 norm a priori estimates for the modeling error and for the

optimization problems and replica symmetry breaking in finite connectivity spin glasses a formalism capable of handling the first step of hierarchical replica symmetry breaking rsb in finite connectivity models is introduced the emerging order parameter is claimed to be a probability distribution over the space of field distributions or equivalently magnetization distributions inside the cluster of states the approach is shown to coincide with previous works in the replica symmetric case and in the two limiting cases m 0 and 1 where m is parisi s break point as an application to the study of optimization problems the gs properties of the random 3 satisfiability problem are investigated and we present a first rsb solution improving replica symmetric results

the travelling salesman problem on a randomly diluted lattice the authors study the problem of a travelling salesman who must visit a randomly chosen subset of sites of a d dimensional lattice the average length of the shortest path per chosen site is alpha q where 1 q is the density of chosen sites for a triangular lattice they show that alpha q differs from 1 only by terms of order q5 for the square lattice they show that to first order in q optimal paths can be found from the dynamics of a model of a one dimensional gas of kinks and antikinks the authors find alpha q or l 3 1 q 1 2 as q tends to 1

central limit theorem for stochastic hamilton jacobi equations we study the asymptotic behavior of where u solves the hamilton jacobi equation u t h x u x 0 with h a stationary ergodic process in the x variable it was shown in rezakhanlou tarver rt that u converges to a deterministic function provided h x p is convex in p and the convex conjugate of h in the p variable satisfies certain growth conditions in this article we establish a central limit theorem for the convergence by showing that for a class of examples u x t can be stochastically represented as where z x t is a suitable random field in particular we establish a central limit theorem when the dimension is one and where is a random function that enjoys some mild regularity

neutrality in fitness landscapes the interplay of ruggedness and neutrality in fitness landscapes plays an important role in explaining the dynamics of evolutionary adaptation while various measures of ruggedness correlation functions adaptive walks or the density of local optima are reasonably well understood and models for constructing landscapes with a desired degree of ruggedness are readily available very little is known about neutrality we introduce the notion of additive random landscapes as a framework for tuning both neutrality and ruggedness at once and we develop a formalism that allows the explicit computation of the most salient parameters that are associated with neutrality in landscapes of this type

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

the perturbed test function method for viscosity solutions of nonlinear pde the method of viscosity solutions for nonlinear partial differential equations pdes justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions we describe in this paper a perturbed test function device which entails various modifications of the test functions by lower order correctors applications include homogenisation for quasilinear elliptic pdes and approximation of quasilinear parabolic pdes by systems of hamilton jacobi equations

critical exponents on fortuin kasteleyn weighted planar maps in this paper we consider random planar maps weighted by the self dual fortuin kastelyn model with parameter q2 0 4 using a bijection due to sheeld and a connection to planar brownian motion in a cone we obtain rigorously the value of critical exponents associated with the length of cluster interfaces which is shown to be 4 arccos p 2 p q 2

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

error estimates and convergence rates for the stochastic homogenization of hamilton jacobi equations we present exponential error estimates and demonstrate an algebraic convergence rate for the homogenization of level set convex hamilton jacobi equations in i i d random environments the first quantitative homogenization results for these equations in the stochastic setting by taking advantage of a connection between the metric approach to homogenization and the theory of first passage percolation we obtain estimates on the fluctuations of the solutions to the approximate cell problem in the ballistic regime away from flat spot of the effective hamiltonian in the sub ballistic regime on the flat spot we show that the fluctuations are governed by an entirely different mechanism and the homogenization may proceed without further assumptions at an arbitrarily slow rate we identify a necessary and sufficient condition on the law of the hamiltonian for an algebraic rate of convergence to hold in the sub ballistic regime and show under this hypothesis that the two rates may be merged to yield comprehensive error estimates and an algebraic rate of convergence for homogenization our methods are novel and quite different from the techniques employed in the periodic setting although we benefit from previous works in both first passage percolation and homogenization the link between the rate of homogenization and the flat spot of the effective hamiltonian which is related to the nonexistence of correctors is a purely random phenomenon observed here for the first time

stochastic homogenization of hamilton jacobi bellman equations we study the homogenization of some hamilton jacobi bellman equations with a vanishing second order term in a stationary ergodic random medium under the hyperbolic scaling of time and space imposing certain convexity growth and regularity assumptions on the hamiltonian we show the locally uniform convergence of solutions of such equations to the solution of a deterministic effective first order hamilton jacobi equation the effective hamiltonian is obtained from the original stochastic hamiltonian by a minimax formula our homogenization results have a large deviations interpretation for a diffusion in a random environment c 2006 wiley periodicals inc

stochastic homogenization of nonconvex hamilton jacobi equations in one space dimension abstract we prove stochastic homogenization for a general class of coercive nonconvex hamilton jacobi equations in one space dimension some properties of the effective hamiltonian arising in the nonconvex case are also discussed

adaptive multi resolution modularity for detecting communities in networks abstract community structure is a common topological property of complex networks which attracted much attention from various fields optimizing quality functions for community structures is a kind of popular strategy for community detection such as modularity optimization here we introduce a general definition of modularity by which several classical multi resolution modularity can be derived and then propose a kind of adaptive multi resolution modularity that can combine the advantages of different modularity by applying the modularity to various synthetic and real world networks we study the behaviors of the methods showing the validity and advantages of the multi resolution modularity in community detection the adaptive modularity as a kind of multi resolution method can naturally solve the first type limit of modularity and detect communities at different scales it can quicken the disconnecting of communities and delay the breakup of communities in heterogeneous networks and thus it is expected to generate the stable community structures in networks more effectively and have stronger tolerance against the second type limit of modularity

gambler s ruin and the first exit position of random walk from large spheres let t r be the first time a sum s n of nondegenerate i i d random vectors in r d leaves the sphere of radius r in some given norm we characterize in terms of the distribution of the individual summands the following probabilistic behavior s tr s tr has no subsequential weak limit supported on a closed half space in one dimension this result solves a very general form of the gambler s ruin problem we also characterize the existence of degenerate limits and obtain analogous results for triangular arrays along any subsequence r k finally we compute the limiting joint distribution of s tr r s tr s tr

statistical mechanics of an np complete problem subset sum we study the statistical properties of an np complete problem the subset sum using the methods and concepts of statistical mechanics the problem is a generalization of the number partitioning problem which is also an npcomplete problem and has been studied in the physics literature the asymptotic expressions for the number of solutions are obtained these results applied to the number partitioning problem as a special case are compared with those which were previously obtained by a different method we discuss the limit of applicability of the techniques of statistical mechanics to the present problem

homogenization for stochastic hamilton jacobi equations homogenization asks whether average behavior can be discerned from partial differential equations that are subject to high frequency fluctuations when those fluctuations result from a dependence on two widely separated spatial scales we prove homogenization for certain stochastic hamilton jacobi partial differential equations the idea is to use the subadditive ergodic theorem to establish the existence of an average in the infinite scale separation limit in some cases we also establish a central limit theorem

replica symmetric solutions of the graph bipartitioning problem with fixed finite valence using the replica method the author presents the replica symmetric solutions of the graph bipartitioning problem with fixed finite valence with the constraint sigma ni 1 si 0 strictly enforced the author is able to find another solution which gives a lower cost function than that given by the spin glass solution

complexity and ultradiffusion the authors present the exact solution to the problem of ultradiffusion in an arbitrary hierarchical space they derive rigorous upper and lower bounds for the dynamic exponent describing the decay of the autocorrelation function they show that the upper bound is saturated by both uniformly and randomly multifurcating hierarchical trees and identify a class of highly unbalanced trees that saturate the lower bound they conclude that the speed of relaxation is a measure of the complexity or lack of self similarity of the underlying tree they point out that complexity may be revealed by the temperature dependence of the dynamic exponent and in particular by the nature of the transition from exponential to power law decay

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

distance fluctuations and lyapounov exponents we associate certain translation invariant random metrics on r d to brownian motion evolving in a truncated poissonian potential these metrics behave over large distances in an appropriate sense like certain deterministic norms the so called lyapounov exponents we prove here upper bounds on the size of fluctuations of the metrics around their mean under an additional assumption of rotational invariance we also derive upper bounds on the difference between the mean of the metrics and the lyapounov norms

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

graph bipartitioning and spin glasses on a random network of fixed finite valence the authors study the problem of bipartitioning a random graph of fixed finite valence using a mean field replica symmetric theory of an ising ferromagnet with zero magnetisation constraint the thermodynamics is determined by the probability distribution of an auxiliary field the expression for the ground state energy agrees with that proposed by mezard and parisi 1986 using a cavity field method but their expression for the fraction of crazy spins is reinterpreted

minimizing unsatisfaction in colourful neighbourhoods colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance here we consider the problem of maximizing the number of different colours available at the nodes and their neighbourhoods given a predetermined number of colours in the analytical framework of a tree approximation carried out at both zero and finite temperatures solutions obtained by population dynamics give rise to estimates of the threshold connectivity for the incomplete to complete transition which are consistent with those of existing algorithms the nature of the transition as well as the validity of the tree approximation are investigated

stochastic homogenization of nonconvex hamilton jacobi equations a counterexample this paper provides a counterexample to hamilton jacobi homogenization in the nonconvex case for general stationary ergodic environments

application of statistical mechanics to combinatorial optimization problems the chromatic number problem and q partitioning of a graph methods of statistical mechanics are applied to two important np complete combinatorial optimization problems the first is the chromatic number problem which seeks the minimal number of colors necessary to color a graph such that no two sites connected by an edge have the same color the second is partitioning of a graph intoq equal subgraphs so as to minimize intersubgraph connections both models are mapped into a frustrated potts model which is related to theq state potts spin glass for the first problem we obtain very good agreement with numerical simulations and theoretical bounds using the annealed approximation the quenched model is also discussed for the second problem we obtain analytic and numerical results by evaluating the groundstate energy of theq 3 and 4 potts spin glass using parisi s replica symmetry breaking we also perform some numerical simulations to test the theoretical result and obtain very good agreement

graph bipartitioning and the bethe spin glass the problem of the equipartitioning of a random graph of fixed finite valence is studied by comparison with a ferromagnetic bethe lattice with random boundary conditions the simplicity of recursion relations for effective fields due to descendents on bethe lattices provides simple approximations for the optimal cost in quite good agreement with simulations

penalized versions of the newman girvan modularity and their relation to normalized cuts and k means clustering two penalized balanced and normalized versions of the newman girvan modularity are introduced and estimated by the non negative eigenvalues of the modularity and normalized modularity matrix respectively in this way the partition of the vertices that maximizes the modularity can be obtained by applying the k means algorithm for the representatives of the vertices based on the eigenvectors belonging to the largest positive eigenvalues of the modularity or normalized modularity matrix the proper dimension depends on the number of the structural eigenvalues of positive sign while dominating negative eigenvalues indicate an anticommunity structure the balance between the negative and the positive eigenvalues determines whether the underlying graph has a community anticommunity or randomlike structure

graph bipartitioning and statistical mechanics the problem of bipartitioning a random graph of fixed finite local valence connectivity so as to minimise the number of cross edges is studied by the application of monte carlo simulation of thermodynamics and annealing this problem is np complete and is relevant as an idealisation of several practical organisation problems evidence is presented for the existence of a multiplicity of metastable states and for non self averaging and ultrametricity in the space of overlaps albeit with a possible critical valence an empirical formula is presented for the optimal cost in excellent accord with values obtained by simulation and those known exactly

bounded correctors in almost periodic homogenization we show that certain linear elliptic equations and systems in divergence form with almost periodic coefficients have bounded almost periodic correctors this is proved under a new condition we introduce which quantifies the almost periodic assumption and includes but is not restricted to the class of smooth quasiperiodic coefficient fields which satisfy a diophantine type condition previously considered by kozlov mat sb n s 107 149 199 217 1978 the proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by armstrong and shen pure appl math 2016 for equations with almost periodic coefficients this yields control on spatial averages of the gradient of the corrector which is converted into estimates on the size of the corrector itself via a multiscale poincare type inequality

graph optimization problems on a bethe lattice thep partitioning andp coloring problems on a bethe lattice of coordination numberz are analyzed it is shown that these two np complete optimization problems turn out to be equivalent to finding the ground state energy ofp state potts models with frustration numerical calculation of the cost function of both problems are carried out for several values ofz andp in the case ofp 2 the results are identical to those obtained by mezard and parisi for the case of the bipartitioning problem a numerical upper bound to the chromatic number is found for several values ofz

first passage percolation has sublinear distance variance let 0 1 the distance dist 0 from the origin to a vertex 2 has variance bounded by c log where c c a b d is a constant which may only depend on a b and d some related variants are also discussed

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

extremal optimization of graph partitioning at the percolation threshold the benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem the performance of this new method called extremal optimization eo is compared with simulated annealing sa in extensive numerical simulations while generally a complex np hard problem the optimization of the graph partitions is particularly difficult for sparse graphs with average connectivities near the percolation threshold at this threshold the relative error of sa for large graphs is found to diverge relative to eo at equalized runtime on the other hand eo based on the extremal dynamics of self organized critical systems reproduces known results about optimal partitions at this critical point quite well