the singular set of solutions to non differentiable elliptic systems we estimate the hausdorff dimension of the singular set of solutions to elliptic systems of the type if the vector fields a and b are holder continuous with respect to the variable x with exponent then the hausdorff dimension of the singular set of any weak solution is at most n 2

unified index to quantifying heterogeneity of complex networks abstract although recent studies have revealed that degree heterogeneity of a complex network has significant impact on the network performance and function a unified definition of the heterogeneity of a network with any degree distribution is absent in this paper we define a heterogeneity index 0 h 1 to quantify the degree heterogeneity of any given network we analytically show the existence of an upper bound of h 0 5 for exponential networks thus explain why exponential networks are homogeneous on the other hand we also analytically show that the heterogeneity index of an infinite power law network is between 1 and 0 5 if and only if its degree exponent is between 2 and 2 5 we further show that for any power law network with a degree exponent greater than 2 5 there always exists an exponential network such that both networks have the same heterogeneity index this may help to explain why 2 5 is a critical degree exponent for some dynamic behaviors on power law networks

probabilistic analysis of a generalization of the unit clause literal selection heuristics for the k satisfiability problem abstract two algorithms for the k satisfiability problem are presented and a probabilistic analysis is performed the analysis is based on an instance distribution which is parametrized to simulate a variety of sample characteristics the algorithms assign values to literals appearing in a given instance of k satisfiability one at a time until a solution is found or it is discovered that further assignments cannot lead to finding a solution one algorithm chooses the next literal from a unit clause if one exists and randomly from the set of remaining literals otherwise the other algorithm uses a generalization of the unit clause rule as a heuristic for selecting the next literal at each step a literal is chosen randomly from a clause containing the least number of literals the algorithms run in polynomial time and it is shown that they find a solution to a random instance of k satisfiability with probability bounded below by a constant greater than zero for two different ranges of parameter values it is also shown that the second algorithm mentioned finds a solution with probability approaching one for a wide range of parameter values

probabilistic analysis of two heuristics for the 3 satisfiability problem an algorithm for the 3 satisfiability problem is presented and a probabilistic analysis is performed the analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics the algorithm assigns values to variables appearing in a given instance of 3 satisfiability one at a time using the unit clause heuristic and a maximum occurring literal selection heuristic at each step a variable is chosen randomly from a subset of variables which is usually large the algorithm runs in polynomial time and it is shown that the algorithm finds a solution to a random instance of 3 satisfiability with probability bounded from below by a constant greater than zero for a range of parameter values the heuristics studied here can be used to select variables in a backtrack algorithm for 3 satisfiability experiments have shown that for about the same range of parameters as above the backtrack algorithm using the heuristics finds a solution in polynomial average time

properties of atypical graphs from negative complexities the one step replica symmetry breaking cavity method is proposed as a new tool to investigate large deviations in random graph ensembles the procedure hinges on a general connection between negative complexities and probabilities of rare samples in spin glass like models this relation between large deviations and replica theory is explicited on different models where it is confronted to direct combinatorial calculations

some large deviation results for sparse random graphs we obtain a large deviation principle ldp for the relative size of the largest connected component in a random graph with small edge probability the rate function which is not convex in general is determined explicitly using a new technique the proof yields an asymptotic formula for the probability that the random graph is connected

solving satisfiability problems by fluctuations the dynamics of stochastic local search algorithms stochastic local search algorithms are frequently used to numerically solve hard combinatorial optimization or decision problems we give numerical and approximate analytical descriptions of the dynamics of such algorithms applied to random satisfiability problems we find two different dynamical regimes depending on the number of constraints per variable for low constraintness the problems are solved efficiently i e in linear time for higher constraintness the solution times become exponential we observe that the dynamical behavior is characterized by a fast equilibration and fluctuations around this equilibrium if the algorithm runs long enough an exponentially rare fluctuation towards a solution appears

many hard examples for resolution for every choice of positive integers c and k such that k 3 and c 2 k 0 7 there is a positive number e such that with probability tending to 1 as n tends to a randomly chosen family of cn clauses of size k over n variables is unsatisfiable but every resolution proof of its unsatisfiability must generate at least 1 e n clauses

rigorous results for random 2 p sat in recent years there has been significant interest in the study of random k sat formulae for a given set of n boolean variables let bk denote the set of all possible disjunctions of k distinct non complementary literals from its variables k clauses a random k sat formula fk n m is formed by selectinguniformly and independently m clauses from bk and takingtheir conjunction motivated by insights from statistical mechanics that suggest a possible relationship between the order of phase transitions and computational complexity monasson and zecchina phys rev e 56 2 1997 1357 proposed the random 2 p sat model for a given p 0 1 a random 2 p sat formula f2 p n m has m randomly chosen clauses over n variables where pm clauses are chosen from b3 and 1 p m from b2 usingthe heuristic replica method of statistical mechanics monasson and zecchina gave a number of non rigorous predictions on the behavior of random 2 p sat formulae in this paper we give the 1rst rigorous results for random 2 p sat including the following surprising fact for p 2 5 with probability 1 o 1 a random 2 p sat formula is satisfiable if its 2 sat subformula is satisfiable that is for p 6 2 5 random 2 p sat behaves like random 2 sat

floppy modes and the free energy rigidity and connectivity percolation on bethe lattices we show that the negative of the number of ital floppy modes behaves as a ital free energy for both connectivity and rigidity percolation and we illustrate this result using bethe lattices the rigidity transition on bethe lattices is found to be first order at a bond concentration close to that predicted by maxwell constraint counting we calculate the probability of a bond being on the infinite cluster and also on the overconstrained part of the infinite cluster and show how a ital specific heat can be defined as the second derivative of the free energy we demonstrate that the bethe lattice solution is equivalent to that of the random bond model where points are joined randomly with equal probability at all length scales to have a given coordination and then subsequently bonds are randomly removed copyright ital 1999 ital the american physical society

optimal partial regularity for sub elliptic systems with dini continuous coefficients under the superquadratic natural growth the paper is concerned with nonlinear sub elliptic systems with dini continuous coefficients under super quadratic m 2 natural growth conditions in the heisenberg group a general criterion for weak solutions to be regular in the neighborhood of a given point is established by a generalization of method of a harmonic approximation in particular our result is optimal in the sense that in the case of holder continuous coefficients we obtain directly the optimal holder exponent for horizontal gradient of weak solutions on its regular set

sampling rare events statistics of local sequence alignments a method to calculate probability distributions in regions where the events are very unlikely e g p approximately 10 40 is presented the basic idea is to map the underlying model on a physical system the system is simulated at a low temperature such that preferably configurations with originally low probabilities are generated since the distribution of such a physical system is known the original unbiased distribution can be obtained as an application local alignment of protein sequences is studied the deviation of the distribution p s of optimum scores from the extreme value distribution is quantified this deviation decreases with growing sequence length

optimal cooperation and submodularity for computing potts partition functions with a large number of states the partition function of the q state potts model with random ferromagnetic couplings in the large q limit is generally dominated by the contribution of a single diagram of the high temperature expansion computing this dominant diagram amounts to minimizing a particular submodular function we provide a combinatorial optimization algorithm the optimal cooperation algorithm which works in polynomial time for any lattice the implementation of the method and its running time are also discussed

relaxation and metastability in a local search procedure for the random satisfiability problem an analysis of the average properties of a local search procedure randomwalksat for the satisfaction of random boolean constraints is presented depending on the ratio alpha of constraints per variable reaching a solution takes a time t res growing linearly t res approximately tau res alpha n alpha alpha d with the size n of the instance the relaxation time tau res alpha in the linear phase is calculated through a systematic expansion scheme based on a quantum formulation of the evolution operator for alpha alpha d the system is trapped in some metastable state and resolution occurs from escape from this state through crossing of a large barrier an annealed calculation of the height zeta alpha of this barrier is proposed the polynomial to exponential cross over alpha d approximately 2 7 is not related to the onset of clustering among solutions occurring at alpha approximately 3 86

on large deviation properties of erdos renyi random graphs we show that large deviation properties of erdos renyi random graphs can be derived from the free energy of the q state potts model of statistical mechanics more precisely the legendre transform of the potts free energy with respect to ln q is related to the component generating function of the graph ensemble this generalizes the well known mapping between typical properties of random graphs and the q 1 limit of the potts free energy for exponentially rare graphs we explicitly calculate the number of components the size of the giant component the degree distributions inside and outside the giant component and the distribution of small component sizes we also perform numerical simulations which are in very good agreement with our analytical work finally we demonstrate how the same results can be derived by studying the evolution of random graphs under the insertion of new vertices and edges without recourse to the thermodynamics of the potts model

generation of uncorrelated random scale free networks uncorrelated random scale free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks we propose and analyze a model capable of generating random uncorrelated scale free networks with no multiple and self connections the model is based on the classical configuration model with an additional restriction on the maximum possible degree of the vertices we check numerically that the proposed model indeed generates scale free networks with no two and three vertex correlations as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree k respectively complex networks constitute a general framework for the topological characterization of many natural and technological systems whose complexity prevents a more detailed microscopic description f1 3g within this framework these systems are represented in terms of networks or graphs f4g in which vertices stand for the units composing the system while edges among vertices represent the interactions or relations between pairs of units the focus is thus shift to the topological characterization of the representative network a task which is largely more feasible and yields nevertheless a noticeable amount of information on the structure and properties of the original system the empirical analysis of many real complex networks has unveiled the presence of several typical properties widely found in systems belonging to a large variety of realms one of the most relevant is given by the scale free nature of the degree distribution pskd f1 3 5g defined as the probability that a randomly chosen vertex has degree k si e it is connected to other k verticesd in mathematical terms the scale free property translates into a power law function of the form psk d k g

network synchronization diffusion and the paradox of heterogeneity many complex networks display strong heterogeneity in the degree connectivity distribution heterogeneity in the degree distribution often reduces the average distance between nodes but paradoxically may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength here we offer a solution to this apparent paradox our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states we show that for a given degree distribution the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum this enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size numerical verification of the main results is provided for representative classes of small world and scale free networks

easy problems are sometimes hard we present a detailed experimental investigation of the easy hard easy phase transition for randomly generated instances of satisfiability problems problems in the hard part of the phase transition have been extensively used for benchmarking satisfiability algorithms this study demonstrates that problem classes and regions of the phase transition previously thought to be easy can sometimes be orders of magnitude more difficult than the worst problems in problem classes and regions of the phase transition considered hard these difficult problems are either hard unsatisfiable problems or are satisfiable problems which give a hard unsatisfiable subproblem following a wrong split whilst these hard unsatisfiable problems may have short proofs these appear to be difficult to find and other proofs are long and hard

exponentially hard problems are sometimes polynomial a large deviation analysis of search algorithms for the random satisfiability problem and its application to stop and restart resolutions a large deviation analysis of the solving complexity of random 3 satisfiability instances slightly below threshold is presented while finding a solution for such instances demands an exponential effort with high probability we show that an exponentially small fraction of resolutions require a computation scaling linearly in the size of the instance only this exponentially small probability of easy resolutions is analytically calculated and the corresponding exponent is shown to be smaller in absolute value than the growth exponent of the typical resolution time our study therefore gives some theoretical basis to heuristic stop and restart solving procedures and suggests a natural cutoff the size of the instance for the restart

probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension d 1 2 3 via a monte carlo procedure in the disorder in order to probe with high precision the tails of the ground state energy distribution of disordered spin systems k orner katzgraber and hartmann have recently proposed an importance sampling monte carlo markov chain in the disorder in this paper we combine their monte carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground state energy distribution p d e 0 for the directed polymer in a random medium of dimension d 1 2 3 in d 1 we check the validity of the algorithm by a direct comparison with the exact result namely the tracy widom distribution in dimensions d 2 and d 3 we measure the negative tail up to ten standard deviations which correspond to probabilities of order p d e 0 ensuremath sim 10 ensuremath 22 our results are in agreement with zhang s argument stating that the negative tail exponent ensuremath eta d of the asymptotic behavior mathrm ln phantom rule 0 2em 0ex p d e 0 ensuremath sim ensuremath ensuremath mid e 0 ensuremath mid ensuremath eta d as e 0 ensuremath rightarrow ensuremath ensuremath infty is directly related to the fluctuation exponent ensuremath theta d which governs the fluctuations ensuremath delta e 0 l ensuremath sim l ensuremath theta d of the ground state energy e 0 for polymers of length l via the simple formula ensuremath eta d 1 1 ensuremath ensuremath theta d throughout the paper we comment on the similarities and differences with spin glasses

entropy optimization of scale free networks robustness to random failures abstract many networks are characterized by highly heterogeneous distributions of links which are called scale free networks and the degree distributions follow p k ck we study the robustness of scale free networks to random failures from the character of their heterogeneity entropy of the degree distribution can be an average measure of a network s heterogeneity optimization of scale free networks robustness to random failures with average connectivity constant is equivalent to maximizing the entropy of the degree distribution by examining the relationship of the entropy of the degree distribution scaling exponent and the minimal connectivity we get the optimal design of scale free networks to random failures we conclude that the entropy of the degree distribution is an effective measure of network s resilience to random failures

results related to threshold phenomena research in satisfiability lower bounds we present a history of results related to the threhold phenomena which arise in the study of random conjunctive normal form cnf formulas in a companion paper d achlioptas theoret comput sci this volume the major ideas used to achieve many of the lower bounds results on the location of the threshold are described in an informal intuitive manner

the dynamics of proving uncolourability of large random graphs i symmetric colouring heuristic we study the dynamics of a backtracking procedure capable of proving uncolourability of graphs and calculate its average running time t for sparse random graphs as a function of the average degree c and the number of vertices n t he analysis is carried out by mapping the history of the search process onto an out of equilibrium multi dimensional surface growth problem the growth exponent of the average running time c ln t n is quantitatively predicted in agreement with simulations

revealing the microstructure of the giant component in random graph ensembles the microstructure of the giant component of the erd s renyi network and other configuration model networks is analyzed using generating function methods while configuration model networks are uncorrelated the giant component exhibits a degree distribution which is different from the overall degree distribution of the network and includes degree degree correlations of all orders we present exact analytical results for the degree distributions as well as higher order degree degree correlations on the giant components of configuration model networks we show that the degree degree correlations are essential for the integrity of the giant component in the sense that the degree distribution alone cannot guarantee that it will consist of a single connected component to demonstrate the importance and broad applicability of these results we apply them to the study of the distribution of shortest path lengths on the giant component percolation on the giant component and spectra of sparse matrices defined on the giant component we show that by using the degree distribution on the giant component one obtains high quality results for these properties which can be further improved by taking the degree degree correlations into account this suggests that many existing methods currently used for the analysis of the whole network can be adapted in a straightforward fashion to yield results conditioned on the giant component

partial regularity for discontinuous sub elliptic systems with subquadratic growth in the heisenberg group abstract we consider nonlinear sub elliptic systems with vmo coefficients in the heisenberg group and prove partial holder continuity for discontinuous sub elliptic problems with subquadratic growth by a generalization of the technique of a harmonic approximation the model case is the sub elliptic p laplace system with vmo coefficients in the form i 1 2 n x i a i 1 x u 2 p 2 2 x i u 0 where a i v m o and 1 p 2

rare events and discontinuous percolation transitions percolation theory characterizing the robustness of a network has applications ranging from biology to epidemic spreading and complex infrastructures percolation theory however only concerns the average response of a network to random damage of its nodes while in real finite networks fluctuations around this average behavior are observable consequently for finite networks there is an urgent need to evaluate the risk of collapse in response to rare configurations of the initial damage here we build a large deviation theory of percolation characterizing the response of a sparse network to rare events this general theory includes the second order phase transition observed typically for random configurations of the initial damage but reveals also discontinuous transitions corresponding to rare configurations of the initial damage for which the size of the giant component is suppressed

ensemble inequivalence in random graphs we present a complete analytical solution of a system of potts spins on a random k regular graph in both the canonical and microcanonical ensembles using the large deviation cavity method ldcm the solution is shown to be composed of three different branches resulting in a non concave entropy function the analytical solution is confirmed with numerical metropolis and creutz simulations and our results clearly demonstrate the presence of a region with negative specific heat and consequently ensemble inequivalence between the canonical and microcanonical ensembles

degree based attacks are not optimal for desynchronization in general networks suppressing harmful synchronization in complex networked systems is receiving increasing interest in various contexts including epileptic seizure and the internet and traffic congestion it is traditionally believed that by disrupting the most important nodes i e those nodes of high degrees synchronization can be effectively mitigated however this approach can perform poorly in more general situations such as homogeneous random or small world networks in this article we investigate how topological properties such as the heterogeneity of the network can have an impact on its desynchronization in particular we propose a topology aware scheme that chooses topologically sufficiently diversified contrarian nodes by taking into account the patterns of connectivity among them hence a maximal number of nodes in the network can be influenced leading to a more global impact and greater disruption to synchronization our scheme demonstrates significantly improved performance on various networked systems including homogeneous networks small world and even scale free networks

factors that predict better synchronizability on complex networks while shorter characteristic path length has in general been believed to enhance synchronizability of a coupled oscillator system on a complex network the suppressing tendency of the heterogeneity of the degree distribution even for shorter characteristic path length has also been reported to see this we investigate the effects of various factors such as the degree characteristic path length heterogeneity and betweenness centrality on synchronization and find a consistent trend between the synchronization and the betweenness centrality the betweenness centrality is thus proposed as a good indicator for synchronizability

c1 partial regularity for sub elliptic systems with dini continuous coefficients in carnot groups abstract in this paper we consider nonlinear sub elliptic systems with dini continuous coefficients for the case 1 m 2 in carnot groups based on a generalization of the technique of a harmonic approximation introduced by duzaar grotowski kronz and a sobolev poincare type inequality established in carnot groups a c 1 partial regularity result for weak solutions of sub elliptic systems with natural growth terms is established in particular our result is optimal in the sense that in the case of holder continuous coefficients it yields directly the optimal holder exponent for horizontal gradients of weak solutions on its regular set

partial regularity for sub elliptic systems with dini continuous coefficients involving natural growth terms in the heisenberg group abstract we consider nonlinear sub elliptic systems with dini continuous coefficients in divergence form in the heisenberg group based on a generalization of the method of a harmonic approximation introduced by duzaar and steffen partial regularity of weak solutions for sub elliptic systems under natural growth conditions is established in particular our result is optimal in the sense that in the case of holder continuous coefficients we obtain directly the optimal holder exponent for the horizontal gradients of weak solutions on the regular set

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

lower bounds for random 3 sat via differential equations it is widely believed that the probability of satisfiability for random k sat formulae exhibits a sharp threshold as a function of their clauses to variables ratio for the most studied case k 3 there have been a number of results during the last decade providing upper and lower bounds for the threshold s potential location all lower bounds in this vein have been algorithmic i e in each case a particular algorithm was shown to satisfy random instances of 3 sat with probability 1 o 1 if the clauses to variables ratio is below a certain value we show how differential equations can serve as a generic tool for analyzing such algorithms by rederiving most of the known lower bounds for random 3 sat in a simple uniform manner

partial regularity for discontinuous sub elliptic systems with vmo coefficients involving controllable growth terms in heisenberg groups abstract in this paper we consider discontinuous sub elliptic systems with vmo coefficients for the controllable growth case of p 2 in the heisenberg group based on a generalization of the technique of a harmonic approximation with the superquadratic growth case a partial holder continuity result for weak solutions of discontinuous sub elliptic systems with vmo coefficients is established in particular the primary model covered by our analysis is the non degenerate p sub laplace system involving super quadratic controllable growth terms

interior partial regularity for nonlinear elliptic systems with dini continuous coefficients for the case 1 m 2 in this article we consider the interior regularity for weak solutions of nonlinear elliptic systems with divergence type and dini continuous coefficients under subquadratic growth conditions we adopt another a harmonic approximation technique and prove the interior partial regularity of weak solutions the proof yields directly the optimal holder exponent for the derivative of the weak solutions on the regular set

typical solution time for a vertex covering algorithm on finite connectivity random graphs we analytically describe the typical solution time needed by a backtracking algorithm to solve the vertex cover problem on finite connectivity random graphs we find two different transitions the first one is algorithm dependent and marks the dynamical transition from linear to exponential solution times the second one gives the maximum computational complexity and is found exactly at the threshold where the system undergoes an algorithm independent phase transition in its solvability analytical results are corroborated by numerical simulations

optimal partial regularity of second order nonlinear elliptic systems with dini continuous coefficients for the superquadratic case abstract we consider the regularity for weak solutions of second order nonlinear elliptic systems with dini continuous coefficients for the superquadratic case under natural growth condition and obtain a general criterion for a weak solution to be regular in the neighborhood of a given point in particular the proof yields directly the optimal holder exponent for the derivative of the weak solutions on the regular set

heuristic average case analysis of the backtrack resolution of random 3 satisfiability instances an analysis of the average case complexity of solving random 3 satisfiability sat instances with backtrack algorithms is presented we first interpret previous rigorous works in a unifying framework based on the statistical physics notions of dynamical trajectories phase diagram and growth process it is argued that under the action of the davis putnam loveland logemann dpll algorithm 3 sat instances are turned into 2 p sat instances whose characteristic parameters ratio of clauses per variable fraction p of 3 clauses can be followed during the operation and define resolution trajectories depending on the location of trajectories in the phase diagram of the 2 p sat model easy polynomial or hard exponential resolutions are generated three regimes are identified depending on the ratio of the 3 sat instance to be solved lower satisfiable sat phase for small ratios dpll almost surely finds a solution in a time growing linearly with the number n of variables upper sat phase for intermediate ratios instances are almost surely satisfiable but finding a solution requires exponential time 2n with 0 with high probability unsat phase for large ratios there is almost always no solution and proofs of refutation are exponential an analysis of the growth of the search tree in both upper sat and unsat regimes is presented and allows us to estimate as a function of this analysis is based on an exact relationship between the average size of the search tree and the powers of the evolution operator encoding the elementary steps of the search heuristic

optimal synchronization of kuramoto oscillators a dimensional reduction approach a recently proposed dimensional reduction approach for studying synchronization in the kuramoto model is employed to build optimal network topologies to favor or to suppress synchronization the approach is based in the introduction of a collective coordinate for the time evolution of the phase locked oscillators in the spirit of the ott antonsen ansatz we show that the optimal synchronization of a kuramoto network demands the maximization of the quadratic function t l where stands for the vector of the natural frequencies of the oscillators and l for the network laplacian matrix many recently obtained numerical results can be reobtained analytically and in a simpler way from our maximization condition a computationally efficient hill climb rewiring algorithm is proposed to generate networks with optimal synchronization properties our approach can be easily adapted to the case of the kuramoto models with both attractive and repulsive interactions and again many recent numerical results can be rederived in a simpler and clearer analytical manner