relaxation dynamics of maximally clustered networks we study the relaxation dynamics of fully clustered networks maximal number of triangles to an unclustered state under two different edge dynamics the double edge swap corresponding to degree preserving randomization of the configuration model and single edge replacement corresponding to full randomization of the erd ifmmode mbox h o else h o fi s r enyi random graph we derive expressions for the time evolution of the degree distribution edge multiplicity distribution and clustering coefficient we show that under both dynamics networks undergo a continuous phase transition in which a giant connected component is formed we calculate the position of the phase transition analytically using the erd ifmmode mbox h o else h o fi s r enyi phenomenology

exponential random graph models for networks with community structure although the community structure organization is an important characteristic of real world networks most of the traditional network models fail to reproduce the feature therefore the models are useless as benchmark graphs for testing community detection algorithms they are also inadequate to predict various properties of real networks with this paper we intend to fill the gap we develop an exponential random graph approach to networks with community structure to this end we mainly built upon the idea of blockmodels we consider both the classical blockmodel and its degree corrected counterpart and study many of their properties analytically we show that in the degree corrected blockmodel node degrees display an interesting scaling property which is reminiscent of what is observed in real world fractal networks a short description of monte carlo simulations of the models is also given in the hope of being useful to others working in the field

an exponential family of probability distributions for directed graphs abstract directed graph or digraph data arise in many fields especially in contemporary research on structures of social relationships we describe an exponential family of distributions that can be used for analyzing such data a substantive rationale for the general model is presented and several special cases are discussed along with some possible substantive interpretations a computational algorithm based on iterative scaling procedures for use in fitting data is described as are the results of a pilot simulation study an example using previously reported empirical data is worked out in detail an extension to multiple relationship data is discussed briefly

exponential random simplicial complexes exponential random graph models have attracted significant research attention over the past decades these models are maximum entropy ensembles subject to the constraints that the expected values of a set of graph observables are equal to given values here we extend these maximum entropy ensembles to random simplicial complexes which are more adequate and versatile constructions to model complex systems in many applications we show that many random simplicial complex models considered in the literature can be casted as maximum entropy ensembles under certain constraints we introduce and analyze the most general random simplicial complex ensemble with statistically independent simplices our analysis is simplified by the observation that any distribution on any collection of objects including graphs and simplicial complexes is maximum entropy subject to the constraint that the expected value of is equal to the entropy of the distribution with the help of this observation we prove that ensemble is maximum entropy subject to the two types of constraints which fix the expected numbers of simplices and their boundaries

a newton cg algorithm with complexity guarantees for smooth unconstrained optimization we consider minimization of a smooth nonconvex objective function using an iterative algorithm based on newton s method and the linear conjugate gradient algorithm with explicit detection and use of negative curvature directions for the hessian of the objective function the algorithm tracks newton conjugate gradient procedures developed in the 1980s closely but includes enhancements that allow worst case complexity results to be proved for convergence to points that satisfy approximate first order and second order optimality conditions the complexity results match the best known results in the literature for second order methods

two weight inequality for the hilbert transform a real variable characterization i let and w be locally finite positive borel measures on r which do not share a common point mass assume that the pair of weights satisfy a poisson a2 condition and satisfy the testing conditions below for the hilbert transform h ih 1i 2dw i ih w1i 2d w i with constants independent of the choice of interval i then h maps l2 to l2 w verifying a conjecture of nazarov treil and volberg the proof uses basic tools of nonhomogeneous analysis with two components particular to the hilbert transform the first component is a global to local reduction which is a consequence of prior work by lacey sawyer shen and uriarte tuero the second component an analysis of the local part is the particular contribution of this article

logarithmic bump conditions and the two weight boundedness of calderon zygmund operators abstract we prove that if a pair of weights u v satisfies a sharp a p bump condition in the scale of all log bumps or certain loglog bumps then haar shifts map l p v into l p u with a constant quadratic in the complexity of the shift this in turn implies the two weight boundedness for all calderon zygmund operators this gives a partial answer to a long standing conjecture we also give a partial result for a related conjecture for weak type inequalities to prove our main results we combine several different approaches to these problems in particular we use many of the ideas developed to prove the a 2 conjecture as a byproduct of our work we also disprove a conjecture by muckenhoupt and wheeden on weak type inequalities for the hilbert transform this is closely related to the recent counterexamples of reguera scurry and thiele

estimates for operator norms on weighted spaces and reverse jensen inequalities we examine the dependence on the a p norm of w of the operator norms of singular integrals maximal functions and other operators in l p w we also examine connections between some fairly general reverse jensen inequalities and the a p and rh p weight conditions

degree based graph construction degree based graph construction is a ubiquitous problem in network modelling newman et al 2006 the structure and dynamics of networks princeton studies in complexity princeton nj princeton university press boccaletti et al 2006 phys rep 424 175 ranging from social sciences to chemical compounds and biochemical reaction networks in the cell this problem includes existence enumeration exhaustive construction and sampling questions with aspects that are still open today here we give necessary and sufficient conditions for a sequence of nonnegative integers to be realized as a simple graph s degree sequence such that a given but otherwise arbitrary set of connections from an arbitrarily given node is avoided we then use this result to present a swap free algorithm that builds all simple graphs realizing a given degree sequence in a wider context we show that our result provides a greedy construction method to build all the f factor subgraphs tutte 1952 can j math 4 314 embedded within kn sk where kn is the complete graph and sk is a star graph centred on one of the nodes

sharp weighted estimates for classical operators abstract we give a general method based on dyadic calderon zygmund theory to prove sharp one and two weight norm inequalities for some of the classical operators of harmonic analysis the hilbert and riesz transforms the beurling ahlfors operator the maximal singular integrals associated to these operators the dyadic square function and the vector valued maximal operator in the one weight case we prove the sharp dependence on the a p constant by finding the best value for the exponent p such that t f l p w c n t w a p p f l p w for the hilbert transform the riesz transforms and the beurling ahlfors operator the sharp value of p was found by petermichl and volberg 2007 2008 2002 47 48 49 their proofs used approximations by the dyadic haar shift operators bellman function techniques and two weight norm inequalities our proofs again depend on dyadic approximation but avoid bellman functions and two weight norm inequalities we instead use a recent result due to a lerner 2010 34 to estimate the oscillation of dyadic operators by applying this we get a straightforward proof of the sharp dependence on the a p constant for any operator that can be approximated by haar shift operators in particular we provide a unified approach for the hilbert and riesz transforms the beurling ahlfors operator and their corresponding maximal singular integrals dyadic paraproducts and haar multipliers furthermore we completely solve the open problem of sharp dependence for the dyadic square functions and vector valued hardy littlewood maximal function in the two weight case we use the very same techniques to prove sharp results in the scale of a p bump conditions for the singular integrals considered above we show they map l p v into l p u 1 p if the pair u v satisfies sup q u 1 p a q v 1 p b q where a b p and b b p are orlicz functions this condition is sharp furthermore this condition characterizes in the scale of these a p bump conditions the corresponding two weight norm inequality for the hardy littlewood maximal operator m and its dual i e m l p v l p u and m l p u 1 p l p v 1 p muckenhoupt and wheeden conjectured that these two inequalities for m are sufficient for the hilbert transform to be bounded from l p v into l p u thus in the scale of a p bump conditions we prove their conjecture we prove similar sharp two weight results for the dyadic square function and the vector valued maximal operator

a decoupled first second order steps technique for nonconvex nonlinear unconstrained optimization with improved complexity bounds in order to be provably convergent towards a second order stationary point optimization methods applied to nonconvex problems must necessarily exploit both first and second order information however as revealed by recent complexity analyses of some of these methods the overall effort to reach second order points is significantly larger when compared to the one of approaching first order ones on the other hand there are other algorithmic schemes initially designed with first order convergence in mind that do not appear to maintain the same first order performance when modified to take second order information into account in this paper we propose a technique that separately computes first and second order steps and that globally converges to second order stationary points it consists in better connecting the steps to be taken and the stationarity criteria potentially guaranteeing larger steps and decreases in the objective our approach is shown to lead to an improvement of the corresponding complexity bound with respect to the first order optimality tolerance while having a positive impact on the practical behavior although the applicability of our ideas is wider we focus the presentation on trust region methods with and without derivatives and motivate in both cases the interest of our strategy

accelerated methods for non convex optimization we present an accelerated gradient method for nonconvex optimization problems with lipschitz continuous first and second derivatives in a time o epsilon 7 4 log 1 epsilon the method f

solution of the two star model of a network the p star model or exponential random graph is among the oldest and best known of network models here we give an analytic solution for the particular case of the two star model which is one of the most fundamental of exponential random graphs we derive expressions for a number of quantities of interest in the model and show that the degenerate region of the parameter space observed in computer simulations is a spontaneously symmetry broken phase separated from the normal phase of the model by a conventional continuous phase transition

reducing degeneracy in maximum entropy models of networks based on jaynes s maximum entropy principle exponential random graphs provide a family of principled models that allow the prediction of network properties as constrained by empirical data observables however their use is often hindered by the degeneracy problem characterized by spontaneous symmetry breaking where predictions fail here we show that degeneracy appears when the corresponding density of states function is not log concave which is typically the consequence of nonlinear relationships between the constraining observables exploiting these nonlinear relationships here we propose a solution to the degeneracy problem for a large class of systems via transformations that render the density of states function log concave the effectiveness of the method is demonstrated on examples

equilibrium distributions in entropy driven balanced processes abstract for entropy driven balanced processes we obtain final states with poisson bernoulli negative binomial and polya distributions we apply this both for complex networks and particle production for random networks we follow the evolution of the degree distribution p n in a system where a node can activate k fixed connections from k possible partnerships among all nodes the total number of connections n is also fixed for particle physics problems p n is the probability of having n particles or other quanta distributed among k states phase space cells while altogether a fixed number of n particles reside on k states

on a general class of models for interaction this paper develops a class of probability models for configurations of interacting points in a domain the distributions depend on a function which may be viewed as giving the potential energy of the configurations examples include models for interaction in a spatial region and on lattices and graphs new models in the general class arise naturally an example being a spatial model for points of different categories some general methods including series expansions and a simulation method known from statistical mechanics can be useful for many of these models several applications of this kind are considered and some connections between the models and with the statistical mechanics are explored

a maximum entropy method for solving the boundary value problem of second order ordinary differential equations we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent

random generation of combinatorial structures from a uniform distribution abstract the class of problems involving the random generation of combinatorial structures from a uniform distribution is considered uniform generation problems are in computational difficulty intermediate between classical existence and counting problems it is shown that exactly uniform generation of efficiently verifiable combinatorial structures is reducible to approximate counting and hence is within the third level of the polynomial hierarchy natural combinatorial problems are presented which exhibit complexity gaps between their existence and generation and between their generation and counting versions it is further shown that for self reducible problems almost uniform generation and randomized approximate counting are inter reducible and hence of similar complexity

a characterization of two weight norm inequalities for fractional and poisson integrals caracterisation de deux inegalites de normes ponderees pour les integrales fractionnaires et de poisson applications aux operateurs differentiels elliptiques degeneres

perfect dyadic operators weighted t 1 theorem and two weight estimates abstract perfect dyadic operators were first introduced in 1 where a local t b theorem was proved for such operators in 3 it was shown that for every singular integral operator t with locally bounded kernel on r n r n there exists a perfect dyadic operator t such that t t is bounded on l p d x for all 1 p in this paper we show a decomposition of perfect dyadic operators on real line into four well known operators two selfadjoint operators paraproduct and its adjoint based on this decomposition we prove a sharp weighted version of the t 1 theorem for such operators which implies a 2 conjecture for such operators with constant which only depends on t 1 bmo d t 1 bmo d and the constant in testing conditions for t moreover the constant depends on these parameters at most linearly in this paper we also obtain sufficient conditions for the two weight boundedness for a perfect dyadic operator and simplify these conditions under additional assumptions that weights are in the muckenhoupt class a d

linear bound for the dyadic paraproduct on weighted lebesgue space l2 w the dyadic paraproduct is bounded in weighted lebesgue spaces lp w if and only if the weight w belongs to the muckenhoupt class apd however the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest l2 w case in this paper we prove that the bound on the norm of the dyadic paraproduct in the weighted lebesgue space l2 w depends linearly on the a2d characteristic of the weight w using bellman function techniques and extrapolate this result to the lp w case

solution for the properties of a clustered network we study strauss s model of a network with clustering and present an analytic mean field solution which is exact in the limit of large network size previous computer simulations have revealed a degenerate region in the model s parameter space in which triangles of adjacent edges clump together to form unrealistically dense subgraphs and perturbation calculations have been found to break down in this region at all orders our solution shows that this region corresponds to a classic symmetry broken phase and that the onset of the degeneracy corresponds to a first order phase transition in the density of the network

prediction of spatiotemporal patterns of neural activity from pairwise correlations we designed a model based analysis to predict the occurrence of population patterns in distributed spiking activity using a maximum entropy principle with a markovian assumption we obtain a model that accounts for both spatial and temporal pairwise correlations among neurons this model is tested on data generated with a glauber spin glass system and is shown to correctly predict the occurrence probabilities of spatiotemporal patterns significantly better than ising models only based on spatial correlations this increase of predictability was also observed on experimental data recorded in parietal cortex during slow wave sleep this approach can also be used to generate surrogates that reproduce the spatial and temporal correlations of a given data set