network geometry with flavor from complexity to quantum geometry network geometry is attracting increasing attention because it has a wide range of applications ranging from data mining to routing protocols in the internet at the same time advances in the understanding of the geometrical properties of networks are essential for further progress in quantum gravity in network geometry simplicial complexes describing the interaction between two or more nodes play a special role in fact these structures can be used to discretize a geometrical d dimensional space and for this reason they have already been widely used in quantum gravity here we introduce the network geometry with flavor s 1 0 1 ngf describing simplicial complexes defined in arbitrary dimension d and evolving by a nonequilibrium dynamics the ngf can generate discrete geometries of different natures ranging from chains and higher dimensional manifolds to scale free networks with small world properties scale free degree distribution and nontrivial community structure the ngf admits as limiting cases both the bianconi barabasi models for complex networks the stochastic apollonian network and the recently introduced model for complex quantum network manifolds the thermodynamic properties of ngf reveal that ngf obeys a generalized area law opening a new scenario for formulating its coarse grained limit the structure of ngf is strongly dependent on the dimensionality d in d 1 ngfs grow complex networks for which the preferential attachment mechanism is necessary in order to obtain a scale free degree distribution instead for ngf with dimension d 1 it is not necessary to have an explicit preferential attachment rule to generate scale free topologies we also show that ngf admits a quantum mechanical description in terms of associated quantum network states quantum network states evolve by a markovian dynamics and a quantum network state at time t encodes all possible ngf evolutions up to time t interestingly the ngf remains fully classical but its statistical properties reveal the relation to its quantum mechanical description in fact the dimensional faces of the ngf have generalized degrees that follow either the fermi dirac boltzmann or bose einstein statistics depending on the flavor s and the dimensions d and

clustering in complex networks i general formalism we develop a full theoretical approach to clustering in complex networks a key concept is introduced the edge multiplicity that measures the number of triangles passing through an edge this quantity extends the clustering coefficient in that it involves the properties of two and not just one vertices the formalism is completed with the definition of a three vertex correlation function which is the fundamental quantity describing the properties of clustered networks the formalism suggests different metrics that are able to thoroughly characterize transitive relations a rigorous analysis of several real networks which makes use of this formalism and the metrics is also provided it is also found that clustered networks can be classified into two main groups the weak and the strong transitivity classes in the first class edge multiplicity is small with triangles being disjoint in the second class edge multiplicity is high and so triangles share many edges as we shall see in the following paper the class a network belongs to has strong implications in its percolation properties

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

a probabilistic model for the nonlinear electromagnetic inverse scattering tm case electromagnetic inverse scattering emis is a noninvasive examination tool which holds the promising potential in science engineering and military applications in contrast to conventional tomography techniques the inverse scattering is a quantitative superresolution imaging method since it is capable of accommodating more realistic interactions between the wavefield and the probed scene in this paper a full probabilistic formulation of the emis is presented for the first time which is then solved by applying the well known expectation maximization method afterward the concept of the complex valued alternating direction method of multipliers has been proposed as an alternative approach to solve the resulting nonlinear optimization problem finally exemplary numerical and experimental results are provided to validate the proposed method

a characterization of certain griesmer codes mmd codes in a more general sense let c be an n k d sub q linear code the defect of c is the parameter s s c n k 1 d if k spl ges m 1 spl ges 2 then by the griesmer bound d spl les q sup m q 1 q sub m 1 s m the author s interest is in those linear codes having the maximum minimum distance i e d q sup m q 1 q sup m 1 s m for m 1 we have d q s 1 and the codes are maximum minimum distance mmd codes in the sense of faldum and willems see ibid vol 44 p 1555 58 1998 thus we consider mmd codes in a more general sense we refer to them simply as mmd codes all mmd codes with m 1 are known up to formal equivalence note that two codes are formally equivalent if they have the same weight distribution the author classifies up to formal equivalence the mmd codes with m spl ges 2

the duals of mmd codes are proper for error detection a linear code when used for error detection on a symmetric channel is said to be proper if the corresponding undetected error probability increases monotonically in spl epsiv the symbol error probability of the channel such codes are generally considered to perform well in error detection a number of well known classes of linear codes are proper e g the perfect codes mds codes macdonald s codes mmd codes and some near mds codes the aim of this work is to show that also the duals of mmd codes are proper

emergence and size of the giant component in clustered random graphs with a given degree distribution standard techniques for analyzing network models usually break down in the presence of clustering here we introduce a new analytic tool the free excess degree distribution which extends the generating function framework making it applicable for clustered networks c 0 the methodology is general and provides a new expression for the threshold point at which the giant component emerges and shows that it scales as 1 c 1 in addition the size of the giant component may be predicted even for more complicated scenarios such as the removal of a fixed fraction of nodes at random

a generalized gaussian image model for edge preserving map estimation the authors present a markov random field model which allows realistic edge modeling while providing stable maximum a posterior map solutions the model referred to as a generalized gaussian markov random field ggmrf is named for its similarity to the generalized gaussian distribution used in robust detection and estimation the model satisfies several desirable analytical and computational properties for map estimation including continuous dependence of the estimate on the data invariance of the character of solutions to scaling of data and a solution which lies at the unique global minimum of the a posteriori log likelihood function the ggmrf is demonstrated to be useful for image reconstruction in low dosage transmission tomography

unsupervised bayesian convex deconvolution based on a field with an explicit partition function this paper proposes a non gaussian markov field with a special feature an explicit partition function to the best of our knowledge this is an original contribution moreover the explicit expression of the partition function enables the development of an unsupervised edge preserving convex deconvolution method the method is fully bayesian and produces an estimate in the sense of the posterior mean numerically calculated by means of a monte carlo markov chain technique the approach is particularly effective and the computational practicability of the method is shown on a simple simulated example

bayesian and regularization methods for hyperparameter estimation in image restoration in this paper we propose the application of the hierarchical bayesian paradigm to the image restoration problem we derive expressions for the iterative evaluation of the two hyperparameters applying the evidence and maximum a posteriori map analysis within the hierarchical bayesian paradigm we show analytically that the analysis provided by the evidence approach is more realistic and appropriate than the map approach for the image restoration problem we furthermore study the relationship between the evidence and an iterative approach resulting from the set theoretic regularization approach for estimating the two hyperparameters or their ratio defined as the regularization parameter finally the proposed algorithms are tested experimentally

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

ml parameter estimation for markov random fields with applications to bayesian tomography markov random fields mrfs have been widely used to model images in bayesian frameworks for image reconstruction and restoration typically these mrf models have parameters that allow the prior model to be adjusted for best performance however optimal estimation of these parameters sometimes referred to as hyperparameters is difficult in practice for two reasons i direct parameter estimation for mrfs is known to be mathematically and numerically challenging ii parameters can not be directly estimated because the true image cross section is unavailable we propose a computationally efficient scheme to address both these difficulties for a general class of mrf models and we derive specific methods of parameter estimation for the mrf model known as generalized gaussian mrf ggmrf we derive methods of direct estimation of scale and shape parameters for a general continuously valued mrf for the ggmrf case we show that the ml estimate of the scale parameter spl sigma has a simple closed form solution and we present an efficient scheme for computing the ml estimate of the shape parameter p by an off line numerical computation of the dependence of the partition function on p we present a fast algorithm for computing ml parameter estimates when the true image is unavailable to do this we use the expectation maximization em algorithm we develop a fast simulation method to replace the e step and a method to improve the parameter estimates when the simulations are terminated prior to convergence experimental results indicate that our fast algorithms substantially reduce the computation and result in good scale estimates for real tomographic data sets

communities clustering phase transitions and hysteresis pitfalls in constructing network ensembles ensembles of networks are used as null models in many applications however simple null models often show much less clustering than their real world counterparts in this paper we study a biased rewiring model where clustering is enhanced by means of a fugacity as in the strauss or triangle model but where the number of links attached to each node is strictly preserved similar models have been proposed previously in milo et al science 298 824 2002 our model exhibits phase transitions as the fugacity is changed for regular graphs identical degrees for all nodes with degree kg2 we find a single first order transition for all nonregular networks that we studied including erd os r enyi scale free and several real world networks multiple jumps resembling first order transitions appear the jumps coincide with the sudden emergence of cluster cores groups of highly interconnected nodes with higher than average degrees where each edge participates in many triangles hence clustering is not smoothly distributed throughout the network once formed the cluster cores are difficult to remove leading to strong hysteresis to study the cluster cores visually we introduce q clique adjacency plots cluster cores constitute robust communities that emerge spontaneously from the triangle generating process rather than being put explicitly into the definition of the model all the quantities we measured including the modularity assortativity clustering and number of four and five cliques exhibit simultaneous jumps and are equivalent order parameters finally we point out that cluster cores produce pitfalls when using the present and similar models as null models for strongly clustered networks due to strong hysteresis which leads to broken ergodicity on realistic sampling time scales

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

deterministic edge preserving regularization in computed imaging many image processing problems are ill posed and must be regularized usually a roughness penalty is imposed on the solution the difficulty is to avoid the smoothing of edges which are very important attributes of the image in this paper we first give conditions for the design of such an edge preserving regularization under these conditions we show that it is possible to introduce an auxiliary variable whose role is twofold first it marks the discontinuities and ensures their preservation from smoothing second it makes the criterion half quadratic the optimization is then easier we propose a deterministic strategy based on alternate minimizations on the image and the auxiliary variable this leads to the definition of an original reconstruction algorithm called artur some theoretical properties of artur are discussed experimental results illustrate the behavior of the algorithm these results are shown in the field of 2d single photon emission tomography but this method can be applied in a large number of applications in image processing

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

generalized network structures the configuration model and the canonical ensemble of simplicial complexes simplicial complexes are generalized network structures able to encode interactions occurring between more than two nodes simplicial complexes describe a large variety of complex interacting systems ranging from brain networks to social and collaboration networks here we characterize the structure of simplicial complexes using their generalized degrees that capture fundamental properties of one two three or more linked nodes moreover we introduce the configuration model and the canonical ensemble of simplicial complexes enforcing respectively the sequence of generalized degrees of the nodes and the sequence of the expected generalized degrees of the nodes we evaluate the entropy of these ensembles finding the asymptotic expression for the number of simplicial complexes in the configuration model we provide the algorithms for the construction of simplicial complexes belonging to the configuration model and the canonical ensemble of simplicial complexes we give an expression for the structural cutoff of simplicial complexes that for simplicial complexes of dimension d 1 reduces to the structural cutoff of simple networks finally we provide a numerical analysis of the natural correlations emerging in the configuration model of simplicial complexes without structural cutoff

random networks with tunable degree distribution and clustering we present an algorithm for generating random networks with arbitrary degree distribution and clustering frequency of triadic closure we use this algorithm to generate networks with exponential power law and poisson degree distributions with variable levels of clustering such networks may be used as models of social networks and as a testable null hypothesis about network structure finally we explore the effects of clustering on the point of the phase transition where a giant component forms in a random network and on the size of the giant component some analysis of these effects is presented

properties of highly clustered networks we propose and solve exactly a model of a network that has both a tunable degree distribution and a tunable clustering coefficient among other things our results indicate that increased clustering leads to a decrease in the size of the giant component of the network we also study susceptible infective recovered type epidemic processes within the model and find that clustering decreases the size of epidemics but also decreases the epidemic threshold making it easier for diseases to spread in addition clustering causes epidemics to saturate sooner meaning that they infect a near maximal fraction of the network for quite low transmission rates

statistical regularization in linearized microwave imaging through mrf based map estimation hyperparameter estimation and image computation the application of a markov random fields mrf based maximum a posteriori map estimation method for microwave imaging is presented in this paper the adopted mrf family is the so called gaussian mrf gmrf whose energy function is quadratic in order to implement the map estimation first the mrf hyperparameters are estimated by means of the expectation maximization em algorithm extended in this case to complex and nonhomogeneous images then it is implemented by minimizing a cost function whose gradient is fully analytically evaluated thanks to the quadratic nature of the energy function of the mrf well posedness and efficiency of the proposed method can be simultaneously guaranteed numerical results also performed on real data show the good performance of the method also when compared with conventional techniques like tikhonov regularization

kinetics of ring formation we study reversible polymerization of rings in this stochastic process two monomers bond and as a consequence two disjoint rings may merge into a compound ring or a single ring may split into two fragment rings this aggregation fragmentation process exhibits a percolation transition with a finite ring phase in which all rings have microscopic length and a giant ring phase where macroscopic rings account for a finite fraction of the entire mass interestingly while the total mass of the giant rings is a deterministic quantity their total number and their sizes are stochastic quantities the size distribution of the macroscopic rings is universal although the span of this distribution increases with time moreover the average number of giant rings scales logarithmically with system size we introduce a card shuffling algorithm for efficient simulation of the ring formation process and we present numerical verification of the theoretical predictions

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

inversion of large support ill posed linear operators using a piecewise gaussian mrf we propose a method for the reconstruction of signals and images observed partially through a linear operator with a large support e g a fourier transform on a sparse set this inverse problem is ill posed and we resolve it by incorporating the prior information that the reconstructed objects are composed of smooth regions separated by sharp transitions this feature is modeled by a piecewise gaussian pg markov random field mrf known also as the weak string in one dimension and the weak membrane in two dimensions the reconstruction is defined as the maximum a posteriori estimate the prerequisite for the use of such a prior is the success of the optimization stage the posterior energy corresponding to a pg mrf is generally multimodal and its minimization is particularly problematic in this context general forms of simulated annealing rapidly become intractable when the observation operator extends over a large support global optimization is dealt with by extending the graduated nonconvexity gnc algorithm to ill posed linear inverse problems gnc has been pioneered by blake and zisserman 1987 in the field of image segmentation the resulting algorithm is mathematically suboptimal but it is seen to be very efficient in practice we show that the original gnc does not correctly apply to ill posed problems our extension is based on a proper theoretical analysis which provides further insight into the gnc the performance of the proposed algorithm is corroborated by a synthetic example in the area of diffraction tomography

trophic coherence determines food web stability why are large complex ecosystems stable both theory and simulations of current models predict the onset of instability with growing size and complexity so for decades it has been conjectured that ecosystems must have some unidentified structural property exempting them from this outcome we show that trophic coherence a hitherto ignored feature of food webs that current structural models fail to reproduce is a better statistical predictor of linear stability than size or complexity furthermore we prove that a maximally coherent network with constant interaction strengths will always be linearly stable we also propose a simple model that by correctly capturing the trophic coherence of food webs accurately reproduces their stability and other basic structural features most remarkably our model shows that stability can increase with size and complexity this suggests a key to may s paradox and a range of opportunities and concerns for biodiversity conservation

a characterization of mmd codes let c be a linear n k d code over gf q with k spl ges 2 if s n k 1 d denotes the defect of c then by the griesmer bound d spl les s 1 q now for obvious reasons we are interested in codes of given defect s for which the minimum distance is maximal i e d s 1 q we classify up to formal equivalence all such linear codes over gf q remember that two codes over gf q are formally equivalent if they have the same weight distribution it turns out that for k spl ges 3 such codes exist only in dimension 3 and 4 with the ternary extended golay code the ternary dual golay code and the binary even weight code as exceptions in dimension 4 they are related to ovoids in pg 3 q except the binary extended hamming code and in dimension 3 to maximal arcs in pg 2 q

entropic origin of disassortativity in complex networks why are most empirical networks with the prominent exception of social ones generically degree degree anticorrelated to answer this long standing question we define the ensemble of correlated networks and obtain the associated shannon entropy maximum entropy can correspond to either assortative correlated or disassortative anticorrelated configurations but in the case of highly heterogeneous scale free networks a certain disassortativity is predicted offering a parsimonious explanation for the question above our approach provides a neutral model from which in the absence of further knowledge regarding network evolution one can obtain the expected value of correlations when empirical observations deviate from the neutral predictions as happens for social networks one can then infer that there are specific correlating mechanisms at work

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

clustering in complex networks ii percolation properties the percolation properties of clustered networks are analyzed in detail in the case of weak clustering we present an analytical approach that allows us to find the critical threshold and the size of the giant component numerical simulations confirm the accuracy of our results in more general terms we show that weak clustering hinders the onset of the giant component whereas strong clustering favors its appearance this is a direct consequence of the differences in the k core structure of the networks which are found to be totally different depending on the level of clustering an empirical analysis of a real social network confirms our predictions