Estimation of on- and off-time distributions in a dynamic Erdős–Rényi random graph

In a large number of microwave remote sensing problems, accurate knowledge of the dielectric constant of natural media such as vegetation and soil is crucial in the estimation of the desired biophysical parameters from the measured data. In the conventional cavity perturbation method, a sample of the test material with dimensions much smaller than the wavelength is placed in a high-quality-factor microwave cavity, and then the resonant frequency and the quality factor of the loaded cavity are measured. Comparing the resonant characteristics of the empty (unloaded and loaded) cavity, one can estimate the real and imaginary parts of the dielectric constant from analytical expressions based on the perturbation theory. Besides practical limitations such as the cavity and sample geometry and size, a major drawback of the cavity perturbation method is its lack of accuracy in measuring the imaginary part of the dielectric constant of low-loss dielectric materials. The change in the quality factor of the cavity is proportional to the total dissipated power in the dielectric sample. When both the dielectric loss tangent and the sample volume are very small, the change in the quality factor becomes extremely small and falls within the measurement errors. To circumvent this difficulty, larger pieces of the material with sizes well beyond the limits of the perturbation theory are needed so that the resulting shift in the cavity characteristics can be measured accurately. In this paper, the authors extend the resonant cavity technique for dielectric constant measurement beyond the limitations of the perturbation method. The new approach incorporates a full-wave simulation of the loaded cavity structure into the inverse measurement problem. Thus, the size of the material sample can be taken arbitrarily large as long as it does not disturb the coupling mechanism of the resonator. For the forward measurement problem, an integral formulation of the related boundary value problem is developed and solved numerically using the method of moments (MoM). The MoM results are validated independently by comparing to the results based on the finite element method (FEM). For the inverse measurement problem, a numerically efficient inversion algorithm based on the Eigen-Analysis of the impedance matrix is employed. In this algorithm the dependence of the complex dielectric constant on the inverse of the impedance matrix is made explicit, thereby establishing simple polynomials relationships between the dielectric constant and the resonant characteristics of the cavity. Both forward and inverse problems are illustrated through simulation examples.

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We consider random graphs with a given degree sequence and show, under weak technical conditions, asymptotic normality of the number of components isomorphic to a given tree, first for the random multigraph given by the configuration model and then, by a conditioning argument, for the simple uniform random graph with the given degree sequence. Such conditioning is standard for convergence in probability, but much less straightforward for convergence in distribution as here. The proof uses the method of moments, and is based on a new estimate of mixed cumulants in a case of weakly dependent variables. The result on small components is applied to give a new proof of a recent result by Barbour and Röllin on asymptotic normality of the size of the giant component in the random multigraph; moreover, we extend this to the random simple graph.

Random key graphs are random graphs induced by the random key predistribution scheme of Es-chenauer and Gligor under the assumption of full visibility. For this class of random graphs we show the existence of a zero-one law for the appearance of triangles, and identify the corresponding critical scaling. This is done by applying the method of first and second moments to the number of triangles in the graph.

The theory of graphons comes with a natural sampling procedure, which results in an inhomogeneous variant of the Erdős–Rényi random graph, called ‐random graphs. We prove, via the method of moments, a limit theorem for the number of ‐cliques in such random graphs. We show that, whereas in the case of dense Erdős–Rényi random graphs the fluctuations are normal of order , the fluctuations in the setting of ‐random graphs may be of order , or . Furthermore, when the fluctuations are of order they are normal, while when the fluctuations are of order they exhibit either normal or a particular type of chi‐square behavior whose parameters relate to spectral properties of . These results can also be deduced from a general setting, based on the projection method. In addition to providing alternative proofs, our approach makes direct links to the theory of graphons.

Microwave tomography is an inverse scattering problem which is formalized by determines the position and dielectric properties distribution of the unknown object from the measured scattered field. The quantitative image obtained from microwave tomography approach provides information directly correlated to the internal structure and composition of the examined object, which is necessity in biomedical and geo-surveying applications. Recently, studies on microwave imaging (MWI) for early breast cancer detection have attracted attentions. The breast cancer detection based on microwave tomography relies on large differences in dielectric properties between healthy and malignant tissues. In MWI, the examined object is successively illuminated by transmitting antenna and the resulting electromagnetic field is measured by the receiving antennas. The image reconstruction process in microwave tomography involves forward problem and inverse problem. For the forward problem, Method of Moment (MoM) is used to obtain the measurement data, which are related to the scattered field resulting from the interaction between the known incident field and the scatterers inside the imaging region. The measurement data is then used in the inverse problem, where Distorted Born Iterative Method (DBIM) is proposed to reconstruct the dielectric properties distribution of the object by solving the non-linear inverse scattering problem. The simulation results show that the ill-posedness of the non-linear problem can be reduced and a more stable solution can be performed by choosing the appropriate solving techniques of the system of linear equations.

Practical applications of mechanical metamaterials often involve solving inverse problems aimed at finding microarchitectures that give rise to certain properties. The limited resolution of additive manufacturing techniques often requires solving such inverse problems for specific specimen sizes. Moreover, the candidate microarchitectures should be resistant to fatigue and fracture. Such a multi-objective inverse design problem is formidably difficult to solve but its solution is the key to real-world applications of mechanical metamaterials. Here, a modular approach titled "Deep-DRAM" that combines four decoupled models is proposed, including two deep learning (DL) models, a deep generative model based on conditional variational autoencoders, and direct finite element (FE) simulations. Deep-DRAM integrates these models into a framework capable of finding many solutions to the posed multi-objective inverse design problem based on random-network unit cells. Using an extensive set of simulations as well as experiments performed on 3D printed specimens, it is demonstrate that: 1) the predictions of the DL models are in agreement with FE simulations and experimental observations, 2) an enlarged envelope of achievable elastic properties (e.g., rare combinations of double auxeticity and high stiffness) is realized using the proposed approach, and 3) Deep-DRAM can provide many solutions to the considered multi-objective inverse design problem.

We study a process of opinion formation in a population of agents whose interaction pattern is defined on the basis of randomly distributed groups of three agents (triplets), in contrast to networks, which are defined on the basis of agent pairs. Results for the time needed to reach full consensus are compared between a triplet-based structure with a given number of triplets and a random network with the same number of triangles. The full-consensus time in the triplet structure is systematically lower than in the network. This discrepancy can be ascribed to differences in the shape of the probability distribution for the number of triplets and triangles per agent in each interaction pattern.

We analyze ensembles of random networks with fixed numbers of edges, triangles, and nodes. In the limit as the number of nodes goes to infinity, this model is known to exhibit sharp phase transitions as the density of edges and triangles is varied. In this paper we study finite size effects in ensembles where the number of nodes is between 30 and 66. Despite the small number of nodes, the phases and phase transitions are clearly visible. Concentrating on 54 nodes and one particular phase transition, we use spectral data to examine the structure of a typical graph in each phase, and see that is it very similar to the graphon that describes the system as n diverges. We further investigate how graphs change their structure from that of one phase to the other under a natural edge flip dynamics in which the number of triangles is either forced downwards or allowed to drift upwards. In each direction, spectral data shows that the change occurs in three stages: a very fast stage of about 100 edge flips, in which the triangle count reaches the targeted value, a slower second stage of several thousand edge flips in which the graph adopts the qualitative features of the new phase, and a very slow third stage, requiring hundreds of thousands of edge flips, in which the fine details of the structure equilibrate.

In this paper we study inverse problems where observations are corrupted by uniform noise. By using maximum a posteriori approach, an $$L_\infty $$ -norm constrained minimization problem can be formulated for uniform noise removal. The main difficulty of solving such minimization problem is how to deal with non-differentiability of the $$L_\infty $$ -norm constraint and how to estimate the level of uniform noise. The main contribution of this paper is to develop an efficient semi-smooth Newton method for solving this minimization problem. Here the $$L_\infty $$ -norm constraint can be handled by active set constraints arising from the optimality conditions. In the proposed method, linear systems based on active set constraints are required to solve in each Newton step. We also employ the method of moments (MoM) to estimate the level of uniform noise for the minimization problem. The combination of the proposed method and MoM is quite effective for solving inverse problems with uniform noise. Numerical examples are given to demonstrate that our proposed method outperforms the other testing methods.

We prove new lower bounds on the likely size of the maximum independent set in a random graph with a given constant average degree. Our method is a weighted version of the second moment method, where we give each independent set a weight based on the total degree of its vertices.KeywordsRandom GraphTotal DegreeMoment MethodWeighted VersionSatisfying AssignmentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let $f_{cut}(n,cn)$ be the expectation of the value of the maximum cut of a given random graph G(n,cn) with n vertices and cn edges. We study the asymptotic change of $f_{cut}(n,cn)/cn$ as n tends to infinity with various densities c. In this paper, we provide new upper bounds by correcting the error items when applying the first moment method. Specifically, we extrapolate the region of c from c>1.386 to c>1.001. Keywords: MAX CUT, Upper Bound, Random Graph, First moment method.

SummarySample covariance matrices play a central role in numerous popular statistical methodologies, for example principal components analysis, Kalman filtering and independent component analysis. However, modern random matrix theory indicates that, when the dimension of a random vector is not negligible with respect to the sample size, the sample covariance matrix demonstrates significant deviations from the underlying population covariance matrix. There is an urgent need to develop new estimation tools in such cases with high‐dimensional data to recover the characteristics of the population covariance matrix from the observed sample covariance matrix. We propose a novel solution to this problem based on the method of moments. When the parametric dimension of the population spectrum is finite and known, we prove that the proposed estimator is strongly consistent and asymptotically Gaussian. Otherwise, we combine the first estimation method with a cross‐validation procedure to select the unknown model dimension. Simulation experiments demonstrate the consistency of the proposed procedure. We also indicate possible extensions of the proposed estimator to the case where the population spectrum has a density.

We provide precise asymptotic estimates for the number of several classes of labeled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky and coworkers. We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact probability of a random cubic planar graph being connected, and we show that the distribution of the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance. To the best of our knowledge, this is the first time one is able to determine the asymptotic distribution for the number of copies of a fixed graph containing a cycle in classes of random planar graphs arising from planar maps.

When solving ill-posed inverse problems, a good choice of the prior is critical for the computation of a reasonable solution. A common approach is to include a Gaussian prior, which is defined by a mean vector and a symmetric and positive definite covariance matrix, and to use iterative projection methods to solve the corresponding regularized problem. However, a main challenge for many of these iterative methods is that the prior covariance matrix must be known and fixed (up to a constant) before starting the solution process. In this paper, we develop hybrid projection methods for inverse problems with mixed Gaussian priors where the prior covariance matrix is a convex combination of matrices and the mixing parameter and the regularization parameter do not need to be known in advance. Such scenarios may arise when data is used to generate a sample prior covariance matrix (e.g., in data assimilation) or when different priors are needed to capture different qualities of the solution. The proposed hybrid methods are based on a mixed Golub–Kahan process, which is an extension of the generalized Golub–Kahan bidiagonalization, and a distinctive feature of the proposed approach is that both the regularization parameter and the weighting parameter for the covariance matrix can be estimated automatically during the iterative process. Furthermore, for problems where training data are available, various data-driven covariance matrices (including those based on learned covariance kernels) can be easily incorporated. Numerical examples from tomographic reconstruction demonstrate the potential for these methods.