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Exponential-family Random Graph Models (ERGMs) constitute a large statistical framework for modeling dense and sparse random graphs with short- or long-tailed degree distributions, covariate effects and a wide range of complex dependencies. Special cases of ERGMs include network equivalents of generalized linear models (GLMs), Bernoulli random graphs, 𝛽-models, 𝑝1-models and models related to Markov random fields in spatial statistics and image processing. While ERGMs are widely used in practice, questions have been raised about their theoretical properties. These include concerns that some ERGMs are near-degenerate and that many ERGMs are non-projective. To address such questions, careful attention must be paid to model specifications and their underlying assumptions, and to the inferential settings in which models are employed. As we discuss, near-degeneracy can affect simplistic ERGMs lacking structure, but well-posed ERGMs with additional structure can be well-behaved. Likewise, lack of projectivity can affect non-likelihood-based inference, but likelihood-based inference does not require projectivity. Here, we review well-posed ERGMs along with likelihood-based inference. We first clarify the core statistical notions of \"sample\" and \"population\" in the ERGM framework, separating the process that generates the population graph from the observation process. We then review likelihood-based inference in finite, super and infinite population scenarios. We conclude with consistency results, and an application to human brain networks.

Dependent phenomena, such as relational, spatial and temporal phenomena, tend to be characterized by local dependence in the sense that units which are close in a well‐defined sense are dependent. In contrast with spatial and temporal phenomena, though, relational phenomena tend to lack a natural neighbourhood structure in the sense that it is unknown which units are close and thus dependent. Owing to the challenge of characterizing local dependence and constructing random graph models with local dependence, many conventional exponential family random graph models induce strong dependence and are not amenable to statistical inference. We take first steps to characterize local dependence in random graph models, inspired by the notion of finite neighbourhoods in spatial statistics and M‐dependence in time series, and we show that local dependence endows random graph models with desirable properties which make them amenable to statistical inference. We show that random graph models with local dependence satisfy a natural domain consistency condition which every model should satisfy, but conventional exponential family random graph models do not satisfy. In addition, we establish a central limit theorem for random graph models with local dependence, which suggests that random graph models with local dependence are amenable to statistical inference. We discuss how random graph models with local dependence can be constructed by exploiting either observed or unobserved neighbourhood structure. In the absence of observed neighbourhood structure, we take a Bayesian view and express the uncertainty about the neighbourhood structure by specifying a prior on a set of suitable neighbourhood structures. We present simulation results and applications to two real world networks with ‘ground truth’.

We propose a camera model for line-scan cameras with telecentric lenses. The camera model assumes a linear relative motion with constant velocity between the camera and the object. It allows to model lens distortions, while supporting arbitrary positions of the line sensor with respect to the optical axis. We comprehensively examine the degeneracies of the camera model and propose methods to handle them. Furthermore, we examine the relation of the proposed camera model to affine cameras. In addition, we propose an algorithm to calibrate telecentric line-scan cameras using a planar calibration object. We perform an extensive evaluation of the proposed camera model that establishes the validity and accuracy of the proposed model. We also show that even for lenses with very small lens distortions, the distortions are statistically highly significant. Therefore, they cannot be omitted in real-world applications.

We propose camera models for cameras that are equipped with lenses that can be tilted in an arbitrary direction (often called Scheimpflug optics). The proposed models are comprehensive: they can handle all tilt lens types that are in common use for machine vision and consumer cameras and correctly describe the imaging geometry of lenses for which the ray angles in object and image space differ, which is true for many lenses. Furthermore, they are versatile since they can also be used to describe the rectification geometry of a stereo image pair in which one camera is perspective and the other camera is telecentric. We also examine the degeneracies of the models and propose methods to handle the degeneracies. Furthermore, we examine the relation of the proposed camera models to different classes of projective camera matrices and show that all classes of projective cameras can be interpreted as cameras with tilt lenses in a natural manner. In addition, we propose an algorithm that can calibrate an arbitrary combination of perspective and telecentric cameras (no matter whether they are tilted or untilted). The calibration algorithm uses a planar calibration object with circular control points. It is well known that circular control points may lead to biased calibration results. We propose two efficient algorithms to remove the bias and thus obtain accurate calibration results. Finally, we perform an extensive evaluation of the proposed camera models and calibration algorithms that establishes the validity and accuracy of the proposed models.

The exponential random graph model (ERGM) plays a major role in social network analysis. However, parameter estimation for the ERGM is a hard problem due to the intractability of its normalizing constant and the model degeneracy. The existing algorithms, such as Monte Carlo maximum likelihood estimation (MCMLE) and stochastic approximation, often fail for this problem in the presence of model degeneracy. In this article, we introduce the varying truncation stochastic approximation Markov chain Monte Carlo (SAMCMC) algorithm to tackle this problem. The varying truncation mechanism enables the algorithm to choose an appropriate starting point and an appropriate gain factor sequence, and thus to produce a reasonable parameter estimate for the ERGM even in the presence of model degeneracy. The numerical results indicate that the varying truncation SAMCMC algorithm can significantly outperform the MCMLE and stochastic approximation algorithms: for degenerate ERGMs, MCMLE and stochastic approximation often fail to produce any reasonable parameter estimates, while SAMCMC can do; for nondegenerate ERGMs, SAMCMC can work as well as or better than MCMLE and stochastic approximation. The data and source codes used for this article are available online as supplementary materials.

Parameter estimation from observable or experimental data is a crucial stage in any modeling study. Identifiability refers to one’s ability to uniquely estimate the model parameters from the available data. Structural unidentifiability in dynamic models, the opposite of identifiability, is associated with the notion of degeneracy where multiple parameter sets produce the same pattern. Therefore, the inverse function of determining the model parameters from the data is not well defined. Degeneracy is not only a mathematical property of models, but it has also been reported in biological experiments. Classical studies on structural unidentifiability focused on the notion that one can at most identify combinations of unidentifiable model parameters. We have identified a different type of structural degeneracy/unidentifiability present in a family of models, which we refer to as the Lambda-Omega (Λ-Ω) models. These are an extension of the classical lambda-omega (λ-ω) models that have been used to model biological systems, and display a richer dynamic behavior and waveforms that range from sinusoidal to square wave to spike like. We show that the Λ-Ω models feature infinitely many parameter sets that produce identical stable oscillations, except possible for a phase shift (reflecting the initial phase). These degenerate parameters are not identifiable combinations of unidentifiable parameters as is the case in structural degeneracy. In fact, reducing the number of model parameters in the Λ-Ω models is minimal in the sense that each one controls a different aspect of the model dynamics and the dynamic complexity of the system would be reduced by reducing the number of parameters. We argue that the family of Λ-Ω models serves as a framework for the systematic investigation of degeneracy and identifiability in dynamic models and for the investigation of the interplay between structural and other forms of unidentifiability resulting on the lack of information from the experimental/observational data.

The codon-degeneracy model (CDM) predicts relative frequencies of substitution for any set of homologous protein-coding DNA sequences based on patterns of nucleotide degeneracy, codon composition, and the assumption of selective neutrality. However, at present, the CDM is reliant on outside estimates of transition bias. A new method by which the power of the CDM can be used to find a synonymous transition bias that is optimal for any given phylogenetic tree topology is presented. An example is illustrated that utilizes optimized transition biases to generate CDM GF-scores for every possible phylogenetic tree for pocket gophers of the genus Orthogeomys. The resulting distribution of CDM GF-scores is compared and contrasted with the results of maximum parsimony and maximum likelihood methods. Although convergence on a single tree topology by the CDM and another method indicates greater support for that particular tree, the value of CDM GF-score as the sole optimality criterion for phylogeny reconstruction remains to be determined. It is clear, however, that the a priori estimation of an optimum transition bias from codon composition has a direct application to differentiating between alternative trees.

This paper is concerned with a complete asymptotic analysis as These boundary layers, which are the main center of interest of the present paper, exhibit several types of peculiar behaviour. First, the size of the boundary layer on the western and eastern boundary, which had already been computed by several authors, becomes formally very large as one approaches northern and southern portions of the boudary, i.e. pieces of the boundary on which the normal is vertical. This phenomenon is known as geostrophic degeneracy. In order to avoid such singular behaviour, previous studies imposed restrictive assumptions on the domain Moreover, when the domain Eventually, the effect of boundary layers is non-local in several aspects. On the first hand, for algebraic reasons, the boundary layer equation is radically different on the west and east parts of the boundary. As a consequence, the Sverdrup equation is endowed with a Dirichlet condition on the East boundary, and no condition on the West boundary. Therefore western and eastern boundary layers have in fact an influence on the whole domain

We present a systematic examination of a real network data set using maximum likelihood estimation for exponential random graph models as well as new procedures to evaluate how well the models fit the observed networks. These procedures compare structural statistics of the observed network with the corresponding statistics on networks simulated from the fitted model. We apply this approach to the study of friendship relations among high school students from the National Longitudinal Study of Adolescent Health (AddHealth). We focus primarily on one particular network of 205 nodes, although we also demonstrate that this method may be applied to the largest network in the AddHealth study, with 2,209 nodes. We argue that several well-studied models in the networks literature do not fit these data well and demonstrate that the fit improves dramatically when the models include the recently developed geometrically weighted edgewise shared partner, geometrically weighted dyadic shared partner, and geometrically weighted degree network statistics. We conclude that these models capture aspects of the social structure of adolescent friendship relations not represented by previous models.

In this paper, we consider the initial Neumann boundary value problem for a degenerate kinetic model of Keller–Segel type. The system features a signal-dependent decreasing motility function that vanishes asymptotically, i.e., degeneracies may take place as the concentration of signals tends to infinity. In the present work, we are interested in the boundedness of classical solutions when the motility function satisfies certain decay rate assumptions. Roughly speaking, in the two-dimensional setting, we prove that classical solution is globally bounded if the motility function decreases slower than an exponential speed at high signal concentrations. In higher dimensions, boundedness is obtained when the motility decreases at certain algebraical speed. The proof is based on the comparison methods developed in our previous work (Fujie and Jiang in J. Differ. Equ. 269:5338–5778, 2020; Fujie and Jiang in Calc. Var. Partial Differ. Equ. 60:92, 2021) together with a modified Alikakos–Moser type iteration. Besides, new estimations involving certain weighted energies are also constructed to establish the boundedness.