Likelihood ratio tests in random graph models with increasing dimensions

We explore the Wilks phenomena in two random graph models: the $\beta$-model and the Bradley-Terry model. For two increasing dimensional null hypotheses, including a specified null $H_0: \beta_i=\beta_i^0$ for $i=1,\ldots, r$ and a homogenous null $H_0: \beta_1=\cdots=\beta_r$, we reveal high dimensional Wilks' phenomena that the normalized log-likelihood ratio statistic, $[2\{\ell(\widehat{\mathbf{\beta}}) - \ell(\widehat{\mathbf{\beta}}^0)\} -r]/(2r)^{1/2}$, converges in distribution to the standard normal distribution as $r$ goes to infinity. Here, $\ell( \mathbf{\beta})$ is the log-likelihood function on the model parameter $\mathbf{\beta}=(\beta_1, \ldots, \beta_n)^\top$, $\widehat{\mathbf{\beta}}$ is its maximum likelihood estimator (MLE) under the full parameter space, and $\widehat{\mathbf{\beta}}^0$ is the restricted MLE under the null parameter space. For the homogenous null with a fixed $r$, we establish Wilks-type theorems that $2\{\ell(\widehat{\mathbf{\beta}}) - \ell(\widehat{\mathbf{\beta}}^0)\}$ converges in distribution to a chi-square distribution with $r-1$ degrees of freedom, as the total number of parameters, $n$, goes to infinity. When testing the fixed dimensional specified null, we find that its asymptotic null distribution is a chi-square distribution in the $\beta$-model. However, unexpectedly, this is not true in the Bradley-Terry model. By developing several novel technical methods for asymptotic expansion, we explore Wilks type results in a principled manner; these principled methods should be applicable to a class of random graph models beyond the $\beta$-model and the Bradley-Terry model. Simulation studies and real network data applications further demonstrate the theoretical results.Comment: This paper supersedes arxiv article arXiv:2211.10055 titled "Wilks' theorems in the $\beta$-model" by T. Yan, Y. Zhang, J. Xu, Y. Yang and J. Zh

A sustainability assessment of five major food crops' water footprints in China from 1978 to 2010

Rice, wheat, corn, soybeans, and sorghum are the five major crops in China, which account for 92% of the country's total grain production and 33% of its water consumption. Combining water footprint analysis tools with sustainability assessment tools, the water sustainability of the five major crops can be analyzed. Based on ecological economics theory, this paper constructs a sustainability evaluation system of China's five major crops' water footprints and analyzes the national and provincial diversity of the sustainability of the five major crops' water footprints using three dimensions: scale, distribution, and equity. We find that the interprovincial distribution equity sustainability divergence is the key bottleneck factor that restricts sustainability (more than scale and configuration). One key strategy is to arrange grain production at the national level, on the basis of considering the differences of water-resource endowment between different provinces to break through the bottleneck of the water-resource distribution sustainability of these five major food crops. This paper determines a general management model that can improve the sustainability of water resource management at the interprovincial level by comparing and analyzing the most sustainable and least sustainable provinces for the water footprint production of these five major crops

Simultaneous profiling of transcriptome and DNA methylome from a single cell.

BackgroundSingle-cell transcriptome and single-cell methylome technologies have become powerful tools to study RNA and DNA methylation profiles of single cells at a genome-wide scale. A major challenge has been to understand the direct correlation of DNA methylation and gene expression within single-cells. Due to large cell-to-cell variability and the lack of direct measurements of transcriptome and methylome of the same cell, the association is still unclear.ResultsHere, we describe a novel method (scMT-seq) that simultaneously profiles both DNA methylome and transcriptome from the same cell. In sensory neurons, we consistently identify transcriptome and methylome heterogeneity among single cells but the majority of the expression variance is not explained by proximal promoter methylation, with the exception of genes that do not contain CpG islands. By contrast, gene body methylation is positively associated with gene expression for only those genes that contain a CpG island promoter. Furthermore, using single nucleotide polymorphism patterns from our hybrid mouse model, we also find positive correlation of allelic gene body methylation with allelic expression.ConclusionsOur method can be used to detect transcriptome, methylome, and single nucleotide polymorphism information within single cells to dissect the mechanisms of epigenetic gene regulation

Exact Calculation of Ring Diagrams and the Off-shell Effect on the Equation of State

The partition function with ring diagrams at finite temperature is exactly caluclated by using contour integrals in the complex energy plane. It contains a pole part with temperature and momentum dependent mass and a phase shift part induced by off-shell effect in hot medium. The thermodynamic potentials for $\phi^4$ and $\phi^3$ interactions are calculated and compared with the quasi-particle (pole) approximation. It is found that the off-shell effect on the equation of state is remarkable.Comment: 7 pages, 11 figures, refereces added, final version to appear in PR

Synthesis, Characterization, and Photocatalysis of Well-Dispersible Phase-Pure Anatase TiO 2

High-purity anatase TiO2 nanoparticles were prepared using an improved sol-hydrothermal method. The as-prepared sample was characterized by X-ray diffraction (XRD), transmission electron microscopy (TEM), Brunauer-Emmett-Teller (BET), and UV-vis diffuse reflectance spectra. TEM results showed that the average particle size of all TiO2 particles was calculated to be (10 ± 1) nm. The XRD analysis indicated that the present sample was fully crystallized and appeared to be highly phase-pure anatase. The BET analysis showed that the as-prepared sample had a very large specific surface area of 186.25 m2/g. The photocatalytic performance of TiO2 nanoparticles was evaluated by photocatalytic degradation of X-3B and X-BR solutions. The degradation results revealed that the as-prepared TiO2 showed slightly higher photocatalytic activities than P25. Whereas, the as-synthesized TiO2 can settle down and be separated easily after the photocatalytic reaction finishes