This paper presents a taxonomy and overview of approaches to the measurement of graph and network complexity. The taxonomy distinguishes between deterministic (e.g., Kolmogorov complexity) and probabilistic approaches with a view to placing entropy-based probabilistic measurement in context. Entropy-based measurement is the main focus of the paper. Relationships between the different entropy functions used to measure complexity are examined; and intrinsic (e.g., classical measures) and extrinsic (e.g., Körner entropy) variants of entropy-based models are discussed in some detail.

A variety of problems in, e.g., discrete mathematics, computer science, information theory, statistics, chemistry, biology, etc., deal with inferring and characterizing relational structures by using graph measures. In this sense, it has been proven that information-theoretic quantities representing graph entropies possess useful properties such as a meaningful structural interpretation and uniqueness. As classical work, many distance-based graph entropies, e.g., the ones due to Bonchev et al. and related quantities have been proposed and studied. Our contribution is to explore graph entropies that are based on a novel information functional, which is the number of vertices with distance \(k\) to a given vertex. In particular, we investigate some properties thereof leading to a better understanding of this new information-theoretic quantity.

Regression models are a form of supervised learning methods that are important for machine learning, statistics, and general data science. Despite the fact that classical ordinary least squares (OLS) regression models have been known for a long time, in recent years there are many new developments that extend this model significantly. Above all, the least absolute shrinkage and selection operator (LASSO) model gained considerable interest. In this paper, we review general regression models with a focus on the LASSO and extensions thereof, including the adaptive LASSO, elastic net, and group LASSO. We discuss the regularization terms responsible for inducing coefficient shrinkage and variable selection leading to improved performance metrics of these regression models. This makes these modern, computational regression models valuable tools for analyzing high-dimensional problems.

Computers are incredibly fast, accurate, and stupid. Human beings are incredibly slow, inaccurate, and brilliant. Together they are powerful beyond imagination (Einstein never said that [1]).

When performing a regression or classification analysis, one needs to specify a statistical model. This model should avoid the overfitting and underfitting of data, and achieve a low generalization error that characterizes its prediction performance. In order to identify such a model, one needs to decide which model to select from candidate model families based on performance evaluations. In this paper, we review the theoretical framework of model selection and model assessment, including error-complexity curves, the bias-variance tradeoff, and learning curves for evaluating statistical models. We discuss criterion-based, step-wise selection procedures and resampling methods for model selection, whereas cross-validation provides the most simple and generic means for computationally estimating all required entities. To make the theoretical concepts transparent, we present worked examples for linear regression models. However, our conceptual presentation is extensible to more general models, as well as classification problems.

The modeling of time to event data is an important topic with many applications in diverse areas. The collective of methods to analyze such data are called survival analysis, event history analysis or duration analysis. Survival analysis is widely applicable because the definition of an ’event’ can be manifold and examples include death, graduation, purchase or bankruptcy. Hence, application areas range from medicine and sociology to marketing and economics. In this paper, we review the theoretical basics of survival analysis including estimators for survival and hazard functions. We discuss the Cox Proportional Hazard Model in detail and also approaches for testing the proportional hazard (PH) assumption. Furthermore, we discuss stratified Cox models for cases when the PH assumption does not hold. Our discussion is complemented with a worked example using the statistical programming language R to enable the practical application of the methodology.