Topological graph theory studies graphs in relation to the surfaces on which they can be drawn. This review presents the main classical ideas of the field in a clear and connected framework. It begins with graph embeddings, Euler’s formula, and the basic principles of planar graphs. It then discusses important classical results, including the theorems of Kuratowski, Whitney, Mac Lane, and Wagner, and explains their role in understanding planarity and graph structure. The review also introduces graph embeddings on more general surfaces and examines key concepts such as genus, rotation systems, dual graphs, and surface colouring. Further topics include maximum genus, genus distributions, excluded minors, and algorithms for deciding whether a graph can be embedded on a fixed surface. Particular attention is given to the relationships among these concepts. Euler-type arguments provide useful numerical restrictions, while subdivisions and graph minors give structural characterisations of embeddability. Rotation systems offer a combinatorial way to describe embeddings, whereas genus-related parameters measure different aspects of their topological complexity. Overall, the review provides an accessible account of the classical foundations of topological graph theory and shows how its major results form a unified framework for studying graphs on surfaces. The historical development of the Heawood map-colouring programme and its relation to complete-graph embeddings is treated explicitly, including the Klein bottle exception and the sphere as the Four Colour Theorem case. Recent work is used to show how classical Kuratowski-type and bounded-genus questions remain active. The final synthesis separates established results from continuing directions in obstruction enumeration, representativity, embedding distributions, and near-planar graph drawing.
Background: Complete childhood immunisation in Nigeria remains uneven across maternal, household, and healthcare contexts. This study identified determinants of complete childhood immunisation and compared classical and Bayesian hierarchical logistic regression models using the 2023–24 Nigeria Demographic and Health Survey. Methods: Weighted descriptive statistics, survey-adjusted chi-square tests, survey-weighted logistic regression, multilevel modelling, and model-adequacy assessments were applied to children aged 12–23 months with complete study information. Results: Overall, 58.0% of children were fully immunised, 32.0% were partially immunised, and 10.0% were unvaccinated. Maternal age, secondary and higher education, urban residence, household wealth, at least four antenatal care visits, and health-facility delivery were positively associated with complete immunisation. Four or more antenatal care visits showed the largest adjusted association (adjusted odds ratio = 2.04; 95% confidence interval: 1.79–2.34). The multilevel model indicated meaningful community-level variation, with an intraclass correlation coefficient of 0.126. Classical and Bayesian estimates were substantively similar; however, the Bayesian model showed slightly better discrimination, calibration, and predictive accuracy, with a higher area under the receiver operating characteristic curve and a lower Brier score. Conclusion: Childhood immunisation in Nigeria is associated with maternal education, household resources, place of residence, and engagement with maternal healthcare services. Bayesian hierarchical modelling provides a useful framework for accounting for clustered survey data and quantifying uncertainty while supporting the identification of underserved populations.
Machine-learning intrusion detectors are typically validated once at deployment and then trusted indefinitely. This is unsafe because the traffic distribution used for training differs from what deployed detectors encounter, with the gap widening as adversaries introduce unseen techniques. We formalise this degradation as model aging and develop a statistical decision framework that identifies, without ground-truth labels, when a cybersecurity model becomes unreliable and when retraining is economically justified. The framework combines the Population Stability Index, Kullback–Leibler and Jensen–Shannon divergences, Wasserstein-1 distance, the two-sample Kolmogorov–Smirnov statistic, and sequential ADWIN, Page–Hinkley, and CUSUM procedures with two bounded state variables: the Model Aging Index (MAI), which accumulates null-calibrated distributional stress under exponential forgetting, and the Cybersecurity Drift Index (CDI), which incorporates the asymmetric costs of missed intrusions and false alerts. We prove that both indices are bounded, monotone, and Lipschitz, while their stationary exceedance probability admits an explicit bound for certified false-alarm control. The underlying marginal transport statistic also lower-bounds the joint Wasserstein radius, establishing alarm soundness. Retraining is formulated as an optimal-stopping problem, yielding a control-limit policy analogous to an inventory replenishment threshold. Experiments on NSL-KDD introduce 17 genuinely novel attack types across 100 batches, five drift regimes, and seven architectures. Novel-family drift reduces attack recall from 0.99 to 0.57–0.66. The CDI controller cuts expected cost by 44.3% versus no retraining and 17.3% versus periodic retraining, while closing 79% of the oracle gap. Notably, periodic retraining can be harmful under adversarial drift.
The main objective of this paper is to evaluate four new image formulas utilizing generalized fractional calculus operators, which include both fractional integration and fractional differentiation operators. Specifically, these operators are applied to the product of a general class of polynomials (also known as Srivastava polynomials) and an extended Mittag-Leffler type function. The analytical results derived in this work are explicitly expressed in terms of the generalized Wright hypergeometric function. Due to the general nature and flexibility of the extended Mittag-Leffler type function, Srivastava polynomials, and Wright hypergeometric function, numerous existing formulas in the literature—as well as several new formulas—can be easily derived as special cases of our main findings. Overall, this work provides simple and useful theoretical tools that can be applied to fractional differential equations and mathematical physics.
Automated detection of false and misleading news circulating on social media has become one of the most heavily researched applications of machine learning in the computational social sciences, yet the practical value of the resulting systems remains contested. Reported classification accuracies on standard benchmarks frequently approach ceiling, while independent assessments of the same model families under temporal, topical and adversarial shift record substantially weaker performance. This review examines the disjunction between benchmark success and operational capability, and asks what the accumulated evidence genuinely supports. The literature was identified through structured searching of openly accessible scholarly indexes and metadata registries, supplemented by backward and forward citation tracking from recent reviews and by targeted retrieval of methodological and institutional sources. Evidence was appraised for construct validity of labels, evaluation design, transparency, replication and external validity, and was synthesised thematically rather than catalogued study by study. Four findings emerge with reasonable confidence. Label provenance, rather than model architecture, is the dominant determinant of what a classifier learns, and source-level labelling propagates publisher-specific stylistic signals that inflate apparent accuracy. Social-context and propagation models achieve stronger discrimination than content-only models but forfeit the early-detection window and depend on platform data whose availability has narrowed. Multimodal and evidence-retrieval systems address genuine failure modes of text-only classification, although gains are reported on heterogeneous benchmarks that resist direct comparison. Large language models function simultaneously as a generative threat that decouples writing style from veracity and as a source of reasoning and rationale that improves small detectors, with the second role better evidenced than autonomous zero-shot verification. Evidence remains concentrated in English-language, politically framed, text-dominant corpora drawn from a small number of platforms, and almost no study measures downstream effects on audiences or on fact-checking workflows. Progress now depends less on architectural novelty than on label construction, temporally honest evaluation and outcome measurement beyond the confusion matrix.
