Thus our heuristic relaxation does give valid upper bounds of the exact Lipschitz constant.

Graph Neural Networks (GNNs) have achieved great success on a node classification task. Despite the broad interest in developing and evaluating GNNs, they have been assessed with limited benchmark datasets. As a result, the existing evaluation of GNNs lacks fine-grained analysis from various characteristics of graphs. Motivated by this, we conduct extensive experiments with a synthetic graph generator that can generate graphs having controlled characteristics for fine-grained analysis. Our empirical studies clarify the strengths and weaknesses of GNNs from four major characteristics of real-world graphs with class labels of nodes, i.e., 1) class size distributions (balanced vs. imbalanced), 2) edge connection proportions between classes (homophilic vs. heterophilic), 3) attribute values (biased vs. random), and 4) graph sizes (small vs. large). In addition, to foster future research on GNNs, we publicly release our codebase that allows users to evaluate various GNNs with various graphs. We hope this work offers interesting insights for future research.

Multi-agent reinforcement learning (MARL) has emerged as a useful approach to solving decentralised decision-making problems at scale. Research in the field has been growing steadily with many breakthrough algorithms proposed in recent years. In this work, we take a closer look at this rapid development with a focus on evaluation methodologies employed across a large body of research in cooperative MARL. By conducting a detailed meta-analysis of prior work, spanning 75 papers accepted for publication from 2016 to 2022, we bring to light worrying trends that put into question the true rate of progress. We further consider these trends in a wider context and take inspiration from single-agent RL literature on similar issues with recommendations that remain applicable to MARL. Combining these recommendations, with novel insights from our analysis, we propose a standardised performance evaluation protocol for cooperative MARL. We argue that such a standard protocol, if widely adopted, would greatly improve the validity and credibility of future research, make replication and reproducibility easier, as well as improve the ability of the field to accurately gauge the rate of progress over time by being able to make sound comparisons across different works. Finally, we release our meta-analysis data publicly on our project website for future research on evaluation accompanied by our open-source evaluation tools repository.

Purpose: The purpose of this study is to map the body of scholarly literature at the intersection of artificial intelligence (AI), analytics and sports and thereafter, leverage the insights generated to chart guideposts for future research. Design/methodology/approach: The study carries out systematic literature review (SLR). Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA) protocol is leveraged to identify 204 journal articles pertaining to utilization of AI and analytics in sports published during 2002 to 2024. We follow it up with extraction of the latent topics from sampled articles by leveraging the topic modelling technique of BERTopic. Findings: The study identifies the following as predominant areas of extant research on usage of AI and analytics in sports: performance modelling, physical and mental health, social media sentiment analysis, and tactical tracking. Each extracted topic is further examined in terms of its relative prominence, representative studies, and key term associations. Drawing on these insights, the study delineates promising avenues for future inquiry. Research limitations/implications: The study offers insights to academicians and sports administrators on transformational impact of AI and analytics in sports. Originality/value: The study introduces BERTopic as a novel approach for extracting latent structures in sports research, thereby advancing both scholarly understanding and the methodological toolkit of the field.

We thank all reviewers for their helpful comments and suggestions! We will address all minor issues. We will add references in the revision. UMG ( π) has cycles. We havn't thought about checking it from data, which is an interesting statistical Theorem 1 is effectively known.