Network science has become a central methodological framework for the study of complex systems, with applications across social, biological, and technological domains. Its strength lies in formalizing such systems as collections of interacting units, whose collective behavior depends not only on the properties of the individual components, but also on the structure, dynamics, and evolution of their interactions. In this Review, we analyze how this framework, rooted in statistical physics, is being applied to the study of sports systems. Sports provide a particularly suitable domain for network modeling, as performance emerges from interactions among agents, whose actions are constrained by spatial embedding and temporal ordering, and evolve under competitive, adaptive, and out-of-equilibrium conditions. We review the main network representations used in sports, from classical graph-based descriptions to more general frameworks, including spatial, temporal, multilayer, and higher-order networks. Particular attention is devoted to soccer, where network science has had its deepest methodological impact, while applications to other sports such as basketball, volleyball, American football, or badminton serve to illustrate the generality of the underlying principles. Network science provides sports analytics with a rigorous language for quantifying coordination, emergence, adaptation, and collective performance. At the same time, the availability of high-resolution event, tracking, and pose-estimation data makes sports an ideal testbed for validating and refining network models in realistic settings. As a result, the relationship between network science and sports, currently driven primarily by the application of network methods to sport, has the potential to become genuinely bidirectional. In this Review, we not only organize the literature on network methodologies and their applications to sports, but also discuss complementary concepts needed to capture the full picture, including entropy and complexity measures, as well as key ideas from statistical physics and nonlinear dynamics. Finally, despite the substantial progress made so far, the field remains under active development, and we highlight the main open questions that should be addressed in future research.
Reconstructing causal dynamic networks from multivariate time series is a foundational problem in complex systems science. Yet, the key scientific issue is not simply causality detection, but causal interpretability. Interpretable causality is the foundation for testable mechanistic hypotheses, transferable forecasting, and principled decision-making for intervention and control. In real-world complex systems, causal inference is often compromised by noise, missing data, high dimensionality, nonlinearity, time delays, heterogeneity, and partial observability. Classic approaches to interpretable causality yield explicit, inspectable quantities such as causal graphs, coefficients, and governing equations. However, this methodological shift has heightened expectations: AI-extended approaches are increasingly required to recover explicit causal mechanisms rather than opaque predictive dependencies, thereby preserving interpretability. This review summarizes four classic methods and their AI-extended counterparts based on time series data, including Granger frameworks, information-theoretic measures, nonlinear state–space/manifold reconstruction, and mechanistic differential-equation learning. Next, we elucidate their motivations, core principles, the origins of interpretability, and the assumptions required for meaningful conclusions. Finally, we highlight representative applications across climate studies, neuroscience, epidemiology, finance, social science, ecology and molecular biology, followed by a discussion of comparative analysis, open challenges, and future research directions.
Poverty mapping is increasingly important for monitoring Sustainable Development Goal 1 (SDG 1) of the United Nations 2030 Agenda, which aims to end poverty in all its forms everywhere. Yet timely and fine-resolution poverty estimation remains difficult because conventional census- and survey-based approaches are costly, infrequent, and often sparse precisely where deprivation is most severe. As poverty emerges from complex socioeconomic systems shaped by human mobility, social interactions, infrastructure, and economic activities, emerging computational methods and nontraditional data sources have created new opportunities for poverty estimation and mapping. At the intersection of statistical physics, complex systems science, and data science, these approaches enable poverty estimation at finer spatial and temporal resolutions. This review summarizes the main concepts of poverty and the principal frameworks used to measure it, and examines recent advances in poverty estimation and mapping using satellite imagery, mobile phone data, social media data, and multisource data fusion. The review also discusses persistent challenges related to representativeness, transferability across regions, interpretability, and uncertainty quantification. Finally, the review clarifies both the analytical promise and the practical limits of contemporary poverty mapping.
The couplings of the strong to electroweak sectors of the Standard Model enable the exploration of each using our growing knowledge of the other. In this review, we will follow the sweep of history. Starting with QED as a precision theory, deep inelastic scattering served as a gateway to the strong interactions, followed by leptonic annihilation and quark–antiquark annihilation in hadron–hadron scattering. In turn, the resulting understanding of QCD helped establish the Standard Model. The same techniques form the basis for many precision electroweak measurements at high energy and searches for signs of new physics.
Electroweak radiative corrections form a crucial ingredient in modern precision calculations for particle processes at high-energy colliders such as the Large Hadron Collider. The salient features of electroweak corrections as well as currently used techniques and concepts for their calculation are reviewed. Recent progress in this enterprise is illustrated in a discussion of electroweak multi-gauge-boson production processes: massive di-boson production, vector-boson scattering, and massive tri-boson production.
