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 on 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.

In the era of information overload, sequential recommender systems have emerged as pivotal tools for modeling user preferences through dynamic behavioral pattern mining. These systems transcend conventional recommendation paradigms by explicitly modeling temporal dependencies in user–item interactions, preference evolution, and contextual dynamics. This study presents a methodologically structured taxonomy of sequential recommender systems through four analytical dimensions: (1) Sequential Modeling, which includes methods ranging from statistical techniques to deep learning architectures to understand user behavior patterns; (2) Temporal Dynamics Modeling, which involves time-aware collaborative filtering and deep temporal modeling; (3) Network-Enhanced Modeling, which leverages graph neural networks, heterogeneous graphs, dynamic graphs, and hypergraphs to explore structural dependencies; and (4) Robust Representation Learning, which encompasses contrastive mechanisms and techniques driven by large language models (LLMs). These algorithms focus on different aspects of sequential recommendation, including but not limited to capturing dynamic interests, modeling long- and short-term preferences, and addressing issues such as data sparsity, noise, and bias, which affect the performance and user experience of recommender systems in practical applications. Furthermore, we summarize and discuss promising future research directions to provide theoretical and methodological insights. The constructed taxonomy not only organizes existing methodological innovations, but also reveals fundamental limitations in current evaluation protocols, providing a roadmap for advancing both theoretical foundations and practical applications in this domain.

Evolutionary game theory traditionally models strategic interactions under static payoff environments, inherently overlooking the dynamic reciprocity between collective human behavior and macroscopic ecological or institutional constraints. To bridge this gap, we propose a closed-loop coevolutionary framework where effective game payoffs dynamically couple with an endogenously evolving environmental state, which is subject to both top-down institutional regulation and bottom-up frequency-dependent exploitation. Through rigorous linear stability and bifurcation analysis, we systematically map how macroscopic trajectories traverse canonical game topologies. We uncover a profound divergence in systemic responses dictated by the environment's baseline resilience. Under weak resilience, institutional intervention acts as a precarious rescuer: moderate regulation enables topological escapes into stable coexistence, whereas over-regulation triggers catastrophic saddle-node annihilations leading to socio-ecological collapse. Conversely, under strong baseline resilience, we identify a counter-intuitive ``Paradox of Prosperity.'' Here, excessive institutional punishment induces a severe feedback phase lag, mathematically shattering peaceful coexistence via destabilizing supercritical Hopf bifurcations and trapping the population in violent, large-amplitude limit cycles. Furthermore, we demonstrate that meticulously calibrated feedback can induce global bifurcations, creating bistable landscapes that lock trajectories into socially optimal utopian attractors. These findings provide a unified topological lens for understanding how socio-ecological feedback fundamentally governs the nonlinear survival of cooperation.

Modern individuals typically participate in multiple distinct social groups, characterizing the structural diversity of their social environment. Empirical observations suggest that individuals are more likely to adopt new ideas when these ideas are validated by multiple distinct groups. Our work shifts the focus from traditional, individual-based higher-order interactions to a novel group-based mechanism, which is determined by the structure of the entire group surrounding a node. We define the structural diversity coefficient based on the number of connected components in the node’s neighborhood, and propose a novel social contagion model that incorporates higher-order effect based on structural diversity. We develop both homogeneous mean-field method and dynamic message passing approach to analyze key dynamical properties and extensive numerical simulations validate the accuracy of the theoretical analyses. The results demonstrate that the introduction of group-based higher-order effect converts the system’s phase transition from continuous to discontinuous. Strengthening higher-order effect leaves the forward threshold unchanged while lowering the backward threshold. Moreover, when only higher-order effect is present, the system exhibits bistability and first-order transition with respect to the higher-order interaction strength, whereas the system exhibits no forward threshold. Our work generalizes the concept of higher-order networks to propose a unified framework for understanding group-based higher-order structures and their associated dynamics.

In the real world, public opinion is inevitably shaped by various forms of message diffusion, and vice versa. However, these two dynamics operate with different timescales and the effect of relative timescale in the coevolution dynamics has not been explored. Here, we propose a coupled noisy threshold voter-UAU model that incorporates an adjustable relative timescale between message diffusion and binary decision-making. We find that the system undergoes a first-order phase transition with respect to the message transmission rate, accompanied by hysteresis and bistability, once the relative timescale exceeds its critical value. Although the system exhibits marked non-monotonic and discontinuous changes with the relative timescale in different situations, there are common features: the prevalence of positive opinion exhibits a slight decline as the relative timescale increases within the low positive opinion phase, whereas it shows an upward trend in the high positive opinion phase. Consequently, the forward critical message transmission rate increases with the relative timescale, while the backward one decreases. Notably, the model illustrates how a minority opinion can abruptly overturn the majority once a critical tipping point is crossed. This study advances our understanding of the role of relative timescale in synergistically coupled dynamics and provides potential strategies for public opinion stabilization.

The Public Goods Game (PGG) serves as the foundational paradigm for modeling collective action, yet existing evolutionary models overwhelmingly treat punishment as a static parameter or rely on discrete algorithmic updates. This structural rigidity severs the continuous macroscopic feedback loops inherent to real-world adaptive systems. In this work, we elevate adaptive punishment in PGGs to a rigorous nonlinear dynamical framework by formulating the collective punitive intensity (beta) as a continuously coevolving macroscopic state variable within a coupled replicator system. Driven by the environmental synergy factor (R) and modulated by redistributive coupling (eta) and self-regulating elasticity (theta), this adaptive mechanism continuously reshapes the effective payoff landscape. Our exact global bifurcation analysis reveals that highly efficient adaptive punishment acts as a non-conservative restoring force. Crucially, this mechanism successfully sustains a stable interior focus of cooperation-even in severely resource-deprived environments (synergy factor R < 1) where static PGG models inevitably collapse to a global boundary attractor of absolute defection. Furthermore, we mathematically demonstrate that the elasticity of adaptive punishment strictly governs the system's topological phase transitions, giving rise to non-dissipative neutral centers, macroscopic heteroclinic cycles, and a Bogdanov-Takens codimension-2 bifurcation hub. Ultimately, these findings provide a purely analytical perspective on how continuous adaptive punishment stabilizes metastable order and drives complex oscillatory dynamics in coevolutionary PGGs, advancing the theoretical taxonomy beyond discrete empirical approximations.

The maximum k-core number of a network, defined as the largest value of k for which a k-core exists, serves as a key metric for identifying core structures and assessing network resilience. Conventional approaches to establishing its relationship with degree distribution rely on computationally intensive processes involving graph generation and subsequent decomposition. To address this limitation, we introduce a fast algorithm that directly estimates the maximum k-core number solely from a given degree sequence, entirely bypassing the need for explicit network instantiation. Extensive validation on both model networks and degree sequences extracted from real-world networks demonstrates that our method achieves accurate estimates within milliseconds, offering a powerful and efficient tool for large-scale network analysis.

We introduce a general framework to solve a class of combinatorial optimization problems, including the matching problem, the Traveling Salesman Problem, and also the minimum weight k-factor problem. By reformulating these problems as an arrangement model, we recast the optimization task into a grand-canonical ensemble, where chemical potentials are used to relax strict topological constraints. The analytical solution found can serve as a polynomial-time algorithm to compute an approximate minimum cost for arbitrary k and link-weight distributions. Our framework is complementary to existing approaches and reveals new connections between combinatorial optimization and the statistical physics of disordered systems.

Abstract Localized oscillatory patterns, confined to finite spatial domains or subsets of nodes in a network, have been observed in cortical neurons, metacommunity ecosystems, and chemical experiments. Previous studies have shown that wave bifurcations (oscillatory Turing instabilities) in undirected networked reaction–diffusion systems can induce such localized oscillations. Here, we propose an alternative mechanism for their emergence, namely Hopf-type instabilities acting on a subcritical Turing branch. Unlike wave bifurcations, this mechanism does not require interactions amongst at least three components. We further demonstrate its robustness in both the Brusselator and FitzHugh-Nagumo models in undirected Barabási–Albert, Erdős–Rényi, and Watts–Strogatz networks. This work extends the universality of such patterns in networked systems and provides a theoretical foundation for the control of collective dynamical phenomena in complex systems.

Spreading dynamics is a central topic in the physics of complex systems and network science, providing a unified framework for understanding how information, behaviors, and diseases propagate through interactions among system units. In many propagation contexts, spreading processes are influenced by multiple interacting factors, such as information expression patterns, cultural contexts, living environments, cognitive preferences, and public policies, which are difficult to incorporate directly into classical modeling frameworks. Recently, large language models (LLMs) have exhibited strong capabilities in natural language understanding, reasoning, and generation, enabling explicit perception of semantic content and contextual cues in spreading processes, thereby supporting the analysis of the different influencing factors. Beyond serving as external analytical tools, LLMs can also act as interactive agents embedded in propagation systems, potentially influencing spreading pathways and feedback structures. Consequently, the roles and impacts of LLMs on spreading dynamics have become an active and rapidly growing research area across multiple research disciplines. This review provides a comprehensive overview of recent advances in applying LLMs to the study of spreading dynamics across two representative domains: digital epidemics, such as misinformation and rumors, and biological epidemics, including infectious disease outbreaks. We first examine the foundations of epidemic modeling from a complex-systems perspective and discuss how LLM-based approaches relate to traditional frameworks. We then systematically review recent studies from three key perspectives, which are epidemic modeling, epidemic detection and surveillance, and epidemic prediction and management, to clarify how LLMs enhance these areas. Finally, open challenges and potential research directions are discussed.

Finding maximum cliques in large networks is a challenging combinatorial problem with many real-world applications. We present a fast algorithm to achieve the exact solution for the maximum clique problem in large sparse networks based on efficient graph decomposition. A bunch of effective techniques is being used to greatly prune the graph and a novel concept called Complete-Upper-Bound-Induced Subgraph (CUBIS) is proposed to ensure that the structures with the potential to form the maximum clique are retained in the process of graph decomposition. Our algorithm first pre-prunes peripheral nodes, subsequently, one or two small-scale CUBISs are constructed guided by the core number and current maximum clique size. Bron-Kerbosch search is performed on each CUBIS to find the maximum clique. Experiments on 50 empirical networks with a scale of up to 20 million show the CUBIS scales are largely independent of the original network scale. This enables an approximately linear runtime, making our algorithm amenable for large networks. Our work provides a new framework for effectively solving maximum clique problems on massive sparse graphs, which not only makes the graph scale no longer the bottleneck but also shows some light on solving other clique-related problems.

The influence of higher-order structures on network dynamic behaviors has gradually emerged as a significant focus in the field of network science. Among these behaviors, synchronization is a pivotal phenomenon. While existing studies primarily examine how higher-order structures affect synchronization types and thresholds, the dynamics of the synchronization process itself remain underexplored. In this work, we study the impact of higher-order BA network structures on ER random networks, specifically analyzing their effect on the synchronization path. Our findings reveal that introducing higher-order BA structures into ER random networks can either promote stepwise aggregation synchronization and hierarchical synchronization or suppress global synchronization, depending on the higher-order coupling strength. Moreover, as the higher-order coupling strength increases, when the higher-order BA network fails to synchronize, the ER structure can be employed to facilitate overall synchronization along the hierarchical synchronization path. The analysis presented in this paper reveals the complex influence of higher-order BA structures on network synchronization, offering new insights into the role of higher-order structures in network dynamics.

The cycle ratio method is designed to define the importance of nodes by the cycles of a network, and a set of important nodes identified by this method has superior control performance than by degree centrality, H-index, and coreness methods in several aspects such as spreading, percolation, and pinning control. Unfortunately, the method is not precise enough to portray the importance of the nodes, so in this paper, we improve the cycle ratio method by reducing the impact of four and larger cycles and adding the effects of the tree structure. Through numerical simulations on several real networks, we find that the set of important nodes discovered by the improved cycle ratio method is more dispersed and has better control in all three aspects of spreading, percolation, and pinning control than the original cycle ratio method. The work in this paper makes it more accurate to use the cycle structure to find a set of important nodes in a network and provides new ideas for a deeper understanding of the effects of local structure on the importance of the nodes.

Complex networks are frequently employed to model physical or virtual complex systems. When certain entities exist across multiple systems simultaneously, unveiling their corresponding relationships across the networks becomes crucial. This problem, known as network alignment, holds significant importance. It enhances our understanding of complex system structures and behaviours, facilitates the validation and extension of theoretical physics research about studying complex systems, and fosters diverse practical applications across various fields. However, due to variations in the structure, characteristics, and properties of complex networks across different fields, the study of network alignment is often isolated within each domain, with even the terminologies and concepts lacking uniformity. This review comprehensively summarizes the latest advancements in network alignment research, focusing on analyzing network alignment characteristics and progress in various domains such as social network analysis, bioinformatics, computational linguistics and privacy protection. It provides a detailed analysis of various methods' implementation principles, processes, and performance differences, including structure consistency-based methods, network embedding-based methods, and graph neural network-based (GNN-based) methods. Additionally, the methods for network alignment under different conditions, such as in attributed networks, heterogeneous networks, directed networks, and dynamic networks, are presented. Furthermore, the challenges and the open issues for future studies are also discussed.

Free-riding severely undermines the sustainability of cooperation in public goods games, yet existing many mechanisms for curbing such behavior often overlook the role of conditional punishment emerging from higher-order interactions. To bridge this gap, we introduce an evolutionary public goods game on hypergraphs, where individuals engage through group-based interactions (hyperedges), and punishment is conditionally triggered when the number of defectors in a group exceeds a predefined threshold phi. Once triggered, defectors retain only a fraction 1-/i of their original payoff, while cooperators benefit from the redistributed penalties. We analytically derive exact critical points for the reduced synergy factor that govern the emergence and saturation of cooperation on uniform random hypergraphs. Specifically, lower thresholds enable cooperation to emerge at a critical reduced synergy factor inversely proportional to the group size g and punishment intensity /i. Conversely, higher thresholds impede cooperation by raising both critical points to 1. Furthermore, introducing heterogeneity in group size, threshold, or punishment intensity amplifies cooperation compared to homogeneous counterparts. By unifying higher-order interactions, threshold-based punishment, and structural heterogeneity into a single analytical framework, this work provides new insights for designing robust cooperation-enhancing mechanisms in complex systems beyond pairwise networks.

Modeling and prediction of dynamical systems are essential in both scientific and engineering fields, but the nonlinearity and chaotic behavior present significant challenges to traditional methods. Next-generation reservoir computing (NGRC) aims to mitigate the randomness of conventional reservoir computing through time-delay feature construction, but it still faces issues such as relying on fixed nonlinear basis functions for feature mapping making it difficult to adapt to varying system dynamics, insufficient sensitivity to historical data, and high-dimensional computational complexity. In this paper, we propose an enhanced NGRC method that improves adaptability and predictive accuracy for complex dynamical systems by integrating temporal decay and kernel functions from the attention mechanism. The decay factor dynamically adjusts the weights of historical data through exponential decay, emphasizing recent temporal dependencies. Meanwhile, a Gaussian kernel function enhances nonlinear mapping capabilities, enabling the model to capture intricate dynamical patterns. Experiments on chaotic systems, including the Lorenz and double-scroll systems, demonstrate that the improved NGRC model significantly enhances prediction accuracy and stability, particularly in long-term forecasting scenarios. Moreover, it exhibits strong generalization capability with respect to variations in initial conditions. The proposed method offers valuable insights into the modeling and prediction of complex dynamical systems and shows great potential for future applications.

This study examines the scale-dependent coupling between economic growth and CO2 emissions in the G7 and the BRICS, two major economic blocs and key emitters, by integrating multifractal detrended cross-correlation analysis (MF-DCCA) and wavelet coherence. We analyze annual per capita GDP and CO2 series from 1950 to 2022 to determine whether these countries exhibit distinct multiscale dependencies. MF-DCCA results show that most G7 economies exhibit nonlinear, intrinsic multifractality, while the BRICS countries display greater heterogeneity. India, China, and Brazil show clear signs of intrinsic multifractality, whereas Russia and South Africa exhibit no such evidence.Wavelet coherence further reveals that in the G7, economic growth typically leads CO2 emissions at business-cycle scales. Among the BRICS, Brazil, Russia, and China exhibit robust coherence at short-and mid-term scales, while India and South Africa remain weakly linked. An event-window comparison of the 1973 Oil Crisis and the 2008 Financial Crisis highlights the starkly contrasting responses of G7 and BRICS countries to these shocks. These findings enhance our understanding of multiscale growth-emission nexus and suggest that the G7 should adopt stable, long-term strategies while the BRICS require agile, scale-aware interventions to meet climate and sustainability challenges.

Synchronization in a network of connected elements is essential to the proper functioning of both natural and engineered systems and is thus of increasing interest across disciplines. In many cases, synchronization phenomena involve not just actions within a single network in isolation, but the coordinated and coherent behaviors of several networks interacting with each other. The interactions between multiple systems play a crucial role in determining the emergent dynamics. One paradigm capable of representing real-world complex systems is that of multiplex networks, where the same set of nodes exists in multiple layers of the network. Recent studies have made significant progress in understanding synchronization in multiplex networks. In this review, we primarily focus on two key aspects: structural complexity and dynamical complexity. From the perspective of structural complexity, we present how the topological setting, such as the interlayer coupling pattern, affects the synchronizability of a multiplex network. The structural characteristics of a multiplex network, in particular, give rise to dynamical complexity, including the emergence of intralayer synchronization (within each layer) and interlayer synchronization (between layers). We also discuss the major methods for studying the stability of complete, intralayer, and interlayer synchronization, as well as synchronization control in multiplex networks. Additionally, we briefly introduce some relevant applications. Lastly, the review provides a comprehensive summary of the notable findings in the study of synchronization in multiplex networks, emphasizing the interplay between their structural and dynamical complexities, and identifies open problems that present opportunities for future research in this field.

A lot of effort has been devoted to network structural interpretations of the economic complexity which can affect the level of economic growth and the activities of economic entities, although it remains unknown how network dynamical interpretations on which the economic complexity index operates affects its performance. Here, we regard export trade flows as the dynamics of resource allocation, and design a tunable resource allocation (TRA) process for interpreting economic complexity. The TRA could degrade into the classical economic complexity index (ECI) when the tunable parameter equals zero. We apply the TRA dynamics to the perfectly nested triangular matrix and the real country-product bipartite network from the world trade data to interpret economic complexity respectively. The parametric interpretations provide conceptual descriptions of the economic complexity index, shedding new light on a pattern of specialization of resource allocation, and the fluctuation of the economic complexity in countries and products. Moreover, the relationship of economic complexity between countries and products in the framework of TRA reveals that higher-ECI countries usually specialize in higher-ECI products while products not so as to countries, and the correlation strength varies with tunable parameters.