Direct reciprocity, based on the repeated interactions, is a fundamental mechanism to promote cooperation. Zero-determinant (ZD) strategies have opened an avenue for unilateral payoff control. However, previous studies neglect internal costs provided what agents do differ from what agents think, which is crucial for decision making of intelligent agents. Motivated by this, we establish a game theoretical framework by assuming that an individual pays the internal cost if the behavior is inconsistent with the internal thought. We prove that ZD strategy does not exist if the cost via behavior-value inconsistency is present. Instead, we find a new class of repeated strategies that enforce a unilateral payoff control, which is termed as positive/negative determinant strategy. The found strategy allows an individual to enforce an affine combination of two individuals' average payoffs above/below zero. Consequently, a focal individual is able to unilaterally control the opponent's payoff below a given value via negative determinant strategy, and a focal individual is able to get more payoff than the opponent via positive determinant strategy. We also find that the control ability of positive/negative determinant strategies is better off than that of ZD strategies. Our work highlights the importance of inconsistency between the behavior and value on payoff control, which is typically absent in classic ZD strategies.

This paper proposes a co-evolutionary model of directed graphs and three opinions, i.e., conservative(+), neutral(O) and liberal(-). Agents update both opinions and social relationships with bias. We find that an emergent game suffices to predict the stability of bi-polarization under fast rewiring, limited neighbor size and a large system size limit. The bi-polarization is stable if and only if the emergent game has an internal Nash equilibrium. The necessary and sufficient condition is explained by both risk dominance and evolutionary stability. This game approach facilitates us to reveal the stability of bi-polarization in empirical systems. Our work fosters the understanding of opinion formation for controversial topics, and shows a deep connection between opinion dynamics and evolutionary game theory. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Cooperation is crucial for both biological and social systems. Yet it is not straightforward from the evolutionary perspective, since individuals are tempted to defect for personal interests. In traditional models, it is usually assumed that cooperation is determined only by external behaviours. Individuals, however, are likely to pay extra psychological costs if what they do differs from what they think, and are likely to pay to know what others think. Motivated by these observations, we extend donation games to psychological games by introducing value-behaviour inconsistency, which creates psychological costs such as guiltiness or regret, and information costs, which individuals pay to know the opponent's values. We find that the key for cooperation under weak rationality is the large guiltiness (the psychological cost for an individual with defective behaviour and cooperative internal value); the heterogeneity of information costs governs how cooperation is further enhanced under moderate rationality; and moderate information costs can effectively favour cooperation under strong rationality, which is typically absent in classical donation games. These findings highlight how guiltiness, regret and information costs together shape cooperative behaviour across the rationality intensity. Our work can be insightful for a novel paradigm of cooperation beyond the classical prisoner's dilemma.

Networked evolutionary games, which integrate network topology and game dynamics, serve as a powerful framework for complex systems. Evolutionary games on large-scale networks, however, have been analytically challenging due to the curse of dimensionality arising from large network size. This paper proposes an aggregation method based on backward equivalence, such that agents within the same equivalence class behave exactly the same over time. We give a necessary and sufficient condition under which the weighted networked evolutionary games (WNEG) are reduced to an equivalent system with low dimension. The aggregation is shown to reduce the computational burden ranging from strategy consensus, strategy optimization, controllability, to optimal control of the WNEG. Examples are provided. Our work opens an avenue to solve the curse of dimensionality on networked evolutionary game systems with mathematical rigor.

Non-pharmaceutical interventions (NPIs), including mask-wearing, physical distancing, and hygiene measures, provide the primary means of reducing transmission in the early stages of an epidemic. Individuals adopt one of two strategies-adherence (A) or non-adherence (N) to NPIs. These strategies influence the transmission rate and thus the number of infections, but they also come with inherent costs and benefits. We propose a model coupling behavior and disease dynamics in adherers and non-adherers based on the SIR framework. This gives rise to six behavioral-epidemiological compartments. Using numerical simulations and analytical considerations, we first examine the case where strategies are fixed. Stronger NPIs and more initial adherers lead to fewer infections, and adherers consistently experience lower infection risk than non-adherers. We then introduce behavioral switching based on the benefits and costs of the two strategies. When NPIs are effective, higher transmission rates promote adherence, resulting in fewer infections. Strikingly, in high-severity outbreaks, even modestly effective NPIs can significantly reduce infections. These findings highlight the critical role of the coupling between behavior and disease dynamics, and underscore how individual choices can compromise or compensate public health interventions.

Traditional methods for processing classical Chinese typically segment language understanding into discrete tasks, which overlook crucial background information and reduce user engagement. Large language models (LLMs) provide integrated solutions, yet they entail high computational costs and risks of generating inaccurate historical information. To tackle these challenges, we propose a novel framework, TEACH (conTrastive knowlEdge Adaptive distillation with enhanCed Historical interpretability), which focuses on classical Chinese understanding by integrating word sense disambiguation with sentence translation. This integration leverages a confidence-annotated knowledge base and a step-by-step Chain-of-Thought prompting mechanism to minimize hallucinations and improve semantic analysis. Moreover, TEACH employs contrastive distillation learning to efficiently transfer capabilities from larger models to smaller ones (e.g., Qwen2-1.5B), addressing overly liberal translations. Additionally, we introduce an innovative generation evaluation metric using iterative word alignment, enhancing LLM performance assessments by distinguishing additional information and addressing excessive translation issues. Experiments conducted on real-world datasets validate TEACH’s efficacy in classical Chinese educational scenarios.

Opinion dynamics in dynamical hypergraphs, mirroring the dynamical nature of high-order social interactions, are attracting increasing attention. Opinion dynamics on high-order interactions lead to non-trivial dynamical patterns compared with that on pairwise networks. How should we systematically understand the intrinsic differences between opinion dynamics in hypergraphs and that in networks? We establish a voter model in a dynamical hypergraph. We find that both opinion formation and transient topology are captured by a single multi-player game, provided that the hypergraphs evolve sufficiently fast. The Nash equilibrium analysis facilitates us to reveal the intrinsic differences between high-order interactions and pairwise interactions in the empirical system. Furthermore, we perform simulations to show how robust our theoretical results are beyond fast rewiring. Our work provides a game route for opinion dynamics and sheds a hidden connection between opinion formation on dynamical high-order interactions and multi-player games.

Understanding nonlinear social contagion dynamics on dynamical networks, such as opinion formation, is crucial for gaining new insights into consensus and polarization. Similar to threshold-dependent complex contagions, the nonlinearity in adoption rates poses challenges for mean-field approximations. To address this theoretical gap, we focus on nonlinear binary-opinion dynamics on dynamical networks and analytically derive local configurations, specifically the distribution of opinions within any given focal individual's neighborhood. This exact local configuration of opinions, combined with network degree distributions, allows us to obtain exact solutions for consensus times and evolutionary trajectories. Our counterintuitive results reveal that neither biased assimilation (i.e., nonlinear adoption rates) nor preferences in local network rewiring – such as in-group bias (preferring like-minded individuals) and the Matthew effect (preferring social hubs) – can significantly slow down consensus. Among these three social factors, we find that biased assimilation is the most influential in accelerating consensus. Furthermore, our analytical method efficiently and precisely predicts the evolutionary trajectories of adoption curves arising from nonlinear contagion dynamics. Our work paves the way for enabling analytical predictions for general nonlinear contagion dynamics beyond opinion formation.

Polarization has been widely observed in social media for controversial topics. Both social relationship adjustments and opinion dynamics are crucial for polarization. Yet, it is challenging to capture the emergence of polarization in such co-evolutionary dynamics due to the curse of dimensionality. In this work, we propose a discrete three-opinion voter-like model on dynamical directed social networks, and successfully derive the polarization probability by solving a partial differential equation (PDE) under fast rewiring. It is found that if all the directed links are broken with the same probability, the polarization probability is proportional to the number of extreme opinions. If the breaking probabilities are slightly different, the polarization probability is not only determined by the total fraction of extreme opinions, but also by the product of the fraction of extreme opinions. Furthermore, we find that if extreme opinions are the majority initially, the polarization probability increases via extending the duration time of individuals who have the same opinion. If extreme opinions are the minority initially, the polarization probability decreases via extending the duration time of individuals who have the same opinion. Our work can be insightful for quantitatively understanding the emergence of polarization.

Unidirectional social interactions are ubiquitous in real social networks whereas undirected interactions are intensively studied. We establish a voter model in a dynamical directed network. We analytically obtain the degree distribution of the evolving network at any given time. Furthermore, we find that the average degree is captured by an emergent game. However, we find that the fate of opinions is captured by another emergent game. Beyond expectation, the two emergent games are typically different due to the unidirectionality of the evolving networks. The Nash equilibrium analysis of the two games facilitates us to give the criterion under which the minority opinion with few disciples initially takes over the population eventually for in-group bias. Our work fosters the understanding of opinion dynamics ranging from methodology to research content.

Opinion dynamics is of paramount importance as it provides insights into the complex dynamics of opinion propagation and social relationship adjustment. It is assumed in most of the previous works that social relationships evolve much faster than opinions. This is not always true in reality. We propose an analytical approximation to study this issue for arbitrary time scales between opinion adjustment and network evolution. To this end, the coefficient of determination in statistics is introduced and a one-dimensional stable manifold is analytically found, i.e., the most likely trajectory. With the aid of the stable manifold, we further obtain the fate of opinions and the consensus time, i.e., fixation probability and fixation time. We find that for in-group bias, the more likely individuals are to adopt the popular opinion, the less likely the majority opinion takes over the population, i.e., conformity inhibits the domination of popular opinions. This counterintuitive result can be interpreted from a game perspective, in which in-group bias refers to a coordination game and rewiring probability refers to a rescaling of the selection intensity. Our work proposes an efficient approximation method to foster the understanding of opinion dynamics in dynamical networks.

The speed of evolution on structured populations is crucial for biological and social systems. The likelihood of invasion is key for evolutionary stability. But it makes little sense if it takes long. It is far from known what population structure slows down evolution. We investigate the absorption time of a single neutral mutant for all the 112 non-isomorphic undirected graphs of size 6. We find that about three-quarters of the graphs have an absorption time close to that of the complete graph, less than one-third are accelerators, and more than two-thirds are decelerators. Surprisingly, determining whether a graph has a long absorption time is too complicated to be captured by the joint degree distribution. Via the largest sojourn time, we find that echo-chamber-like graphs, which consist of two homogeneous graphs connected by few sparse links, are likely to slow down absorption. These results are robust for large graphs, mutation patterns as well as evolutionary processes. This work serves as a benchmark for timing evolution with complex interactions, and fosters the understanding of polarization in opinion formation.

Polarization has received much attention in recent years. Mechanisms of polarization are found and control protocols for depolarization have been proposed. In spite of these, there is no clear definition of the stability of polarization. To this end, we introduce a co-evolutionary model of discrete opinions and directed networks. We propose an explicit definition of stable polarization based on the mean-field equations. We give a necessary and sufficient condition for the stable polarization, provided that the network evolves much faster than the opinions. In addition, we prove that the polarization is not stable if all individuals are allowed to learn all other opinions, which implies that learning the similar is necessary for the stable polarization. Our work is insightful for the depolarization control.

Voluntary vaccination is crucial for public health. The likelihood that the vaccinated get fully immune is typically referred to as the vaccine efficacy. In addition to this, the efficacy of vaccines also depends on inhibiting the transmission rates and protecting patients from critical symptoms. It has yet to be answered how the multi-efficacy of vaccines affects the vaccination behaviour. Here, we propose a game theoretical model of vaccination behaviour with a heterogeneous-transmission epidemiological process. Intuitively, individuals are likely to take vaccination for either high-efficient vaccines or epidemics with a large basic reproductive number. We find, however, a low economic discount rate can promote the uptake level for low-effectiveness vaccines and epidemics with small basic reproductive numbers. This result suggests that the perceived cost is more important than the risk of infection and the effectiveness of vaccines. In addition, we develop an analytical approximation to address the challenge arising from the coupled dynamics of vaccination and disease spread. This study shows how the interplay among three types of vaccine efficacies (transmission blocking, symptom mitigating and full immunity protecting) affects individual vaccination decisions and epidemic outcomes, and the model can be insightful for epidemic control.

The evolution of cooperation is a theme commonly studied in biology, psychology, sociology, and economics. Mechanisms that promote cooperative behavior in structured populations have been intensively studied. However, individuals’ values, specifically, their opinions have been rarely taken into account so far. Inspired by cognition dissonance theory, we assume that individuals pay the cost of guiltiness if the behavior is defection but the opinion deviates from defection, and pay the cost of regret if the behavior is cooperation but the opinion deviates from cooperation. For all general stochastic evolutionary dynamics on arbitrary static networks with multiple opinions, we prove in the weak selection limit that: (i) value-behavior inconsistency cost promotes cooperative behavior if and only if the average cost of regret is less than that of guiltiness; (ii) individuals with value-behavior consistency are more abundant than that with value-behavior inconsistency. This is in contrast with other mechanisms that are at work for cooperation for one population structure but not others. Furthermore, it is also validated on an empirical network and for non-weak selection intensity. The value-behavior inconsistency is thus a robust mechanism to promote cooperative behavior in structured populations. Our results shed light on the importance of the co-evolutionary dynamics of opinion and behavior, which opens an avenue for cooperation.

The voter model on networks is crucial to understand opinion formation. Uni-directional social interactions are ubiquitous in real social networks whereas undirected interactions are intensively studied. We establish a voter model on a dynamical directed network. We show that the opinion invasion is captured by a replicator equation of an emergent four-player two-strategy game, and the average in(out)-degree for the two opinions is fully captured by an emergent three-player two-strategy game. Interestingly, it is shown that the difference between the two emergent games arises from the uni-directionality of the network. The difference implies that the opinion with a small number of disciples can take over the population for in-group bias, provided that the network is directed. Our work makes an explicit connection between opinion dynamics and evolutionary games.

Opinion dynamics is crucial for unraveling the complexities of human interaction in the information age. How to speed up consensus without disturbing the fate of the system is key for opinion dynamics. We propose a voter model on adaptive networks, which resembles the coevolutionary process between opinions and social relationships. We prove the existence of a one-dimensional stable manifold for the system, which facilitates us to study both the fate of the system and the consensus time it takes. Surprisingly, we find the adjustment of social relations speeds up consensus but does not affect the fate of the system. For echo-chamber-like networks which consist of two homogeneous subnetworks connected by few sparse links, a small probability of adaptive edge dynamics is sufficient to accelerate consensus formation, which is counterintuitive. If the network structure makes consensus much slower than that of the regular networks, minor random rewiring makes a discontinuous drop in consensus time. Our work opens up an avenue for speeding up consensus without disturbing the fate of the system. It can be insightful for crowd control.

Chinese classical poetry occupies an important position in ancient Chinese literature. However, the existing research on Chinese classical poetry is usually limited to a certain poet or dynasty to analyze its historical and cultural influence and lacks comprehensive research on the process of ancient poetry from the perspective of time and space. We integrate multisource data and information, including poets’ biographies and Chinese classical poetry, to build a relatively complete social network of 41,310 poets. Based on this network, we use natural language processing and social network analysis techniques to research the relationships between poets. For example, how poets of different dynasties and different schools relate to each other. In order to quantitatively analyze the changing process of poets’ influence over a period of time, we propose a new method—time-series entropy weight method—to calculate the dynamic changing process of poets’ influence over time. Besides, we evaluate and discuss the method of calculating the influence of poets by means of propagation dynamics model and verify the effectiveness of the proposed method. Through the study of the complex social relations of poets and the quantification of their influence, we can assist in the study of the development of different schools and styles of ancient poetry. Our work offers a new, data-driven, long-run perspective on the evolution of Chinese poetry for historical researchers and enthusiasts to understand the complex relationships among historical figures.

Environmental change and human behavior are co-depended. The quality of the environment affects human's welfare, and the human's behavior in turn changes the environment. Yet the co-dependent nature seems to give a single individual few capabilities to change the environment. Intuitively, it is the collective actions that matter. What is a single individual able to do with the population welfare and the environment? We set up a toy model to explicitly address this issue. We take into account the eco-evolutionary nature of the feedback between environment and human behavior. One strategy, termed as Welfare Time strategy, is found, using which one individual suffices to set a linear relationship between collective welfare and environmental quality, no matter what the opponent does. This linear relationship can be either positively or negatively correlated, which is also unilaterally set by a single individual. It indicates that collective welfare can be higher even if it takes longer in a poor environment. Furthermore, we prove that the Welfare-Time strategy is able to dominate Win-Stay-Lose-Shift strategy, which is evolutionary stable against many strategies in repeated games. Our work reveals a hidden relationship between population welfare and the environment quality, which can be controlled unilaterally by a single individual. In addition, it implies that a single individual is able to control the environmental quality, provided that the rule of the environmental dynamics is known. (c) 2022 Elsevier Inc. All rights reserved.