Indirect reciprocity unveils how social cooperation is founded upon moral systems. Within the frame of dyadic games based on individual reputations, the ‘leading-eight’ strategies distinguish themselves in promoting and sustaining cooperation. However, in real-world societies, there are widespread interactions at the group level, where individuals need to make a singular action choice when facing multiple individuals with different reputations. Here, through introducing the assessment of collective reputations, we develop a framework that embeds group-level reputation structure into public goods games to study the evolution of group-level indirect reciprocity. We show that changing the criteria of group assessment destabilizes the reputation dynamics of leading-eight strategies. In a particular range of social assessment criteria, all leading-eight strategies can break the social dilemma in public goods games and sustain cooperation. Specifically, there exists an optimal, moderately set assessment criterion that is most conducive to promoting cooperation. Moreover, in the evolution of assessment criteria, the preference of the leading-eight strategies for social strictness is inversely correlated with the payoff level. Our work reveals the impact of social strictness on prosocial behaviour, highlighting the importance of group-level interactions in the analysis of evolutionary games and complex social dynamics.

Network modeling characterizes the underlying principles of structural properties and is of vital significance for simulating dynamical processes in real world. However, bridging structure and dynamics is always challenging due to the multiple complexities in real systems. Here, through introducing the individual’s activity rate and the possibility of group interaction, we propose a probabilistic activity-driven (PAD) model that could generate temporal higher-order networks with both power-law and high-clustering characteristics, which successfully links the two most critical structural features and a basic dynamical pattern in extensive complex systems. Surprisingly, the power-law exponents and the clustering coefficients of the aggregated PAD network could be tuned in a wide range by altering a set of model parameters. We further provide an approximation algorithm to select the proper parameters that can generate networks with given structural properties, the effectiveness of which is verified by fitting various real-world networks. Finally, we construct the co-evolution framework of the PAD model and higher-order contagion dynamics and derive the critical conditions for phase transition and bistable phenomenon using theoretical and numerical methods. Results show that tendency of participating in higher-order interactions can promote the emergence of bistability but delay the outbreak under heterogeneous activity rates. Our model provides a basic tool to reproduce complex structural properties and to study the widespread higher-order dynamics, which has great potential for applications across fields.

Background/Objectives: Cocaine use disorder (CUD) poses a worldwide health challenge, with severe consequences for brain function. However, the phase dynamics underlying CUD and the transitions between CUD and health remain poorly understood. Methods: Here, we used resting-state functional magnetic resonance imaging (fMRI) data from 43 CUD patients and 45 healthy controls (HCT). We performed empirical analysis to identify phase-coherence states and compared their probabilities of occurrence between conditions. To further explore the underlying mechanism, we employed computational modeling to replicate the observed state probabilities for each condition. These generated whole-brain models enabled us to simulate external perturbations and identify optimal brain regions mediating transitions between HCT and CUD. Results: We found that CUD was associated with a reduced occurrence probability of the state dominated by the default mode network (DMN). Perturbing the nucleus accumbens, thalamus, and specific regions within the default mode, limbic and frontoparietal networks drives transitions from HCT to CUD, while perturbing the hippocampus and specific regions within the visual, dorsal attention, and DMN facilitates a return from CUD to HCT. Conclusions: This study revealed altered DMN-related dynamics in CUD from the phase perspective and provides potential regions critical for state transitions. The results contribute to understanding the pathogenesis of CUD and the development of therapeutic stimulation strategies.

Identifying the most influential spreaders in online social networks plays a prominent role in affecting information dissemination and public opinions. Researchers propose many effective identification methods, such as k-shell. However, these methods are usually validated by simulating propagation models, such as epidemic-like models, which rarely consider the Push-Republish mechanism with attenuation characteristic, the unique and widely-existing spreading mechanism in online social media. To address this issue, we first adopt the Push-Republish (PR) model as the underlying spreading process to check the performance of identification methods. Then, we find that the performance of classical identification methods significantly decreases in the PR model compared to epidemic-like models, especially when identifying the top 10% of superspreaders. Furthermore, inspired by the local tree-like structure caused by the PR model, we propose a new identification method, namely the Local-Forest (LF) method, and conduct extensive experiments in four real large networks to evaluate it. Results highlight that the Local-Forest method has the best performance in accurately identifying superspreaders compared with the classical methods.