The adaptive adjustment of node weights based on reputation and memory promotes fairness

In real-world social networks, individuals exhibit varying degrees of influence that evolve during their interactions. To address this dynamic nature of social influence, we provide a mechanism for adaptive node weight adjustment based on reputation within the ultimatum game model. In this model, an individual's social influence is quantified through their node weight, which adapts in response to the comparison between their reputation and the average reputation of their network neighbors. In each game round, individual reputation is determined by the ratio of successfully satisfying the responder in the proposer role. Moreover, individuals maintain a memory of historical game information, leading to the consideration of an individual's historical reputation. The calculation of individual reputation incorporates a modifiable parameter termed "memory strength", representing the reputation ratio between current and previous rounds. Through Monte Carlo simulations, it is revealed that the coevolution of node weight and game strategy, combined with reputation-based adaptive adjustments, significantly enhances fairness. For a wide range of specific reputation strengths, an expanded node weight range fosters the emergence of fairness. However, without considering historical reputation, smaller node weight ranges result in higher fairness levels. Furthermore, memory strength proves pivotal in fairness evolution: smaller memory strength promotes fairness in smaller node weight ranges, while larger memory strength positively influences fairness evolution in larger weight ranges. Therefore, it is the combined influence of reputation memory strength and node weight range that affects fairness. Our findings shed light on the promotion mechanism of fairness from the perspective of node weight evolution, and emphasize the role of reputation in the spread of altruistic behaviors.