Trust has played a pivotal role in the evolution of modern human societies, and it continues to be an essential underpinning of our social interactions. It is therefore important that we develop rigorous mathematical foundations that will enable us to better understand what promotes and what erodes trust and how to best preserve trustworthiness. To that effect we here propose a trust game, wherein investors, trustworthy trustees, and untrustworthy trustees compete for assets subject to a third-party evaluation system that oversees and modifies each individual reputation. We use Monte Carlo simulations on social networks to determine critical values of the degree of rationality and the reputation threshold that warrant high levels of trust and social wealth. We show that if investors have access to the reputation scores of trustees, the fraction of untrustworthy trustees drops if only the degree of rationality is sufficiently large, and this irrespective of the reputation threshold that determines the cutoff for untrustworthiness. But even though investors are allowed irrational investments, trust can still proliferate if the reputation threshold is sufficiently high. Our results thus formalize essential mechanisms of trust in social networks, which also outline policies to diminish untrustworthiness that can be employed in real life.

Influence, as an inherently special attribute, is bound to profoundly affect a player’s behavior. Meanwhile, a growing body of studies suggests that interactions among networks may be more important than isolated ones. Thus, we try our best to research whether such a setup can stimulate the sense of cooperation in spatial prisoner’s dilemma games through the co-evolution of strategy imitation and interdependence networks structures. To be specific, once a player’s influence exceeds the critical threshold $\tau$, they will be permitted to build a connection with the corresponding partner on another network in a self-organized way, thus gaining additional payoff. However, a player’s influence changes dynamically with the spread of strategy, resulting in time-varying connections between networks. Our results show that influence-based self-organization can facilitate cooperation, even under quite poor conditions, where cooperation cannot flourish in a single network. Furthermore, there is an optimal threshold $\tau$ to optimize the evolution of cooperation. Through microcosmic statistical analysis, we are surprised to find that the spontaneous emergence of connections between interdependence networks, especially those between cooperators, plays a key role in alleviating social dilemmas. Finally, we uncover that if the corresponding links between interdependence networks are adjusted to random ones, the evolution of cooperation will be blocked, but it is still better than relying on simple spatial reciprocity on an isolated lattice.

Most studies on the evolution of cooperation are based on the assumption of symmetry. However, symmetry represents only a subset or special case, whereas asymmetry is far more prevalent in reality. This discrepancy makes it difficult for existing research to comprehensively explain the diversity and complexity of cooperative behaviors in the real world. In this regard, some scholars in recent years have been studying the impact of asymmetry on the evolution of cooperation and have achieved a series of important research findings. Here in this short perspective, we overview the recent process of asymmetric effects on the evolution of cooperation. At last, to foster cooperative behavior in the dynamically evolving world, we highlight some key future research.