The Structure-Coefficient Theorem for Evolutionary Games on Temporal Networks

In large-scale networked systems composed of autonomous decision-making agents, the structure of the network determines who interacts with and learns from whom, thereby playing a pivotal role in shaping collective behaviors such as cooperation and coordination. Importantly, both interaction and learning structures are dynamic, with connections and their intensities evolving over time. Understanding how these time-varying structures influence strategy adoption and, in turn, drive collective dynamics is a challenging yet essential research problem with significant practical implications. To address this, we introduce a general framework for evolutionary games on temporal networks. This framework is highly flexible, accommodating games with any number of available strategies, temporal networks with any number of network snapshots and arbitrary structures within each snapshot. Within this framework, we establish the first mathematical condition—a structure-coefficient theorem for temporal networks—that succinctly captures the effects of all structural factors on evolutionary outcomes using only three coefficients. This theorem provides a rigorous foundation for analyzing the dynamic adoption of strategies in time-varying networks. To demonstrate the utility of our approach, we apply the framework to public goods games on temporal networks. Our results reveal that temporal network structures can significantly enhance collective cooperation compared to each static snapshot. These findings offer novel theoretical insights into how time-varying network structures drive the emergence of collective intelligence, paving the way for a deeper understanding of adaptive systems in complex and dynamic environments.