Scale-Free Networks Enhance the Spread of Better Strategy

In this study, we mathematically demonstrate that heterogeneous networks accelerate the social learning process, using a mean-field approximation of networks. Network heterogeneity, characterized by the variance in the number of links per vertex, is effectively measured by the mean degree of nearest neighbors, denoted as \(\langle k_{nn}\rangle \) $⟨k_{\mathit{nn}}⟩$ . This mean degree of nearest neighbors plays a crucial role in network dynamics, often being more significant than the average number of links (mean degree). Social learning, conceptualized as the imitation of superior strategies from neighbors within a social network, is influenced by this network feature. We find that a larger mean degree of nearest neighbors \(\langle k_{nn}\rangle \) $⟨k_{\mathit{nn}}⟩$ correlates with a faster spread of advantageous strategies. Scale-free networks, which exhibit the highest \(\langle k_{nn}\rangle \) $⟨k_{\mathit{nn}}⟩$ , are most effective in enhancing social learning, in contrast to regular networks, which are the least effective due to their lower \(\langle k_{nn}\rangle \) $⟨k_{\mathit{nn}}⟩$ . Furthermore, we establish the conditions under which a general strategy A proliferates over time in a network. Applying these findings to coordination games, we identify the conditions for the spread of Pareto optimal strategies. Specifically, we determine that the initial probability of players adopting a Pareto optimal strategy must exceed a certain threshold for it to spread across the network. Our analysis reveals that a higher mean degree \(\langle k \rangle \) $⟨k⟩$ leads to a lower threshold initial probability. We provide an intuitive explanation for why networks with a large mean degree of nearest neighbors, such as scale-free networks, facilitate widespread strategy adoption. These findings are derived mathematically using mean-field approximations of networks and are further supported by numerical experiments.