Replicator-Mutator Dynamics on Networks with Spatial Nonlocality: Theory and Applications
The replicator-mutator model has been widely studied to understand the emergence of social norms and dominant behavioral patterns in social networks. In particular, it provides a framework to explain how “hubs” arise across diverse systems. Within this setting, agents adapt their behaviors through interactions with others in the network.
In the traditional replicator-mutator model, mutation spreads only through nearest-neighbor interactions, and the network’s interaction structure is constructed accordingly. In realistic settings, however, agents are also influenced by others beyond their immediate neighbors, with the strength of influence typically decaying with distance. This motivates the question of how the evolutionary dynamics change when such nonlocal interactions are explicitly incorporated. In this work, we investigate the role of long-range interactions within the replicator-mutator framework by introducing a generalized interaction structure based on the d-path Laplacian, which captures interactions across all hop distances within a single operator. We analyze how the strength of long-range coupling affects the system’s evolutionary behavior across different network topologies. Notably, we observe that cultural globalization increases diversity while making behaviors more homogeneous. The combination of analytical results and numerical experiments provides a comprehensive picture of how nonlocality reshapes the spectrum of possible behaviors.
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