A current challenge in evolutionary ecology is to assess how the spatial structure of interacting species shapes coevolution. Previous work on the geographic mosaic of coevolution has shown that coevolution depends on the spatial structure, the strength of selection, and gene flow across populations. We used spatial subgraphs and coevolutionary models to evaluate how spatial structure and the location of coevolutionary hotspots (sites in which reciprocal selection occurs) and coldspots (sites in which unidirectional selection occurs) contribute to the dynamics of coevolution and the maintenance of polymorphisms. Specifically, we developed a new approach based on the Laplacian matrices of spatial subgraphs to explore the tendency of interacting species to evolve toward stable polymorphisms. Despite the complex interplay between gene flow and the strength of reciprocal selection, simple rules drive coevolution in small groups of spatially structured interacting populations. Hotspot location and the spatial organization of coldspots are crucial for understanding patterns in the maintenance of polymorphisms. Moreover, the degree of spatial variation in the outcomes of the coevolutionary process can be predicted from the network pattern of gene flow among sites. Our work provides us with novel tools that can be used in the field or the laboratory to predict the effects of spatial structure on coevolutionary trajectories.