Networks rarely compete in isolation. Firms, infrastructures, biological populations, social groups, and institutions interact with rivals while being embedded in physical or functional spaces. Previous work on competing networks has shown that the outcome of a linear spreading process depends on the spectral properties of the interacting networks and on the centrality of the connector nodes. However, these models usually assume that all nodes of a network have the same ability to exploit the interaction once the inter-network link is created, disregarding spatial limitations. Here we introduce a spatial extension of the competing-networks framework in which two networks occupy adjacent regions separated by a boundary. The key ingredient is an energy vector assigned to the nodes, whose components decrease with the Euclidean distance to the boundary. This vector transforms the evolution operator of the system and couples topology with space. Now, a node is not only important because of its topological position, but also because of its spatial access to the competing region. Using pairs of Barabási–Albert networks, we compare central–central and peripheral–peripheral connection strategies while progressively relocating the nodes of the weaker network with respect to the boundary. We show that a topologically weaker network can overtake a stronger competitor when its most central nodes are placed sufficiently close to the boundary, whereas placing only peripheral nodes near the boundary is typically insufficient except under extreme spatial sensitivity. We also develop an analytical treatment of how the difference in spectral strength between the two competing networks affects the transition, obtaining results that agree with the numerical simulations.
