Small interconnected quantum processors can collaborate to tackle quantum computational problems that typically demand more capable devices. These linked processors, referred to as quantum nodes, can use shared entangled states to execute nonlocal operations. As a consequence, understanding how to distribute entangled states among nodes is essential for developing hardware and software. We analyze a protocol where entanglement is continuously distributed among nodes that are physically arranged in a regular pattern: a chain, a honeycomb lattice, a square grid, and a triangular lattice. These regular patterns allow for the modular expansion of networks for large-scale distributed quantum computing. Within the distribution protocol, we investigate how nodes can optimize the frequency of attempting entanglement swaps, trading off multiple entangled states shared with neighboring nodes for fewer states shared with non-neighboring nodes. We evaluate the protocol's performance using the virtual neighborhood size - a metric indicating the number of other nodes with which a given node shares entangled states. Employing numerical methods, we find that nodes must perform more swaps to maximize the virtual neighborhood size when coherence times are short. In a chain network, the virtual neighborhood size's dependence on swap attempt frequency differs for each node based on its distance from the end of the chain. Conversely, all nodes in the square grid exhibit a qualitatively similar dependence of the virtual neighborhood size on the swap frequency.