  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 entanglement distribution protocol, nodes can fix the
probability of attempting entanglement swaps to trade off multiple entangled
states shared with neighboring nodes for fewer entangled 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 probability 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 probability.
