  We study the structure of stationary patterns in bistable lattice dynamical
systems posed on rings with a symmetric coupling structure in the regime of
small coupling strength. We show that sparse coupling (for instance,
nearest-neighbour or next-nearest-neighbour coupling) and all-to-all coupling
lead to significantly different solution branches. In particular, sparse
coupling leads to snaking branches with many saddle-node bifurcations, whilst
all-to-all coupling leads to branches with six saddle nodes, regardless of the
size of the number of nodes in the graph.
