  Interactions in natural communities can be highly heterogeneous, with any
given species interacting appreciably with only some of the others, a situation
commonly represented by sparse interaction networks. We study the consequences
of sparse competitive interactions, in a theoretical model of a community
assembled from a species pool. We find that communities can be in a number of
different regimes, depending on the interaction strength. When interactions are
strong, the network of coexisting species breaks up into small subgraphs, while
for weaker interactions these graphs are larger and more complex, eventually
encompassing all species. This process is driven by emergence of new allowed
subgraphs as interaction strength decreases, leading to sharp changes in
diversity and other community properties, and at weaker interactions to two
distinct collective transitions: a percolation transition, and a transition
between having a unique equilibrium and having multiple alternative equilibria.
Understanding community structure is thus made up of two parts: first, finding
which subgraphs are allowed at a given interaction strength, and secondly, a
discrete problem of matching these structures over the entire community. In a
shift from the focus of many previous theories, these different regimes can be
traversed by modifying the interaction strength alone, without need for
heterogeneity in either interaction strengths or the number of competitors per
species.
