  Percolation on networks is a common framework to model a wide range of
processes, from cascading failures to epidemic spreading. Standard percolation
assumes short-range interactions, implying that nodes can merge into clusters
only if they are nearest-neighbors. Cumulative Merging Percolation (CMP) is an
new percolation process that assumes long-range interactions, such that nodes
can merge into clusters even if they are topologically distant. Hence in CMP
percolation clusters do not coincide with the topological connected components
of the network. Previous work has shown that a specific formulation of CMP
features peculiar mechanisms for the formation of the giant cluster, and allows
to model different network dynamics such as recurrent epidemic processes. Here
we develop a more general formulation of CMP in terms of the functional form of
the cluster interaction range, showing an even richer phase transition scenario
with competition of different mechanisms resulting in crossover phenomena. Our
analytic predictions are confirmed by numerical simulations.
