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

Cumulative Merging Percolation: A long-range percolation process in networks