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Given a weighted graph, we introduce a partition of its vertex set such thatthe distance between any two clusters is bounded from below by a power of theminimum weight of both clusters. This partition is obtained by recursivelymerging smaller clusters and cumulating their weights. For several classicalrandom weighted graphs, we show that there exists a phase transition regardingthe existence of an infinite cluster.  The motivation for introducing this partition arises from a connection withthe contact process as it roughly describes the geometry of the sets where theprocess survives for a long time. We give a sufficient condition on a graph toensure that the contact process has a non trivial phase transition in terms ofthe existence of an infinite cluster. As an application, we prove that thecontact process admits a sub-critical phase on d-dimensional random geometricgraphs and on random Delaunay triangulations. To the best of our knowledge,these are the first examples of graphs with unbounded degrees where thecritical parameter is shown to be strictly positive.

Percolation by cumulative merging and phase transition for the contact process on random graphs