  We study perturbations of the Erdos-Renyi model for which the statistical
weight of a graph depends on the abundance of certain geometrical patterns.
Using the formal correspondance with an exactly solvable effective model, we
show the existence of a percolation transition in the thermodynamical limit and
derive perturbatively the expression of the threshold. The free energy and the
moments of the degree distribution are also computed perturbatively in that
limit and the percolation criterion is compared with the Molloy-Reed criterion.
