On certain perturbations of the Erdös-Renyi random graph

We study perturbations of the Erdös-Renyi model for which the statistical weight of a graph depends on the abundanceof 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 theexpression 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.