Scaling and universality for percolation in random networks. A unified view

Percolation processes on random networks have been the subject of intense research activity over the last decades: the overall phenomenology of standard percolation on uncorrelated and unclustered topologies is well known. Still some critical properties of the transition, in particular for heterogeneous substrates, have not been fully elucidated and contradictory results appear in the literature. In this paper we present, by means of a generating functions approach, a thorough and complete investigation of percolation critical properties in uncorrelated locally tree-like random networks. We determine all critical exponents, the associated critical amplitude ratios and the form of the cluster size distribution for networks of any level of heterogeneity. We uncover, in particular for highly heterogeneous networks, subtle crossover phenomena, nontrivial scaling forms and violations of hyperscaling. In this way we clarify the origin of inconsistencies in the previous literature.