Complete set of types of phase transition in generalized heterogeneous k -core percolation

We study heterogeneous k-core (HKC) percolation with a general mixture of the threshold k, with kmin=2 on random networks. Based on the local tree approximation, the scaling behaviors of the percolation order parameter P(p) are analytically obtained for general distributions of the threshold k. The analytic calculations predict that the generalized HKC percolation is completely described by the series of continuous transitions with order parameter exponents βn=2/n, discontinuous hybrid transitions with βH=1/2 or βA4=1/4, and three kinds of multiple transitions. Simulations of the generalized HKC percolations are carried out to confirm analytically predicted transition natures. Specifically, the exponents of the series of continuous transitions are shown to satisfy the hyperscaling relation 2βn+γn= ν̄n.