We study heterogeneous $k$-core (HKC) percolation with a general mixture of the threshold $k$, with $k_{min}=2$ on random networks. Based on the local tree approximation, the scaling behaviors of the percolation order parameter $P_{\infty }\left(p\right)$ 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 ${\beta }_n=2/n$, discontinuous hybrid transitions with ${\beta }_H=1/2$ or ${\beta }_{A_4}=1/4$, 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 $2{\beta }_n+{\gamma }_n={\overline{\nu }}_n$.

• Received 16 December 2013

DOI:https://doi.org/10.1103/PhysRevE.89.052134