  $k$-core percolation is a percolation model which gives a notion of network
functionality and has many applications in network science. In analysing the
resilience of a network under random damage, an extension of this model is
introduced, allowing different vertices to have their own degree of resilience.
This extension is named heterogeneous $k$-core percolation and it is
characterized by several interesting critical phenomena. Here we analytically
investigate binary mixtures in a wide class of configuration model networks and
categorize the different critical phenomena which may occur. We observe the
presence of critical and tricritical points and give a general criterion for
the occurrence of a tricritical point. The calculated critical exponents show
cases in which the model belongs to the same universality class of facilitated
spin models studied in the context of the glass transition.
