Critical phenomena in heterogeneous k-core percolation
  $k$-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. 
Davide Cellai, Aonghus Lawlor, Kenneth A. Dawson and James P. Gleeson
cond-mat.dis-nn cond-mat.stat-mech physics.soc-ph