Dynamics of k-core percolation
  In many network applications nodes are stable provided they have at least k neighbors, and a network of k-stable nodes is called a k-core. The vulnerability to random attack is characterized by the size of culling avalanches which occur after a randomly chosen k-core node is removed. Simulations of lattices in two, three and four dimensions, as well as small world networks, indicate that power-law avalanches occur in first order k-core systems, while truncated avalanches are characteristic of second order cases. 
C. L. Farrow and P. Shukla and P. M. Duxbury