The k -Core and Branching Processes

Source abstract

The k-core of a graph G is the maximal subgraph of G having minimum degree at least k . In 1996, Pittel, Spencer and Wormald found the threshold λ c for the emergence of a non-trivial k -core in the random graph G ( n , λ/ n ), and the asymptotic size of the k -core above the threshold. We give a new proof of this result using a local coupling of the graph to a suitable branching process. This proof extends to a general model of inhomogeneous random graphs with independence between the edges. As an example, we study the k -core in a certain power-law or ‘scale-free’ graph with a parameter c controlling the overall density of edges. For each k ≥ 3, we find the threshold value of c at which the k -core emerges, and the fraction of vertices in the k -core when c is ϵ above the threshold. In contrast to G ( n , λ/ n ), this fraction tends to 0 as ϵ→0.