The k-core and branching processes

Journal article

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 $\lambda_c$ for the emergence of a non-trivial k-core in the random graph $G(n,\lambda/n)$, 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 at least 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 \epsilon above the threshold. In contrast to $G(n,\lambda/n)$, this fraction tends to 0 as \epsilon tends to 0.

Authors: Riordan, O

• Publication status: Published
• Journal: Probability and Computing
• Volume: 17
• Issue: 1
• Pages: 111-136
• Publication date: 2005-11-03
• DOI: 10.1017/S0963548307008589
• EISSN: 1469-2163
• ISSN: 0963-5483
• Language: English
• Keywords: math.PR; math.CO; 05C80