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W2035720072 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 $lambda_c$ 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." @default.
W2035720072 created "2016-06-24" @default.
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W2035720072 date "2008-01-01" @default.
W2035720072 modified "2023-10-18" @default.
W2035720072 title "The<i>k</i>-Core and Branching Processes" @default.
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W2035720072 doi "https://doi.org/10.1017/s0963548307008589" @default.
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