K-core in percolated dense graph sequences

Abstract

We determine the size of $k$-core in a large class of dense graph sequences. Let $G_n$ be a sequence of undirected, $n$-vertex graphs with edge weights $\{a^n_{i,j}\}_{i,j \in [n]}$ that converges to a kernel $W:[0,1]^2\to [0,+\infty)$ in the cut metric. Keeping an edge $(i,j)$ of $G_n$ with probability $\min \{ {a^n_{i,j}}/{n},1 \}$ independently, we obtain a sequence of random graphs $G_n(\frac{1}{n})$. Denote by $\mathcal{A}$ the property of a branching process that the initial particle has at least $k$ children, each of which has at least $k-1$ children, each of which has at least $k-1$ children, and so on. Using branching process and the theory of dense graph limits, under mild assumptions we obtain the size of $k$-core of random graphs $G_n(\frac{1}{n})$, \begin{align*} \text{size of $k$-core of } G_n\left(\frac{1}{n}\right) =n \mathbb{P}_{X^W}\left(\mathcal{A}\right) +o_p(n). \end{align*} Our result can also be used to obtain the threshold of appearance of a $k$-core of order $n$.