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تسجيل مستخدم جديد$k$-core in percolated dense graph sequences
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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 $W:[0,1]^2to [0,+infty)$ 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 $C_k(G)$ the size of $k$-core in graph $G$, by $X^W$ the branching process associated with the kernel $W$, 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 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*} C_kleft(G_nleft(frac{1}{n}right)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 $n$. In addition, we obtain a probabilistic result concerning cut-norm and branching process which might be of independent interest.
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