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- W2894624719 abstract "A uniform attachment graph (with parameter $k$), denoted $G_{n,k}$ in the paper, is a random graph on the vertex set $[n]$, where each vertex $v$ makes $k$ selections from $[v-1]$ uniformly and independently, and these selections determine the edge set. We study several aspects of this graph. Our motivation comes from two similarly constructed, well-studied random graphs: $k$-out graphs and preferential attachment graphs. In this paper, we find the asymptotic distribution of its minimum degree and connectivity, and study the expansion properties of $G_{n,k}$ to show that the conductance of $G_{n,k}$ is of order $(log n)^{-1}$. We also study the bootstrap percolation on $G_{n,k}$, where, each vertex is either initially infected with probability $p$, independently of others, or gets infected later as a result of having $r$ infected neighbors at some point. We show that, for $2le rle k-1$, if $pll (log n)^{-r/(r-1)}$, then, with probability approaching 1, the process ends before all vertices get infected. On the other hand, if $pge omega(log n)^{-r/(r-1)}$, where $omega$ is a certain very slowly growing function, then all the vertices get infected with probability approaching 1." @default.
- W2894624719 created "2018-10-12" @default.
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- W2894624719 date "2018-10-04" @default.
- W2894624719 modified "2023-09-27" @default.
- W2894624719 title "On connectivity, conductance and bootstrap percolation for a random k-out, age-biased graph" @default.
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