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- W2963511747 abstract "We consider the following activation process in undirected graphs: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least $r$ active neighbors. A contagious set is a set whose activation results with the entire graph being active. Given a graph $G$, let $m(G,r)$ be the minimal size of a contagious set. We study this process on the binomial random graph $G:=G(n,p)$ with $p:=frac{d}{n}$ and $1ll dll (frac{nloglog n}{log^{2}n})^{frac{r-1}{r}}$. Assuming $r>1$ to be a constant that does not depend on $n$, we prove that [m(G,r)=Theta (frac{n}{d^{frac{r}{r-1}}log d}),] with high probability. We also show that the threshold probability for $m(G,r)=r$ to hold is $p^{*}=Theta (frac{1}{(nlog^{r-1}n)^{1/r}})$." @default.
- W2963511747 created "2019-07-30" @default.
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- W2963511747 date "2017-10-01" @default.
- W2963511747 modified "2023-09-23" @default.
- W2963511747 title "Contagious sets in random graphs" @default.
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- W2963511747 doi "https://doi.org/10.1214/16-aap1254" @default.
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