Stable sets of threshold-based cascades on the erdős-rényi random graphs
IWOCA'11 Proceedings of the 22nd international conference on Combinatorial Algorithms
Triggering cascades on undirected connected graphs
Information Processing Letters
Bounding the sizes of dynamic monopolies and convergent sets for threshold-based cascades
Theoretical Computer Science
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Consider the following model on the spreading of messages. A message initially convinces a set of vertices, called the seeds, of the Erdős-Rényi random graph G(n,p). Whenever more than a ρ∈(0,1) fraction of a vertex v’s neighbors are convinced of the message, v will be convinced. The spreading proceeds asynchronously until no more vertices can be convinced. This paper derives lower bounds on the minimum number of initial seeds, $\mathrm{min\hbox{-}seed}(n,p,\delta,\rho)$, needed to convince a δ∈{1/n,…,n/n} fraction of vertices at the end. In particular, we show that (1) there is a constant β0 such that $\mathrm{min\hbox{-}seed}(n,p,\delta,\rho)=\Omega(\min\{\delta,\rho\}n)$ with probability 1−n −Ω(1) for p≥β (ln (e/min {δ,ρ}))/(ρ n) and (2) $\mathrm{min\hbox{-}seed}(n,p,\delta,1/2)=\Omega(\delta n/\ln(e/\delta))$ with probability 1−exp (−Ω(δ n))−n −Ω(1) for all p∈[ 0,1 ]. The hidden constants in the Ω notations are independent of p.