Spreading of Messages in Random Graphs

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 β>0 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.