Spreading of Messages in Random Graphs |
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Authors: | Ching-Lueh Chang Yuh-Dauh Lyuu |
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Affiliation: | 1.Department of Computer Science and Information Engineering,National Taiwan University,Taipei,Taiwan;2.Department of Computer Science and Information Engineering & Department of Finance,National Taiwan University,Taipei,Taiwan |
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Abstract: | 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, min-seed(n,p,d,r)\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 min-seed(n,p,d,r)=W(min{d,r}n)\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) min-seed(n,p,d,1/2)=W(dn/ln(e/d))\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. |
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Keywords: | |
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