Unformatted text preview: eterogeneous thresholds seem to enhance global cascades. • Heterogeneous node degrees seem to reduce global cascades. Part I. C. Master Equation approaches
• SIS and SIR dynamics on networks
• Rather then thresholds, each infected neighbor has a
probability of infecting a healthy Following PastorSatorras and A. Vespignani
• ρk (t) is density of infected nodes of degree k at time t.
(Hence [1 − ρk (t)] is probability a node of degree k is NOT
infected.)
• λ = β/γ , the effective spreading rate. Set γ = 1.
(Recall β is infection rate, γ is recovery.) • The time evolution (a “master equation”):
dρk (t)
= −ρk (t) + λk [1 − ρk (t)] Θ(ρ(t))
dt
• First term: nodes recover with unit rate (γ = 1)
• Second term: Infection rate λ, times number of neighbors k ,
times prob node of degree k is healthy, times prob of being
connected to an infected node Θ(ρ(t)). Searching for more solutions to last equation,
in interval 0 < Θ ≤ 1
• Taking derivative w.r.t....
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 Winter '11
 RaissaD'Souza
 Graph Theory

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