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Unformatted text preview: 1/kj , or equivalently
degree kj ≤ Kj = 1/Φj Using GFs can reduce a complicated dynamics to a static
percolation problem
• As usual, degree distribution Pk .
• A node is vulnerable / early adopter if it’s threshold Φ ≤ 1/k .
The probability a given node of degree k is vulnerable is thus
ρk = P [Φ ≤ 1/k ] = 1/k
f (Φ)dΦ.
0 • The probability a node drawn uniformly at random from all
nodes has 1) degree k , and 2) is vulnerable is thus: ρk Pk .
• Generating function for this (our base GF)
G0(x) = ρk Pk xk .
k “Propagation” of a cascade is edge following from a
vulnerable node
• As with the basic framework, probability of following edge to
node of degree k is proportional to k .
• GF for following a random edge to a vulnerable node of degree
k . (Again, observe building up process.):
G1(x) = k (kρk Pk ) / = G0(x)/G0(1) k kPk = G0(x)/ k k
and the distribution of the sum of the sizes of the k components is generated by H1 (as
previously explained for the sum of degrees),...
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This document was uploaded on 03/12/2014 for the course CSCI 289 at UC Davis.
 Winter '11
 RaissaD'Souza

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