Generating function for this our base gf g0x k pk xk

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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.

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