Recall the most basic gf g0x p k xk k and the

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Unformatted text preview: Recall prob of following random edge to node of degree k : qk = kPk / k kPk = kPk / k . • Define a corresponding GF called G1(x): G1(x) = = q k xk k kPk xk / k k d = x dx ( d Pk xk )/ dx G0(x) k d d = x dx G0(x)/ dx G0(x) x=1 x=1 ≡ xG0(x)/G0(1). • (Recall the most basic GF: G0(x) = P k xk ) 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), i.e. H1(x), Generating kfunction for size S ) · x S = ( H1 (x ))component Prob(union of components has distribution in k . (A2.8) S sizes reached by following random edge A B H1(x) = xq0 + xq1H1(x) + xq2[H1(x)]2 + xq3[H1(x)]3 · · · (A self-consistency equation) Fig. A2.1. Diagrammatic visualization of property, (A2.9) random variable (Note here we used the “Powers”(A) Equation that a and (B) Equation (A2.10). Each square corresponds to an arbitrary tree-like cluster, while the kcircle is a node of the network. summed over m independent realizations of th...
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