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 selfconsistency 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 treelike cluster, while the
kcircle is a node of the network.
summed over m independent realizations of th...
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 Winter '11
 RaissaD'Souza
 Graph Theory

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