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lecture14

# Easiest to see if m 2 sum over two realizations 2 g0x

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Unformatted text preview: e object is generated by the mth power of that generating function, hence the [H1(x)]m factors above.) Aside: Powers property GF for a random variable k summed over m independent realizations of the object is generated by the mth power of that generating function. • Easiest to see if m = 2 (sum over two realizations) 2 • [G0(x)] = = k Pk x k2 pj pk xj +k jk = p0p0x0 + (p0p1 + p1p0)x + (p0p2 + p1p1 + p2p0)x2 + · · · • The coefﬁcient multiplying power n is the sum of all products pipj such that i + j = n. H0(x), Generating function for distribution in component sizes starting from arbitrary node B g. A2.1. Diagrammatic visualization of (A) Equation (A2.9) and (B) Equan (A2.10).0(x) =square corresponds + xP2arbitrary2tree-likeH1(x)]3 while the H Each xP0 + xP1H1(x) to an [H1(x)] + xP3[ cluster, · · · cle is a node of the network. = xG0(H1(x)). • Can take derivatives of H0(x) to ﬁnd moments of component sizes! • Note we have assumed a tree-like topology. The expected size of a component starting from arbitrary edge •s= d dx H0(x) x=1 = d dx xG0(H1(x)) x=1 d d = G0(H1(1)) + dx G0(H1(1)) · dx H1(1) • From H1(x) deﬁnition ﬁnd that s =1+ G0(1) 1−G1(1) =1+ k2 2 k − k2 Most im...
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