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component is composed by the node initially reached, plus k other treelike components,
which have the same size distribution, where k is the number of outgoing links of the
node, whose distributionFollowing a random edge
is qk . The probability that the global component Q S has size S
is thus
(A2.6)
Q S likely to follow components a size S − of
• k times more = qk Prob(union of kedge to has node 1) degree k than a
k
node of degree 1:
(counting the initially reached node in S ). The generating function H1 is by deﬁnition
qk = kPk / k kPk .
H1 (x ) = QS x S, (A2.7) S • There are k − 1 other edges outgoing from this node.
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. • Each one of those k − 1 edges has probability qk of leading to
Prob(union of k components has size S ) · x S = ( H1 (x ))k .
(A2.8)
node of degree k .
S
A B (Circles denote isolated nodes, squares components of unknown size.) For convenience, deﬁne the GF for random edge following
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 Winter '11
 RaissaD'Souza
 Graph Theory

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