and then v
C
1
n
p
−→ χ
v
π :=
∞
X
j=1
π
j
p
j
1 − ξ
j
0, 3.9
e C
1
n
p
−→ µ
v
π := 1 − ξ
∞
X
j=1
j π
j
p
j
− 1 − ξ
2
2
∞
X
j=1
j p
j
. 3.10
Furthermore, v C
2
n
p
−→ 0 and eC
2
n
p
−→ 0. ii If 3.7 does not hold, then v
C
1
n
p
−→ 0 and eC
1
n
p
−→ 0. Proof. We apply Theorem 3.1 to G
∗
˜ n, ˜
d as discussed above. Note that ˜ p
1
0 by 2.15 and the assumption 1
− π
j
p
j
0 for some j. By 2.15, ζ E ˜
D ˜ D
− 2 = ζ
∞
X
j=0
j j − 2˜p
j
=
∞
X
j=1
j j − 2π
j
p
j
−
∞
X
j=1
j1 − π
j
p
j
=
∞
X
j=1
j j − 1π
j
p
j
−
∞
X
j=1
j p
j
. Hence, the condition E ˜
D ˜ D
− 2 0 is equivalent to 3.7. In particular, it follows that v
C
2
= o
p
n in i and vC
1
= o
p
n in ii. That also eC
2
= o
p
n in i and e
C
1
= o
p
n in ii follows by Remark 2.2 applied to G
∗
˜ n, ˜
d.
It remains only to verify the formulas 3.8–3.10. The equation g
′ ˜
D
ξ = ˜ λξ is by 2.18 equivalent
to ζg
′ ˜
D
ξ = λξ, which can be written as 3.8 by 2.15 and a simple calculation. By 3.5, using 2.10, 2.14 and 2.19,
v C
1
n
p
−→ ζ − ζg
˜ D
ξ − 1 − ξ
∞
X
j=1
j1 − π
j
p
j
=
∞
X
j=0
π
j
p
j
−
∞
X
j=0
π
j
p
j
ξ
j
+ j1 − π
j
p
j
ξ + ξ
∞
X
j=0
j1 − π
j
p
j
=
∞
X
j=0
π
j
p
j
1 − ξ
j
. Similarly, by 3.6, 2.18, 2.14 and 2.10,
e C
1
n
p
−→
1 2
λ1 − ξ
2
− 1 − ξ
∞
X
j=1
j1 − π
j
p
j
= 1 − ξ
∞
X
j=1
j π
j
p
j
− 1 − ξ
2
2 λ.
In the standard case when all π
d
= π, this leads to a simple criterion, which earlier has been shown by Britton, Janson and Martin-Löf [4] and Fountoulakis [7] by different methods. A modification
of the usual branching process argument for G
∗
n, d in [4] and a method similar to ours in [7].
97
Corollary 3.6 [4; 7]. Suppose that Condition 2.1 holds and 0 π 1. Then there exists w.h.p. a
giant component in G
∗
n, d
π,v
if and only if π π
c
:= E
D E
DD − 1
. 3.11
Remark 3.7. Note that π
c
= 0 is possible; this happens if and only if E D
2
= ∞. Recall that we assume 0
E D ∞, see Condition 2.1. Further, π
c
≥ 1 is possible too: in this case there is w.h.p. no giant component in G
∗
n, d except possibly in the special case when p
j
= 0 for all j 6= 0, 2, and consequently none in the subgraph G
∗
n, d
π
. Note that by 3.11,
π
c
∈ 0, 1 if and only if E D E DD − 1 ∞, i.e., if and only if 0 E
DD − 2 ∞.
Remark 3.8. Another case treated in [4] there called E1 is π
d
= α
d
for some α ∈ 0, 1. Theo-
rem 3.5 gives a new proof that then there is a giant component if and only if P
∞ j=1
j j − 1α
j
p
j
λ, which also can be written
α
2
g
′′ D
α λ = g
′ D
1. The cases E2 and A in [4] are more complicated and do not follow from the results in the present paper.
For edge percolation we similarly have the following; this too has been shown by Britton, Janson and Martin-Löf [4] and Fountoulakis [7]. Note that the percolation threshold
π is the same for site and bond percolation, as observed by Fountoulakis [7].
Theorem 3.9 [4; 7]. Consider the bond percolation model G
∗
n, d
π,e
, and suppose that Condi- tion 2.1 holds and that 0
π 1. Then there is w.h.p. a giant component if and only if π π
c
:= E
D E
DD − 1
. 3.12
i If 3.12 holds, then there is a unique ξ = ξ
e
π ∈ 0, 1 such that π
1 2
g
′ D
1 − π
1 2
+ π
1 2
ξ + 1 − π
1 2
λ = λξ, 3.13
and then v
C
1
n
p
−→ χ
e
π := 1 − g
D
1 − π
1 2
+ π
1 2
ξ 0,
3.14 e
C
1
n
p
−→ µ
e
π := π
1 2
1 − ξλ −
1 2
λ1 − ξ
2
. 3.15
Furthermore, v C
2
n
p
−→ 0 and eC
2
n
p
−→ 0. ii If 3.12 does not hold, then v
C
1
n
p
−→ 0 and eC
1
n
p
−→ 0. Proof. We argue as in the proof of Theorem 3.5, noting that ˜
p
1
0 by 2.27. By 2.28, ζ E ˜
D ˜ D
− 2 = ζg
′′ ˜
D
1 − ζg
′ ˜
D
1 = πg
′′ D
1 − π
1 2
g
′ D
1 − 1 − π
1 2
λ = π E DD − 1 − λ,
which yields the criterion 3.12. Further, if 3.12 holds, then the equation g
′ ˜
D
ξ = ˜ λξ, which by
2.18 is equivalent to ζg
′ ˜
D
ξ = ζ˜ λξ = λξ, becomes 3.13 by 2.28.
98
By 3.5, 2.26, 2.22 and 2.28, v
C
1
n
p
−→ ζ − ζg
˜ D
ξ − 1 − ξ1 − π
1 2
λ = 1 − g
D
1 − π
1 2
+ π
1 2
ξ ,
which is 3.14. Similarly, 3.6, 2.26, 2.18 and 2.22 yield e
C
1
n
p
−→
1 2
λ1 − ξ
2
− 1 − ξ1 − π
1 2
λ = π
1 2
1 − ξλ −
1 2
λ1 − ξ
2
, which is 3.15. The rest is as above.
Remark 3.10. It may come as a surprise that we have the same criterion 3.11 and 3.12 for site and bond percolation, since the proofs above arrive at this equation in somewhat different
ways. However, remember that all results here are consistent with the standard branching process approximation in Remark 3.4 even if our proofs use different arguments and it is obvious that both
site and bond percolation affect the mean number of offspring in the branching process in the same way, namely by multiplication by
π. Cf. [4], where the proofs are based on such branching process approximations.
Define ρ
v
= ρ
v
π := 1 − ξ
v
π and
ρ
e
= ρ
e
π := 1 − ξ
e
π; 3.16
recall from Remark 3.4 that ξ
v
and ξ
e
are the extinction probabilities in the two branching processes defined by the site and bond percolation models, and thus
ρ
v
and ρ
e
are the corresponding survival probabilities. For bond percolation, 3.13–3.15 can be written in the somewhat simpler forms
π
1 2
g
′ D
1 − π
1 2
ρ
e
= λπ
1 2
− ρ
e
, 3.17
v C
1
n
p
−→ χ
e
π := 1 − g
D
1 − π
1 2
ρ
e
π ,
3.18 e
C
1
n
p
−→ µ
e
π := π
1 2
λρ
e
π −
1 2
λρ
e
π
2
. 3.19
Note further that if we consider site percolation with all π
j
= π, 3.8 can be written π λ − g
′ D
1 − ρ
v
= λρ
v
3.20 and it follows by comparison with 3.17 that
ρ
v
π = π
1 2
ρ
e
π. 3.21
Furthermore, 3.9, 3.10, 3.18 and 3.19 now yield χ
v
π = π 1 − g
D
ξ
v
π = π 1 − g
D
1 − ρ
v
π = πχ
e
π, 3.22
µ
v
π = πλρ
v
π −
1 2
λρ
v
π
2
= πµ
e
π. 3.23
We next consider how the various parameters above depend on π, for both site percolation and
bond percolation, where for site percolation we in the remainder of this section consider only the case when all
π
j
= π. We have so far defined the parameters for
π ∈ π
c
, 1 only; we extend the definitions by letting ξ
v
:= ξ
e
:= 1 and ρ
v
:= ρ
e
:= χ
v
:= χ
e
:= µ
v
:= µ
e
:= 0 for π ≤ π
c
, noting that this is compatible with the branching process interpretation of
ξ and ρ in Remark 3.4 and that the equalities in 3.8– 3.23 hold trivially.
99
Theorem 3.11. Assume Condition 2.1. The functions ξ
v
, ρ
v
, χ
v
, µ
v
, ξ
e
, ρ
e
, χ
e
, µ
e
are continuous func- tions of
π ∈ 0, 1 and are analytic except at π = π
c
. Hence, the functions are analytic in 0, 1 if and only if
π
c
= 0 or π
c
≥ 1. Proof. It suffices to show this for
ξ
v
; the result for the other functions then follows by 3.16 and 3.21–3.23. Since the case
π ≤ π
c
is trivial, it suffices to consider π ≥ π
c
, and we may thus assume that 0
≤ π
c
1. If
π ∈ π
c
, 1, then, as shown above, g
′ ˜
D
ξ
v
= ˜ λξ
v
, or, equivalently, G ξ
v
, π = 0, where Gξ, π :=
g
′ ˜
D
ξξ− ˜λ is an analytic function of ξ, π ∈ 0, 1
2
. Moreover, G ξ, π is a strictly convex function
of ξ ∈ 0, 1] for any π ∈ 0, 1, and Gξ
v
, π = G1, π = 0; hence
∂ Gξ,π ∂ ξ
¯ ¯
ξ=ξ
v
0. The implicit function theorem now shows that
ξ
v
π is analytic for π ∈ π
c
, 1. For continuity at
π
c
, suppose π
c
∈ 0, 1 and let ˆ ξ = lim
n →∞
ξ
v
π
n
for some sequence π
n
→ π
c
. Then, writing ˜
D π and ˜
λπ to show the dependence on π, g
′ ˜
D π
n
ξ
v
π
n
= ˜ λπ
n
ξ
v
π
n
and thus by continuity, e.g. using 2.28, g
′ ˜
D π
c
ˆ ξ = ˜
λπ
c
ˆ ξ. However, for π ≤ π
c
, we have E ˜ D ˜
D − 2 ≤ 0
and then ξ = 1 is the only solution in 0, 1] of g
′ ˜
D
ξ = ˜ λξ; hence ˆ
ξ = 1. This shows that ξ
v
π → 1 as
π → π
c
, i.e., ξ
v
is continuous at π
c
.
Remark 3.12. Alternatively, the continuity of ξ
v
in 0, 1 follows by Remark 3.4 and continuity of the extinction probability as the offspring distribution varies, cf. [4, Lemma 4.1]. Furthermore, by
the same arguments, the parameters are continuous also at π = 0 and, except in the case when
p + p
2
= 1 and thus ˜ D = 1 a.s., at
π = 1 too. At the threshold
π
c
, we have linear growth of the size of the giant component for slightly larger π, provided E D
3
∞, and thus a jump discontinuity in the derivative of ξ
v
, χ
v
, . . . . More precisely, the following holds. We are here only interested in the case 0
π
c
1, which is equivalent to E DD − 2 ∞, see Remark 3.7.
Theorem 3.13. Suppose that 0 E DD − 2 ∞; thus 0 π
c
1. If further E D
3
∞, then as ǫ ց 0,
ρ
v
π
c
+ ǫ ∼ 2 E DD
− 1 π
c
E DD
− 1D − 2 ǫ =
2 E DD − 1
2
E D
· E DD − 1D − 2 ǫ
3.24 χ
v
π
c
+ ǫ ∼ µ
v
π
c
+ ǫ ∼ π
c
λρ
v
π
c
+ ǫ ∼ 2 E D
· E DD − 1 E
DD − 1D − 2
ǫ. 3.25
Similar results for ρ
e
, χ
e
, µ
e
follow by 3.21–3.23. Proof. For
π = π
c
+ ǫ ց π
c
, by g
′′ D
1 = E DD − 1 = λπ
c
, see 3.11, and 3.20, ǫ g
′′ D
1ρ
v
= π − π
c
g
′′ D
1ρ
v
= πg
′′ D
1ρ
v
− λρ
v
= π g
′′ D
1ρ
v
− λ + g
′ D
1 − ρ
v
. 3.26
Since E D
3
∞, g
D
is three times continuously differentiable on [0, 1], and a Taylor expansion yields g
′ D
1 − ρ
v
= λ − ρ
v
g
′′ D
1 + ρ
2 v
g
′′′ D
12 + oρ
2 v
. Hence, 3.26 yields, since ρ
v
0, ǫ g
′′ D
1 = πρ
v
g
′′′ D
12 + oρ
v
= π
c
ρ
v
g
′′′ D
12 + oρ
v
.
100
Thus, noting that g
′′ D
1 = E DD − 1 and g
′′′ D
1 = E DD − 1D − 2 0 since E DD − 2 0, ρ
v
∼ 2g
′′ D
1 π
c
g
′′′ D
1 ǫ =
2 E DD − 1
π
c
E DD
− 1D − 2 ǫ,
which yields 3.24. Finally, 3.25 follows easily by 3.22 and 3.23. If E D
3
= ∞, we find in the same way a slower growth of ρ
v
π, χ
v
π, µ
v
π at π
c
. As an example, we consider D with a power law tail, p
k
∼ ck
−γ
, where we take 3 γ 4 so that E D
2
∞ but E
D
3
= ∞.
Theorem 3.14. Suppose that p
k
∼ ck
−γ
as k → ∞, where 3 γ 4 and c 0. Assume further that
E DD
− 2 0. Then π
c
∈ 0, 1 and, as ǫ ց 0, ρ
v
π
c
+ ǫ ∼ E
DD − 1
c π
c
Γ2 − γ
1 γ−3
ǫ
1 γ−3
, χ
v
π
c
+ ǫ ∼ µ
v
π
c
+ ǫ ∼ π
c
λρ
v
π
c
+ ǫ ∼ π
c
λ E
DD − 1
c π
c
Γ2 − γ
1 γ−3
ǫ
1 γ−3
. Similar results for
ρ
e
, χ
e
, µ
e
follow by 3.21–3.23. Proof. We have, for example by comparison with the Taylor expansion of 1
− 1 − t
γ−4
, g
′′′ D
1 − t =
∞
X
k=3
kk − 1k − 2p
k
1 − t
k −3
∼ cΓ4 − γt
γ−4
, t
ց 0, and thus by integration
g
′′ D
1 − g
′′ D
1 − t ∼ cΓ4 − γγ − 3
−1
t
γ−3
= c|Γ3 − γ|t
γ−3
, and, integrating once more,
ρ
v
g
′′ D
1 − λ − g
′ D
1 − ρ
v
∼ cΓ2 − γρ
γ−2 v
. Hence, 3.26 yields
ǫ g
′′ D
1ρ
v
∼ cπ
c
Γ2 − γρ
γ−2 v
and the results follow, again using 3.22 and 3.23.
4 k
-core
Let k ≥ 2 be a fixed integer. The k-core of a graph G, denoted by Core
k
G, is the largest induced subgraph of G with minimum vertex degree at least k. Note that the k-core may be empty. The
question whether a non-empty k-core exists in a random graph has attracted a lot of attention for various models of random graphs since the pioneering papers by Bollobás [2], Łuczak [14] and
Pittel, Spencer and Wormald [17] for Gn, p and Gn, m; in particular, the case of Gn,
d and
101
G
∗
n, d with given degree sequences have been studied by several authors, see Janson and Luczak
