On the normality of giant components
Taral Guldahl Seierstad
University of Oslo
November 19, 2008
Abstract
We consider a general family of random graph processes, which begin with an empty
graph, and where at every step an edge is added at random according to some rule. We
show that when certain general conditions are satisfied, the order of the giant component
tends to a normal distribution.
1
Introduction
We consider discrete random graph processes, where the initial state typically is an empty
graph on n vertices, and where, at every step, an edge is added at random, according to
some given rule. The standard example of such graph processes is the model introduced
by Erdős and Rényi [5], where at every step the edge to be added is chosen uniformly at
random from all edges which are not already present in the graph. We use the notation
GER
n,m to denote the state in the Erdős–Rényi graph process on n vertices after m edges have
been added.
An interesting phenomenon in this graph process is the emergence of the so-called giant
component: Erdős and Rényi [5] showed that when cn edges have been added and c > 1/2,
then asymptotically almost surely (i.e. with probability tending to 1 as n tends to infinity,
abbreviated a.a.s.) there is a unique component with Θ(n) vertices, whereas the second
largest component has Θ(ln n) vertices. In fact they showed that if Γn,m is the order of
p
−1
the largest component in GER
Γn,cn → d, where d is the number
n,m , and c > 1/2, then n
between 0 and 1 such that d + e−2cd = 1.
Later, Pittel [10] showed that Γn,cn tends to a normal distribution; that is, there is a
constant σ, depending on c, such that
Γn,cn − dn d
√
→ N (0, σ 2 ),
n
where N (0, σ 2 ) denotes the probability distribution of a normal random variable with mean 0
and variance σ 2 . Such central limit theorems for the giant component have interested several
authors, in particular in relation to the similar random graph model G(n, p). For that graph
model a central limit theorem was first found by Stepanov [16], and it has been reproved
several times. In fact, Pittel’s proof in [10] also works for this model. Another proof is due
to Barraez, Boucheron and Fernandez de la Vega [2] who used a random walk approach. For
hypergraphs, the problem has been studied by Behrisch, Coja-Oghlan and Kang [3], who
proved a similar theorem.
1
In this paper we consider a wide family of random graph processes, which includes GER
n,m ,
and we show that a central limit theorem holds for these processes as well, provided that
certain conditions are satisfied. The main requirement of our method is that at every
stage in the graph process, the probability that the next edge to be added is incident to
a particular component depends only on certain features of the component, such as the
number of vertices in the component, or the degrees of the vertices in the component. This
is satisfied by several random graph processes which have been studied earlier; in particular,
in Section 4, we will consider the so-called minimum-degree graph process [18, 7, 8] in detail.
In Section 5 we will discuss some other graph processes for which this technique may work.
Note that we will not prove that these graph processes actually have a giant component; we
will merely show that if there is a giant component, then the number of vertices it contains
is asymptotically normally distributed.
In order to prove a central limit theorem for the giant component in this more general
setting, we will use an approach similar to that of Pittel [10]. A central ingredient in
our proof is the so-called differential equation method. Wormald [17, 18] used differential
equations as a general way to prove laws of large numbers for random variables defined on
graph processes. This was extended to a central limit theorem by Seierstad [13], and this
theorem will be used to prove the main theorem of the present paper.
In Section 2 we present the differential equation method to be used in the proof. The
theorem we present is a generalization of the main theorem in [13]; the proof is found
in Section 6. In Section 3 a rather general method to prove that the giant component has
asymptotically normal distribution is presented. The method requires a number of conditions
to be satisfied in order to function. Therefore, in Section 4, we apply it to the minimum
degree graph process, thereby showing that it actually can be used in real problems. In
Section 5 we briefly discuss some other processes to which the method can be applied.
2
The differential equation method
Let (Ωn , Fn , Pn ) be a sequence of probability spaces, and assume that a filtration Fn,0 ⊆
Fn,1 ⊆ · · · ⊆ Fn,mn ⊆ Fn is given for every n ≥ 1, where mn = O(n). Let q ≥ 1
be a fixed number and let Xn,m,1 , . . . , Xn,m,q be Fn,m -measurable random variables. Let
X n,m = [Xn,m,1 , . . . , Xn,m,q ]0 , where v 0 denotes the transpose of v, and let ∆Xn,m,k =
Xn,m,k − Xn,m−1,k for 1 ≤ m ≤ mn and 1 ≤ k ≤ q. If Σ is a q × q-matrix, N (0, Σ)
denotes the multivariate normal distribution with mean 0 and covariance matrix Σ; if σ is a
constant, N (0, σ 2 ) denotes the univariate normal distribution with mean 0 and variance σ 2 .
Asymptotic statements are meant to hold as n → ∞.
Wormald [17, 18] developed the differential equation method: provided that the random
variables satisfy certain conditions, they converge in probability to the solution of some
differential equations. Seierstad [13] extended this method by showing that if further conditions are satisfied, then the random variables converge jointly to a multivariate normal
distribution. The following theorem is a combination of Theorem 5.1 of Wormald [18] and
the main theorem of Seierstad [13], with a slight generalization to allow for random initial
states.
(n)
Theorem 1. Assume that there is a constant C0 such that Xm,k ≤ C0 n for all n, 0 ≤ m ≤
mn and 1 ≤ k ≤ q. Let fk , gij : Rq → R, 1 ≤ i, j, k ≤ q, be functions. Assume that there is
a constant vector µ0 and a constant q × q-matrix Σ0 such that
X 0 − nµ0 d
√
→ N (0, Σ0 ).
n
2
(1)
Let D be a bounded connected open set containing the closure of
(n)
{(z1 , . . . , zq ) : P[X0,k = zk n, 1 ≤ k ≤ q] 6= 0
for some n},
and let HD = HD (X n,m ) be the random variable denoting the minimum m such that
n−1 X n,m ∈
/ D. Assume that the following conditions hold.
(i) For some function β = β(n), with 1 ≤ β = o(n1/12−ε ) for some ε > 0, we have
(n)
∆Xm,i ≤ β, for 1 ≤ m < HD and 1 ≤ i ≤ q.
(ii) For some function λ1 (n) = o(n−1/2 ), and all i ≥ 1 and 1 ≤ m < HD ,
(n)
E[∆Xm,i | Fm−1 ] − fi (n−1 Xm−1,1 , . . . , n−1 Xm−1,q ) ≤ λ1 (n).
(iii) For some function λ2 (n) = o(1), and all i, j ≥ 1 and 1 ≤ m < HD ,
(n)
(n)
E[∆Xm,i ∆Xm,j | Fm−1 ] − gij (n−1 Xm−1,1 , . . . , n−1 Xm−1,q ) ≤ λ2 (n).
(iv) Each function fi is continuous and differentiable, with continuous derivatives, and
satisfies a Lipschitz condition on D.
(v) Each function gij is continuous and satisfies a Lipschitz condition on D.
Then the following are true.
(a) For (ẑ1 , . . . , ẑq ) ∈ D, the system of differential equations
dzk
= fk (z1 , . . . , zq ),
dt
k = 1, . . . , q,
has a unique solution in D for zk : R → R passing through zk (0) = ẑk , 1 ≤ k ≤ q, and
which extends to points arbitrarily close to the boundary of D.
(b) Let F : Rq → Rq be the function F (z) = [f1 (z), . . . , fq (z)]0 , and let G : Rq → Rq×q be
the function G(z) = {gij (z)}ij . Let α : R → Rq be the unique function satisfying
d
α(t) = F (α(t))
dt
∂fi
with α(0) = µ0 and let J(z) = { ∂z
}i,j be the Jacobian matrix of F . Let T (t) be the
j
q × q-matrix satisfying the differential equation
d
T (t) = −T (t)J(α(t)),
dt
Let
Z
Ξ(t) = Σ0 +
T (0) = I.
(2)
t
T (s)(G(α(s)) − F (α(s))2 )T (s)0 ds
(3)
0
and let Σ(t) = T (t)−1 Ξ(t)(T (t)0 )−1 . If
(n)
(n)
Ym,k =
Xm,k − nαk (m/n)
√
n
(n)
(n)
0
for 1 ≤ k ≤ q, and Y (n)
m = [Ym,1 , . . . , Ym,q ] , then
d
Y (n)
m → N (0, Σ(m/n)),
3
(4)
for 0 ≤ m ≤ σn ≤ mn , where σ = σ(n) is the supremum of those m to which the
solution α can be extended before reaching within L∞ -distance Cλ of the boundary
of D, for a sufficiently large constant C. Moreover, the convergences (1) and (4)
occur jointly.
This theorem is proved in [13] for the case that Σ0 is the zero matrix. A modification
of the proof is necessary to allow for nonzero initial covariance matrices. The reason that
we include this generalization is that it is necessary for studying the minimum-degree graph
process in Section 4; for most typical graph processes however, the generalization is not
needed. Since this is not the central theme of the paper, we defer the proof of Theorem 1
to Section 6.
3
General theorem
Let {Gn,m }m≥0 be a Markov process, whose states are graphs on n vertices.
Let T1 , T2 , . . . be classes of trees such that every tree is a member of Tk for exactly one k.
We require that all the trees in a given class have the same number of vertices, and we let
ck be the number of vertices in the trees contained in Tk . Furthermore, we assume that the
sequence (c1 , c2 , . . .) is non-decreasing and that the number of integers i such that ci = k is
at most k γ for some γ ≥ 0.
(n)
Let Tm,k be the random variable denoting the number of components in Gn,m which are
(n)
(n)
(n)
members of Tk . Let ∆Tm,k = Tm,k − Tm−1,k .
For δ > 0, let Lδ = Lδ,{ck } be the Banach space of all sequences x = (x1 , x2 , . . .) with
the norm
X
kxk =
cδk |xk | < ∞.
k≥1
(q)
Theorem 2. For every q ≥ 1, let functions fi
(q)
: Rq → R and gij : Rq×q → R be given
(q)
(q 0 )
for 1 ≤ i, j ≤ q. Assume that if q > q 0 ≥ i, then fi (z1 , . . . , zq ) = fi
(z10 , . . . , zq0 0 ) and if
(q 0 )
gij (z10 , . . . , zq0 0 ),
(q)
gij (z1 , . . . , zq )
q > q 0 ≥ max(i, j), then
=
whenever zk = zk0 for 1 ≤ k ≤ q.
Assume furthermore that for every fixed q ≥ 1, the conditions of Theorem 1 are satisfied,
(n)
(n)
with Xm,k = Tm,k and for some D = Dq ⊆ Rq . Let t be a constant, and let m = btnc.
Assume that there are constants K, C 0 and ρ with 0 < ρ < 1, possibly depending on t, such
that
(n)
ETm,k ≤ Knρck
and
(n)
Var Tm,k ≤ Knρck .
Then there exist continuous functions µi (t) and σij (t) for i, j ≥ 1, and jointly normal
variables W1 (t), W2 (t), . . . such that EWi (t) = 0 and Cov(Wi (t), Wj (t)) = σij (t) and such
that the following hold.
a. Let
(n)
(n)
Um,k
Tm,k − nµk (t)
√
=
n
for k ≥ 1. If q ≥ 1 is fixed, then
(n)
d
Um,k → Wk (t)
jointly for 1 ≤ k ≤ q and 0 ≤ m ≤ σn ≤ mn , where σ = σq is as in Theorem 1.
4
(n)
(n)
∗
b. Let U (n)
m = (Um,1 , Um,2 , . . .) and W (t) = (W1 (t), W2 (t), . . .) and σ = inf q σq . Then,
(n)
for 0 ≤ m ≤ σ ∗ n, W (t) ∈ Lδ a.s., and U m converges to W (t) in distribution; that
is, for every bounded continuous functional h on Lδ , h(U (n)
m ) converges in distribution
to h(W (t)).
(n)
c. Let Zm be the number of vertices contained in components which are not trees. Then
there are continuous functions µ(t) and σ(t) such that for 0 ≤ m ≤ σ ∗ n,
(n)
Zm − nµ(t) d
√
→ N (0, σ(t)).
n
(n)
d. Let Γm be the random variable denoting the number of vertices in the largest compo(n)
nent in Gn,m , and let Cm be the random variable denoting the number of vertices in
√
(n)
cyclic components, except for the largest component. Assume that Cm = o( n) a.a.s.
Then, for 0 ≤ m ≤ σ ∗ n,
(n)
Γm − nµ(t) d
√
→ N (0, σ(t)).
n
Proof. In our proof, (a) is proved by the differential equation method, while the proof of (b)
follows the proof by Pittel [10]. Finally, (c) and (d) follow from (b).
(q)
(q)
Henceforth we will suppress the dependence on n. The conditions imply that fi and gij
are essentially the same for all values of q; we therefore refer to these functions as fi and gij
and we note that fi only depends on z1 , . . . , zi and gij only depends on z1 , . . . , zmax(i,j) .
Thus, the functions µi (t) and σij (t), which we obtain by the differential equation method
will be the same, regardless of the value of q. Conclusion (a) then follows directly from
Theorem 1.
In order to prove (b) we first have to show that W (t) ∈ Lδ a.s. for δ > 0. By Jensen’s
d
2
inequality, E[|Um,k |]2 ≤ E[Um,k
]. Moreover, by (a), Um,k → N (0, σk2 ), for some σk ≥ 0, so
E[Um,k ] = o(1) as n → ∞. Then
(n)
E[|Um,k |]2 ≤ Var[Um,k ] + E[Um,k ]2
= Var
Tm,k − nµk (m/n)
√
+ o(1)
n
= n−1 Var Tm,k + o(1) ≤ Kρck + o(1).
√
ρ < ρ0 < 1. Then, by Markov’s inequality,
√ c /2
E|Um,k |
Kρ k + o(1)
0 ck
P[|Um,k | > ρ ] ≤
≤
= Kθck + o(1),
ρ0ck
ρ0ck
√
where 0 < θ = ρ/ρ0 < 1. By (a), for every a, P[|Wm,k | > a] = limn→∞ P[|Um,k | > a], so
Let ρ0 be a constant such that
X
c
P[|Wk (t)| > ρ0 k ] ≤ K
X
θ ck ≤ K
k≥1
k≥1
X
k γ θk < ∞.
k≥1
c
By the Borel–Cantelli lemma, there is a.s. a k0 such that for all k ≥ k0 , |Wk (t)| ≤ ρ0 k .
Hence,
X
X
k
cδk |Wk (t)| ≤
k δ+γ ρ0 < ∞,
k≥k0
k≥ck0
5
so W (t) ∈ Lδ a.s.
In order to prove that U m tends to W (m/n) in distribution, we have to show the
following. (See Billingsley [4].)
(i) Convergence of the finite-dimensional distributions. For any selection of k positive
integers i1 , . . . , ik , where k ≥ 1, the vector [Um,i1 , . . . , Um,ik ] converges in distribution
to [Wi1 (t), . . . , Wik (t)] as n tends to infinity.
(ii) Tightness. For every ε > 0 there exists a compact subset K(ε) of Lδ such that
P[U 6∈ K(ε)] < ε for every n.
Convergence of the finite-dimensional distributions follows from (a), so it remains to
show that (ii) holds. Let
√
K(ε) = {x ∈ Lδ : |xk | ≤ ε−1 3 ρck },
which is a compact set. By Chebyshev’s inequality, for ν > 0,
P[|Um,k | > ν] ≤
Var Tm,k
Var Um,k
Kρck
=
.
≤
ν2
ν2n
ν2
Hence,
P[U m ∈
/ K(ε)] ≤
X
X
√
√
P[|Um,k | > ε−1 3 ρck ] ≤
Kε2 3 ρck = O(ε2 ).
k≥1
k≥1
P
Therefore the tightness condition is met, and (b) follows. If we define h(x) = k≥1 ck xk ,
this function is clearly bounded on L1 , so h(U (n)
m ) converges in distribution to h(W (t)).
Since
X
Zm = n −
ck Tm,k ,
k≥1
we see that (c) follows directly
from (b).
√
Finally, since Cm = o( n) a.a.s. and Γm = Zm − Cm ,
Γm − nµ(t)
Zm − nµ(t)
√
√
P
>a =P
> a + o(1),
n
n
so n−1/2 (Γm − nµ(t)) and n−1/2 (Zm − nµ(t)) converge to the same distribution.
Theorem 2 is adequate for graph processes such as GER
n,m , where the probability that
the next edge chosen is incident to a particular component depends only on the type of
the component. However, we can also allow the probability to depend on other random
variables, such as the total number of vertices of degree k, say, in the graph, provided that
these random variables also behave nicely, and can be subjected to the differential equation
method.
(n)
(n)
Theorem 3. Let Dm,1 , Dm,2 , . . . be Fn,m -measurable random variables, and let
(n)
(n)
(n)
(n)
Φ = {Dm,1 , Dm,2 , . . .} ∪ {Tm,1 , Tm,2 , . . .}.
(n)
(n)
Order the elements of Φ, and write Φ = (Xm,1 , Xm,2 , . . .), so that each Xm,k is equal to
Dm,i or Tm,i for some i. Assume that the random variables in Φ have been ordered in such
(n)
(n)
a way that Theorem 1 can be applied to any initial segment Φq := (Xm,1 , . . . , Xm,q ) of Φ.
Assume furthermore, that the conditions of Theorem 2 are otherwise satisfied. Then the
conclusions of Theorem 2 hold.
6
Proof. We observe that the differential equation can be applied to the random variables
in Φq . Thus, the proof of Theorem 2 can be applied virtually unchanged.
A number of conditions must be satisfied in order for Theorem 2 or Theorem 3 to be
applied. In the next section we consider a concrete example — the minimum degree graph
process — in order to demonstrate that the procedure explained in this section actually is
practical to apply to real graph processes.
The differential equation method provides an explicit formula for the mean µ(t) and the
covariance matrix Σ(t). In principle it is calculate the functions µ(t) and σ(t) from µ(t) and
Σ(t), but in many cases this will be unfeasible. So although we can show that the order of
the giant component is asymptotically normally distributed, our method is not necessarily
suitable for calculating the actual parameters of the normal distribution.
4
The minimum degree graph process
We will now apply the method of the previous section to the minimum degree graph process,
which was introduced by Wormald [18]. Let Gmin
n,0 be the empty graph on n vertices, and
for
m
≥
1
be
obtained
in
the
following
way: Choose first a vertex v in Gmin
let Gmin
n,m
n,m−1
uniformly at random from the set of vertices of minimum degree; then choose a vertex w
uniformly at random from the set of vertices distinct from v. Note that w may have any
min
degree. Let Gmin
n,m be the graph on n vertices consisting of all edges in Gn,m−1 in addition
to the edge vw.
Let h1 = ln 2 ≈ 0.69 and h2 = ln 2 + ln(1 + ln 2) ≈ 1.22, and let t = n/m. Kang et al. [7]
showed that if t is a fixed number with 0 ≤ t < h1 , then the minimum degree is a.a.s. 0; if
h1 < t < h2 , then the minimum degree is a.a.s. 1; and if t > h2 , then the minimum degree
is a.a.s. at least 2. Moreover, they showed that if t > h2 , then the largest component a.a.s.
has n − o(n) vertices, and the graph is in fact connected with probability bounded away
from 0.
Moreover, it was shown by Kang and Seierstad [8] that there is a phase transition at the
2−2
≈ 0.86. That is, when t < hg , the largest component
point hg = ln 2 ln 2−1+16√ln27−16
ln 2 ln 2
a.a.s. has order Θ(ln n), whereas when t > hg , the largest component a.a.s. has order Θ(n),
while every other component a.a.s. has order O(ln n). Although not explicitly stated, it is
implied by the proof in [8] that there is a function µ(t) such that if Γn,tn is the number
p
−1
of vertices in the largest component in Gmin
Γn,tn → µ(t) when t ∈ (hg , h2 ).
n,tn , then n
(See Seierstad [14].) Moreover, a formula for µ(t) was given in [8]. Here we will show the
following theorem.
Theorem 4. There are functions µ(t) and σ(t), which are continuous on (hg , h2 ), such that
Γn,tn − nµ(t) d
√
→ N (0, σ(t))
n
for hg < t < h2 .
Due to some complications explained below, specific to this graph process, we cannot
apply Theorem 2 or Theorem 3 directly. However, the general method explained in Section 3
can be used. For reasons explained in Section 4.1 we will split the trees into classes depending
on their order and number of leaves; thus Tk,l is the set of trees of order k with l leaves, and
we let Tm,k,l be the random variable denoting the number of trees in Gmin
n,m with k vertices
and l leaves. The fact that we use two indices instead of one is only for convenience, and
7
has no effect on the argument. By considering the proof of Theorem 2, we see that in order
to prove Theorem 4 it is sufficient to show that the following three conditions are satisfied.
1. Any finite number of the random variables Tm,k,l converge jointly to a multivariate
normal distribution.
2. There are constants K > 0 and ρ with 0 < ρ < 1 such that ETm,k,l ≤ Knρk and
Var Tm,k,l ≤ Knρk .
3. Let Cn,m be the random
variable denoting the number of vertices in cyclic components.
√
Then ECn,m = o( n).
These three conditions will be shown to hold in the next three sections.
4.1
Applying the differential equation method
From now on, we will write Gn,m for Gmin
n,m . Assume that t is a fixed number with 0 ≤ t < h2 .
If 0 ≤ t < hg , then the graph consists a.a.s. of small components of order O(ln n). If
hg < t < h2 , there is a.a.s. a giant component, but a positive proportion of the vertices is
contained in components of order O(ln n).
Let C be a component in Gn,m , and suppose vw is the next edge to be added to the
process, with v chosen first and w second. The probability that w is contained in C is
proportional to the number of vertices in C. When the minimum degree is 0, v must be an
isolated vertex; when the minmum degree is 1, the probability that v is contained in C is
proportional to the number of leaves in C. Thus, we should define the classes Tk in terms
of both the number of vertices and the number of leaves. It is then most practical to use
two indices. We define a leaf to be a vertex of degree 0 or 1; thus, an isolated vertex is
considered to be a leaf. We let Λ be the set of pairs (k, l) for which there exists at least one
tree with k vertices and l leaves; in other words
Λ = {(1, 1)} ∪ {(2, 2)} ∪ {(k, l); k ≥ 3, 2 ≤ l ≤ k − 1}.
For (k, l) ∈ Λ, let Tk,l be the set of trees which have k vertices and l leaves. Clearly, every
tree is in exactly one class, and there are max(1, k − 2) classes of trees with k vertices, so
this partition satisfies our conditions.
For (k, l) ∈ Λ, let Tm,k,l be the number of tree components in Gn,m which are of type Tk,l .
We also define Tm,k,l = 0 whenever (k, l) ∈
/ Λ, so that we do not need to be so careful with
the bounds on the summation indices. In order to use the differential equation method,
we should now express the expected change ∆Tm,k,l by functions fk,l ; however, here we
encounter a problem, since the functions fk,l are different when there are vertices of degree 0
in the graph, and when the minimum degree is 1. Thus, the functions fk,l are not necessarily
continuous at t = h1 .
We therefore consider the graph process in two phases: the first when the minimum
degree is 0, and the second when the minimum degree is 1.
This graph process was studied in [7] and [8], and we will use some of the calculations
from those papers, which we summarize here:
Lemma 5. Let H be the random variable denoting the smallest integer for which Gn,H
has minimum degree 1, and let Lm be the random variable denoting the number of vertices
p
p
of degree 1 in Gn,m . Then n−1 H → ln 2, n−1 LH → ln 2 and there are constants µk,l
p
for (k, l) ∈ Λ such that n−1 TH,k,l → µk,l , with µ1,1 = 0 and µk,l > 0 for (k, l) 6= (1, 1).
Moreover, there is a continuous function b(t) with b(t) > 0 for t ∈ (0, h2 ) and b(t) = 0
p
otherwise, such that for t = m/n, n−1 Lm → b(t).
8
p
p
Proof. In [7] it was shown that n−1 H → ln 2 and n−1 LH → ln 2. Let Rm,k be the number
of trees in Gn,m of order k. In [8] it was shown that there are constants ρk with ρ1 = 0 and
p
ρk > 0 for k > 1, such that n−1 RH,k → ρk . Moreover it was shown that there are constants
pk,l > 0 for (k, l) ∈ Λ such that if C is an arbitrary component with order k, then it has
p
exactly l leaves with probability pk,l . It follows that n−1 TH,k → µk,l where µk,l = pk,l ρk ,
and that µk,l = 0 for (k, l) = (1, 1) and µk,l > 0 for (k, l) ∈ Λ\{(1, 1)}. Finally, the existence
of b(t) was shown in [7].
The next lemma shows that these random variables also satisfy a central limit theorem:
they converge jointly to normal random variables.
Lemma 6. Let Λ∗ be a finite subset of Λ. There are jointly normal random variables
d
d
η, χ and {Uk,l }(k,l)∈Λ∗ such that n−1/2 (H − n ln 2) → η, n−1/2 (LH − n ln 2) → χ and
d
n−1/2 (TH,k,l − µk,l n) → Uk,l for all (k, l) ∈ Λ∗ jointly as n → ∞.
Proof. We can assume without loss of generality that Λ∗ is on the form {(k, l) ∈ Λ; k ≤ q}
for some fixed integer q.
The proof uses Theorem 1; however, there is a complication, since we want to determine
the distribution of the random variables with respect to the random stopping time H. In
order to make sure that Theorem 1 can be used, we would prefer that the process continues
◦
which are
smoothly also some time after H. Below we will introduce random variables Tm,k,l
identical to Tm,k,l when m ≤ H, but which continue smoothly also some time after m = H.
∗
∗
∗
We will now find functions f(k,l) : RΛ → R and g(k,l),(k0 ,l0 ) : RΛ ×Λ → R such that
E[∆Tm,k,l | Fm−1 ] = f(k,l) ({n−1 Tm,κ,λ }(κ,λ)∈Λ∗ ) + o(n−1/2 )
(5)
E[∆Tm,k,l ∆Tm,k0 ,l0 | Fm−1 ] = g(k,l),(k0 ,l0 ) ({n−1 Tm,κ,λ }(κ,λ)∈Λ∗ ) + o(1).
(6)
and
Assume that m ≤ H, and let vm wm be the mth edge added in the graph process. When
we talk about the properties of vm and wm , we always mean their properties as vertices
in Gn,m−1 . Then vm is necessarily an isolated vertex, while wm can be in any type of
component. The probability that wm is in a particular component is proportional to the
order of the component. Let Wm ∈ Λ × {1, 2} be the random vector containing the relevant
characteristics of wm : We let Wm = (κ, λ, d) if wm is contained in a component of order κ
with λ leaves and d = 1 if wm is a leaf and d = 2 otherwise. Then, for (κ, λ) ∈ Λ∗ and
d = 1, 2,
λTm−1,κ,λ − δκ1
P[Wm = (κ, λ, 1) | Fm−1 ] =
(7)
n−1
and
(κ − λ)Tm−1,κ,λ
P[Wm = (κ, λ, 2) | Fm−1 ] =
.
(8)
n−1
Define

 −δkκ δlλ + δk,κ+1 δl,λ+d−1 if (k, l) ∈ Λ \ {(2, 2)},
(d)
−δκ2 δλ2 + δκ1 δλ1
if (k, l, d) = (2, 2, 1),
ζk,l (κ, λ) =

0
otherwise.
If Wm = (κ, λ, d), then
(d)
∆Tm,k,l = −δk1 + ζk,l (κ, λ).
(9)
To see that this holds, note that if w is contained in a (κ, λ)-component with κ ≥ 2, then
we lose a component of type (κ, λ). On the other hand, we gain a component with κ + 1
9
vertices, where the number of leaves is λ if w is a leaf, and λ + 1 if w is not a leaf. The
case (κ, λ) = (1, 1) is a bit different, since we then get a component with two vertices and
λ + 1 = 2 leaves, even though w is a leaf itself. Finally, δk1 is subtracted, since an isolated
vertex disappears at every step. Thus,
X
(d)
E[∆Tm,k,l | Fm−1 ] = − δk1 +
P[Wm = (κ, λ, d) | Fm−1 ]ζk,l (κ, λ)
κ,λ,d
= − δk1 +
X λTm−1,κ,λ
n
κ,λ
+
X (κ − λ)Tm−1,κ,λ
n
κ,λ
(1)
ζk,l (κ, λ)
(10)
(2)
ζk,l (κ, λ) + o(n−1/2 ).
Note that only a finite number of the summands are nonzero, so the sums are well-defined.
(d)
Furthermore ζk,l (κ, λ) = 0 whenever κ > k or λ > l, so we obtain an expression on the
form (5). In a similar manner, we obtain
Tm,1,1
Tm,1,1
2
(−2) + 1 −
(−1)2 + o(1)
E[∆Tm,1,1 ∆Tm,1,1 | Fm−1 ] =
n
n
(11)
3Tm−1,1,1
=
+ 1 + o(1),
n
and if k > 1,
E[∆Tm,1,1 ∆Tm,k,l | Fm−1 ]
X λTm,κ,λ
X (κ − λ)Tm,κ,λ
(1)
(2)
=
(−1 − δκ1 )ζk,l (κ, λ) +
(−1)ζk,l (κ, λ) + o(1).
n
n
κ,λ
(12)
κ,λ
Finally, if k, k 0 > 1,
E[∆Tm,k,l Tm,k0 ,l0 | Fm−1 ]
(13)
X
X λTm−1,κ,λ (1)
(κ − λ)Tm−1,κ,λ (2)
(1)
(2)
ζk,l (κ, λ)ζk0 ,l0 (κ, λ) +
ζk,l (κ, λ)ζk0 ,l0 (κ, λ) + o(1).
=
n
n
κ,λ
κ,λ
Hence, we also have found expressions on the form (6), so Conditions (ii) and (iii) of
∗
Theorem 1 are satisfied. We then have to find a domain D ∈ RΛ on which the functions f
and g satisfy a Lipschitz condition. Since the functions are all linear, this is simple: we always
∗
have 0 ≤ n−1 Tm,k,l ≤ 1, so we can define D = {{z(k,l) }(k,l)∈Λ∗ ∈ RΛ ; −ε < zk,l < 1 + ε} for
any ε > 0. The functions f(k,l) and g(k,l),(k0 ,l0 ) then clearly satisfy Conditions (iv) and (v).
Finally, since |∆Tk,l | ≤ 2 always, Condition (i) is satisfied. Hence, we can apply Theorem 1.
We conclude that for every t with 0 < t < h1 there are functions µk,l (t) and jointly normal
d
∗
∗
random variables Uk,l
(t) for (k, l) ∈ Λ∗ such that n−1/2 (Ttn,k,l − nµk,1 (t)) → Uk,l
(t) jointly.
Our aim is to show that Tm,k,l also converges to a normal distribution for the random
◦
time m = H. To this end we define random variables Tm,k,l
for m ≥ 0 and (k, l) ∈ Λ∗ which
behave like Tm,k,l when m ≤ H, but whose behaviour does not change abruptly as soon as
◦
m > H. We assume that m ≤ hn + o(n). We first define a random variable Wm
as follows.
◦
For 1 ≤ m ≤ H, let Wm = Wm . For m > H, we let
◦
P[Wm
= (κ, λ, 1) | Fm−1 ] =
10
◦
λ|Tm−1,κ,λ
| − δκ1
n−1
(14)
and
◦
P[Wm
= (κ, λ, 2) | Fm−1 ] =
◦
(κ − λ)|Tm−1,κ,λ
|
.
n−1
(15)
◦
◦
We define Tm,k,l
such that if Wm
= (κ, λ, d), then
(d)
◦
◦
∆Tm,k,l
= −δk1 + ζk,l (κ, λ) sgn(Tm−1,κ,λ
).
Then
◦
E[∆Tm,k,l
| Fm−1 ]
X
(d)
◦
◦
= − δk1 +
P[Wm
(κ, λ, d)] | Fm−1 ]ζk,l (κ, λ) sgn(Tm−1,κ,λ
)
κ,λ,d
= − δk1 +
◦
X λ|Tm−1,κ,λ
| − δκ1
n−1
κ,λ
+
◦
X (κ − λ)|Tm−1,κ,λ
|
n−1
κ,λ
= − δk1 +
◦
X λTm−1,κ,λ
κ,λ
n
(1)
◦
ζk,l (κ, λ) sgn(Tm−1,κ,λ
)
(2)
◦
ζk,l (κ, λ) sgn(Tm−1,κ,λ
) + o(n−1/2 )
(1)
ζk,l (κ, λ) +
◦
X (κ − λ)Tm−1,κ,λ
κ,λ
n
(2)
ζk,l (κ, λ) + o(n−1/2 ),
◦
which is the same as (10) except that Tm−1,κ,λ is exchanged with Tm−1,κ,λ
. Similarly, (11)
◦
◦
and (12) hold when Tm−1,κ,λ is exchanged with Tm−1,κ,λ , since |Tm−1,κ,λ
| is multiplied
◦
by sgn(Tm−1,κ,λ
) whenever it appears. It is not so clear that (13) holds, since the terms
◦
◦
sgn(Tm−1,κ,λ
) will be squared; thus Tm−1,κ,λ in (13) must be exchanged with |Tm−1,κ,λ
|,
p
◦
rather than Tm−1,κ,λ
. However, according to Lemma 5, we have n−1 Tm,k,l → µk,l > 0
whenever κ > 1 and (k, l) ∈ Λ, so for these values of k and l, Tm−1,k,l is a.a.s. positive. The
p
only problem therefore lies in the terms where κ = 1, since, by Lemma 5, n−1 Tm−1,1,1 → 0, so
Tm−1,1,1 may be either positive or negative. But when m = hn+o(n), we have Tm,1,1 = o(1)
a.a.s., so this case is consumed by the error term in (13). Hence all the equations (10) to (13)
hold for T ◦ as well, and we can apply Theorem 1. Hence, there are jointly normal random
d
∗
◦
◦
◦
◦
(t) = Uk,l
(t)
(m/n) for 0 < t < h1 +o(1). Clearly Uk,l
variables Uk,l
(t) such that Tm,k,l
→ Uk,l
when t < h1 .
◦
The random variables we are looking for are TH,k,l = TH,k,l
, and we will now show
that these random variables also converge jointly to a jointly normal distribution. Since we
d
◦
2
have limt→h− µ1,1 (t) = 0, we see that n−1/2 Th◦1 n,1,1 → U1,1
∼ N (0, σ1,1
), for some constant
1
√
0
σ11 > 0. Let m = h1 n + c √n, where c is a constant, either positive or negative. Then
◦
we have a.a.s. |Tm
n). Let m1 = min(m0 , h1 n) and m2 = max(m0 , h1 n). Let
0 ,1,1 | = O(
νk,l,d be the random variable denoting the number of values m between m1 and m2 for
◦
◦
which Wm
= (k, l, d); in other words, νk,l,d = |{m : m1 ≤ m ≤ m2 and Wm
= (k, l, d)}|.
◦
√ |Tm,1,1
|
is bounded in probability, and we have Tm2 ,1,1 − T√m1 ,1,1 =
Then E[ν1,1,1 ] = c n √n
◦
◦
m1 − m2 − ν1,1,1 ∼ −|c| n a.a.s., and it follows that Tm
n a.a.s.
0 ,1,1 = Th n,1,1 − (c + o(1))
1
√
H−h
n
◦
◦
◦
Thus, if ηn = √n1 , then TH,1,1 = Th1 n,1,1 − (ηn + o(1)) n. Since TH,1,1
= 0 by the
√
d
◦
definition of H, we have ηn = Th◦1 n,1,1 / n + o(1) → U1,1
(h1 ).
◦
When m = h1 n + o(n), we have Tm,k,l = µk,l (h1 )n + o(n) a.a.s. Hence, for these
◦
values of m, P[Wm
= (κ, λ, d)] = aκ,λ,d + o(1), where we define aκ,λ,1 = λµκ,λ (h), and
11
√
aκ,λ,2 = (κ − λ)µκ,λ (h). If m1 and m2 are as above, then νk,l,d = (ηn ak,l,d + o(1)) n a.a.s.
√ d
◦
Hence νk,l,d / n → ak,l,d U1,1
(h). We have
◦
TH,k,l
− Th◦1 n,k,l = −
X
(d)
νκ,λ,d ζk,l (κ, λ),
κ,λ,d
so it follows that
X νκ,λ (d)
Th◦ n,k,l − nµk,l (h1 )
TH,k,l − nµk,l (h)
√
√
√ ζk,l (κ, λ)
= 1
−
n
n
n
κ,λ,d
X
d
(d)
◦
◦
→ Uk,l
(h1 ) −
ak,l,d U1,1
(h1 )ζk,l (κ, λ) =: Uk,l .
κ,λ,d
Again only a finite number of the summands are nonzero, so the sum is well-defined. Since
◦
{Uk,l
}(k,l)∈Λ∗ are jointly normal random variables and {Uk,l }(k,l)∈Λ∗ are linear combinations
◦
of {Uk,l
}(k,l)∈Λ∗ , it follows that also {Uk,l }(k,l)∈Λ∗ are jointly normal random variables.
Finally we tie in the variable Lm , denoting the number of vertices of degree 1 in Gn,m .
Since vm is an isolated vertex in Gn,m−1 and has degree 1 in Gn,m , it contributes an increase
of 1 to Lm . Thus, if wm is an isolated vertex, we have ∆Lm = 2, if wm has degree 1, then
∆Lm = 0, and in all other cases ∆Lm = 1. All in all,
E[∆Lm | Fm−1 ] = 1 +
Tm−1,1,1 − 1 Lm−1
−
n−1
n−1
(16)
and
Tm−1,1,1
Lm−1
−
+ O(1/n).
n
n
Then we should find the correlation between Lm and Tm,k,l . Thus,
E[(∆Lm )2 | Fm−1 ] = 1 + 3
E[∆Lm ∆Tm,k,l | Fm−1 ] = 2
Tm−1,k−1,l−1
Tm−1,k,l
Tm−1,1,1
δk1 δl1 +
−
+ o(1).
n
n
n
(17)
(18)
Hence, it follows from Theorem 1 that Ln,tn converges to some normal variable χ(t), and
that the convergence is joint with the convergence of Ttn,k,l to Uk,l (t). By a repetition of
the above argument, we can extend this to the variables LH as well; thus there is a normal
d
variable χ such that n−1 LH → χ jointly with the convergence of TH,k,l to Uk,l .
We now consider the second phase of the graph process, when the minimum degree is 1.
The following lemma establishes that the finite-dimensional distributions of {Tm,k,l } are
asymptotically normal in this phase.
Lemma 7. Let Λ∗ be a finite subset of Λ. There are functions µk,l (t) and σ(k,l),(k0 ,l0 ) (t),
which are continuous on (hg , h2 ) such that the following holds. For hg < t < h2 , let
{Uk,l (t)}(k,l)∈Λ∗ be jointly normal random variables such that E[Uk,l (t)] = 0 and such that
E[Uk,l (t)Uk0 ,l0 (t)] = σ(k,l),(k0 ,l0 ) (t). If m = btnc, then
Tm,k,l − µk,l (t)n d
√
→ Uk,l (t)
n
jointly for all (k, l) ∈ Λ∗ .
12
Proof. Again we may assume that Λ∗ is on the form {(k, l) ∈ Λ; k ≤ q}. In order to prove
this lemma, we consider the graph process {Gn,m }m≥H ; that is, we consider Gn,H to be the
initial state. Let m0 = m + H − n ln 2. Then the number of steps from Gn,H to G0n,m is
deterministically equal to m − hn. Lemma 6 then asserts that there are random variables χ
d
d
and Uk,l such that n−1/2 (LH − n ln 2) → χ and n−1/2 (TH,k,l − µk,l n) → Uk,l jointly.
We want to show that Tm,k,l converge jointly to jointly normal random variables; however, in this phase the evolution of these variables also depends on the number of vertices
of degree 1, which is why we included the random variable Lm in the analysis. Thus, we
have to find equations analogue to (10) and to (11), (12) and (13), as well as (16), (17) and
(18). There are fewer details to take care of in this case, since the number of steps made in
the process is determined beforehand as m − H. We will therefore be satisfied with showing
that equations analogue to (10–13) and (16–18) exist, without giving their explicit forms.
We now define Vm ∈ Λ to be the random vector denoting the type of component which
vm is contained in; that is, Vm = (κ, λ) if vm lies in a component with κ vertices and λ leaves.
We let Wm ∈ Λ × {1, 2} be as in the proof of the previous lemma: Wm = (κ, λ, d) if wm is
contained in a (κ, λ)-component, and d = 1 if wm is a leaf, and d = 2 otherwise. Then
P[Vm = (κ, λ) | Fm−1 ] =
while
P[Wm = (κ, λ, 1) | Fm−1 ] =
λTm−1,κ,λ
,
Lm
λTm−1,κ,λ − IVm =(κ,λ),deg(vm )=1
n−1
(19)
(20)
and
(κ − λ)Tm−1,κ,λ − IVm =(κ,λ),deg(vm )>1
.
(21)
n−1
Suppose that Vm = (κ, λ) and that Wm = (κ0 , λ0 , d). Then we lose one (κ, λ)-component and
one (κ0 , λ0 )-component, while we gain one (κ+κ0 , λ+λ0 +d−3)-component. The probability
of this event is obtained by multiplying (19) with (20) or (21) depending on the value of d,
and is therefore on the form φ({n−1 Tm−1,κ,λ }κ≤k+k0 ∪ {n−1 Lm }) + o(n−1/2 ). Moreover,
there is only a finite number of choices of (κ, λ) and (κ0 , λ0 , d) which cause ∆Tm,k,l to be
nonzero. Therefore functions fk,l and g(k,l),(k0 ,l0 ) can be found such that
P[Wm = (κ, λ, 2) | Fm−1 ] =
E[∆Tm,κ,λ | Fm−1 ] = fκ,λ ({n−1 Tm−1,k,l }k<κ ∪ {n−1 Lm }) + o(n−1/2 )
(22)
and
E[∆Tm,κ,λ ∆Tm,κ0 ,λ0 | Fm−1 ] = g(κ,λ),(κ0 ,λ0 ) ({n−1 Tm−1,k,l }k<max(κ,κ0 ) ∪ {n−1 Lm }) + o(1).
Then we consider the number of leaves. By definition, vm is always a leaf, while wm has
m −1
degree 1 with probability Ln−1
, so
E[∆Lm | Fm−1 ] = −1 −
and
Lm−1
+ O(1/n)
n
Lm−1
+ 1 + O(1/n).
n
Finally we have to find an expression for E[∆Lm ∆Tm,κ,λ | Fm−1 ]. Clearly the product
∆Lm ∆Tm,κ,λ is nonzero only if ∆Tm,κ,λ is nonzero. If wm is a leaf, then ∆Lm = −2, while
if wm is not a leaf, then ∆Lm = −1. Thus, an expression for E[∆Lm ∆Tm,κ,λ | Fm−1 ] (up
to an error term of o(1)) can be obtained by multiplying each term in (22) with −1 or −2.
E[∆Lm ∆Lm | Fm−1 ] = 3
13
We have one random variable for every element in Λ∗ , in addition to Lm . Let us rename
the random variables Xm,0 , Xm,1 , . . . , Xm,|Λ∗ | such that Xm,0 = Lm and {Xm,k }1≤k≤|Λ∗ | =
{Tm,k,l }(k,l)∈Λ∗ . We write X n,m = [Xn,m,0 , . . . , Xn,m,|Λ∗ | ]0 . Thus, the functions fk,l and
∗
∗
g(k,l),(k0 ,l0 ) are functions from R|Λ |+1 to R and we have to find a set D ⊂ R|Λ |+1 such
that the conditions of Theorem 1 are satisfied. Unlike in Lemma 6, where all the functions
are linear and automatically satisfy a Lipschitz condition, we have to be careful since Lm
appears in the denominator of (19). Thus, to ensure that a Lipschitz condition is satisfied,
we have to choose D such that n−1 Xm,0 = n−1 Lm is bounded away from 0 whenever
p
X n,m ∈ D. By Lemma 5, there is a continuous function b(t) such that n−1 Xm,0 → b(t)
for all t, and such that b(t) > 0 whenever t ∈ (0, h2 ), and b(h2 ) = 0. Then, for every
ε > 0, there is a δ = δ(ε) > 0 such that b(t) > ε whenever t ∈ [h1 , h2 − δ). We then define
Dε = {z = (z0 , . . . , z|Λ∗ | ); z0 > δ(ε), −ε < zk < 1 + ε for 1 ≤ k ≤ |Λ∗ |}. Then X n,m ∈ Dε
a.a.s. whenever t < h2 −ε, and the functions fk,l and g(k,l),(k0 ,l0 ) satisfy a Lipschitz condition
on Dε .
Let ε > 0. From Theorem 1 it follows that there are continuous functions µk,l (t) and
∗
∗
σ(k,l),(k
0 ,l0 ) (t), such that for all t ∈ (h1 , h2 − ε) the following holds. Let {Uk,l (t)}(k,l)∈Λ be
∗
∗
∗
∗
jointly normal random variables with E[Uk,l (t)] = 0 and E[Uk,l (t)Uk0 ,l0 (t)] = σ(k,l),(k
0 ,l0 ) (t).
Then
Tm0 ,k,l − µk,l (t)n d ∗
√
→ Uk,l (t).
n
We let ε → 0, and see that the convergence holds for all t ∈ (h1 , h2 ).
The only thing that remains is to exchange m0 with m. Let m1 = min(m, m0 ) and
m2 = max(m, m0 ), and let ν(κ,λ),(κ0 ,λ0 ,d)√= |{m : m1 ≤ m ≤ m2 and Vm = (κ, λ) and Wm =
(κ0 , λ0 , d)}|. Assume that m2 − m1 = c n. Since Tm,k,l = µk,l (t)n + o(n) a.a.s. for m1 ≤
m ≤ m2 , we see from (19–21) that a.a.s.
ν(κ,λ),(κ0 ,λ0 ,1) = cλ
√
√
√
√
µκ,λ (t) 0
λ µκ0 ,λ0 (t) n + o( n) =: cr(k,l),(k0 ,l0 ,1) n + o( n)
β1 (t)
and
ν(κ,λ),(κ0 ,λ0 ,2) = cλ
√
√
√
√
µκ,λ (t) 0
(κ − λ0 )µκ0 ,λ0 (t) n + o( n) =: cr(k,l),(k0 ,l0 ,2) n + o( n).
β1 (t)
(d)
Define ζk,l (κ, λ, κ0 λ0 ) to be the value of ∆Tk,l,m in the event that Vm = (κ, λ) and Wm =
(d)
(κ0 , λ0 , d). Then ζk,l (κ, λ, κ0 , λ0 ) is well-defined and equal to an integer between −2 and 1,
inclusive, and
X
X
(d)
(d)
Tm0 ,k,l − Tm,k,l =
ν(κ,λ),(κ0 ,λ0 ,d) ζk,l (κ, λ, κ0 , λ0 ) =
ηr(κ,λ),(κ0 ,λ0 ,d) ζk,l (κ, λ, κ0 , λ0 ),
κ,λ,κ0 ,λ0 d
κ,λ,κ0 ,λ0 ,d
d
√
where η is the random variable found in Lemma 6, such that H−hn
→ η. We know that η is
n
∗
jointly normal with Uk,l for (k, l) ∈ Λ. By Theorem 1, Uk,l (t) are jointly normal with Uk,l ,
so we actually have
X
Tm,k,l − nµk,l (t)
Tm0 ,k,l − nµk,l (t)
(d)
√
√
=
+
ηr(κ,λ),(κ0 ,λ0 ,d) ζk,l (κ, λ, κ0 , λ0 )
n
n
0
0
κ,λ,κ ,λ ,d
d
→
∗
Uk,l
(t)
+ ηsk,l =: Uk,l (t),
∗
where sk,l is a constant for all (k, l) ∈ Λ. Since η and Uk,l
(t) are jointly normal random
variables, the linear combinations are also jointly normal, which completes the proof.
14
4.2
Exponential bound
In order to show that the expectation and variance of Tm,k,l are exponentially bounded, we
will use a branching process argument. In [8] a branching process was introduced in order to
approximate the manner in which one might expose the components of the graph. We will
first need some results from the theory of branching processes. An ordinary Galton–Watson
branching process starts with a single particle. At every step every particle still alive gets
a number of children which is distributed as a random variable Z, and then dies. The size
of the Galton–Watson process is the total number of particles produced by the branching
process. This value may be finite or infinite; if it is finite we say that the branching process
dies out, and the probability that this happens is called the extinction probability of the
branching process.
The following theorem summarizes several facts about branching processes, regarding the
extinction probability and the generating functions associated with the branching process.
It is a combination of results from Chapter I of Athreya and Ney [1] and Theorem VI.6 of
Flajolet and Sedgewick [6].
Theorem 8. Let
P Z be a nonnegative integer-valued random variable. Let pk = P[Z = k]
and let p(z) = k≥0 pk z k be the probability generating function of Z. Let S be the random
variable denoting the size of the Galton–Watson process, where the number P
of offspring of
a given particle is distributed according to Z. Let sk = P[S = k] and s(z) = 0≤k<∞ sk z k .
Let q denote the smallest positive solution to the equation p(z) = z, let τ satisfy
p(τ ) − τ p0 (τ ) = 0,
and let ρ = τ /p(τ ). Then
P[S < ∞] = q,
s(z) is given implicitly by s(z) = zp(s(z)) and sk ∼ ck −3/2 ρ−k , for a constant c.
We now present the branching process from [8]. In order to define the branching process,
we colour the edges in the graph. For d ≥ 1, we let Hd be the smallest value of m such that
the minimum degree of Gn,m is at least d; thus H1 = H. We let all the edges in Gn,H1 be
coloured red, and if m ≤ H2 , we colour the edges in Gn,m \ Gn,H blue. If m > H2 , the edges
in Gn,m \ Gn,H2 do not receive a colour; we will, however, consider the graph process when
m ≤ H2 , so that all edges are coloured. When m ≤ H1 , a step in the graph process consists
of attaching an isolated vertex to another vertex; thus, no cycle can be formed in this phase,
and so Gn,m contains no red cycle. When m > H1 , any vertex of degree 1 must necessarily
be incident to a red edge, and is therefore not incident to any blue edges; when m ≤ H2 , a
step consists of attaching such a vertex to another vertex, and no blue cycle can therefore
be formed either. Thus, when H1 < m ≤ H2 , the graph Gn,m is a union of a red and a blue
forest. Moreover, whenever a blue tree grows with one vertex, the new vertex must have
degree 1 and be incident to one red edge; hence, the only way a vertex incident to more
than one red edge can become part of a blue tree is if at some stage in the graph proces an
edge vm wm is added such that neither vm nor wm is incident to a blue edge in Gn,m−1 , and
wm has degree larger than 1. Hence, every blue tree contains at most one vertex incident
to more than one red edge. We call a vertex light if it is incident to exactly one red edge,
and heavy if it is incident to more than one red edge.
For a vertex v, let Cred (v) and Cblue (v) be the maximal red and blue tree, respectively,
incident to v. Let k and l be positive integers. In [8] the asymptotic probability that Cred (v),
respectively Cblue (v), consists of exactly k vertices of which exactly l are light, is calculated.
15
To be more precise, there are constants Px,y,k,l such that a.a.s.
P[Cx (v) is of type (k, l) | v is y] = (1 + o(1))Px,y,k,l ,
(23)
where x is ‘red’ or ‘blue’ and y is ‘light’ or ‘heavy’. Since (23) holds a.a.s., we can in the
following condition on the event that it holds.
Moreover, it is observed that the order and number of light vertices in Cred (v) is essentially independent of the order and number of light vertices in Cblue (v), when we condition
on whether v itself is light or heavy. What this means is that if we know whether v is light
or heavy, then (23) holds for Cred (v), even if we condition on the type of Cblue (v), and vice
versa.
We let Gn,m be an instance of the graph process such that (23) holds. We let πx,y,k,l =
P[Cx (v) is of type (k, l) | v is y], where v is a randomly selected vertex. Then πx,y,k,l =
(1 + o(1))Px,y,k,l .
We now define a multitype branching process mimicking the graph process in the following way. We start with a randomly chosen vertex v, which may be light or heavy, and
generate a red tree incident to it, whose type is chosen at random according to the probabilities πx,y,k,l . By ‘type’ we mean the number of vertices and the number of light vertices. At
every vertex in this red tree we add a blue tree, again with probabilities πx,y,k,l . At any new
vertex generated this time, we add a red tree, and we continue in this manner, alternatingly
generating red and blue trees. Since a vertex is not necessarily incident to a blue edge, it is
possible that no new vertices are generated when a blue tree is to be added; thus, there is
a chance that the branching process dies out. On the other hand, every vertex is incident
to at least one red edge, so every time a red tree is to be added, at least one new vertex is
generated. It was found in [8] that the probability that v is in a component of order O(log n)
is asymptotically equal to the probability that this branching process dies out, which is 1
when t < hg and less than 1 when t > hg .
The branching process above is a multitype branching process: a vertex in the process
may either be light or heavy, and it can be generated either when a red tree or a blue tree
is created. We give the vertices colour as well, and say that a vertex is red (blue) if it
was created when a red (blue) tree was generated. For our purposes, it is easier to work
with an ordinary branching process with only one type of vertices. We therefore consider
a variant of this branching process, where we are only interested in one of the types of
vertices, say the light, red vertices. Whenever we make a step in the branching process and
acquire some vertices which are not light, red vertices, we continue the branching process
from these vertices, until we only have light, red vertices. Then we get a Galton–Watson
process. Let Z be the random variable denoting
of red light vertices generated
P the number
from one red light vertex, and let pZ (z) =
P[Z = k]z k be the corresponding probability
generating function. Since the original multitype branching process is supercritical, this
Galton–Watson process must also be supercritical, so EZ > 1.
We will now use Theorem 8. The number of vertices created in one step is finite with
0
0
probability 1, so p(1) = 1, while
Pp (1) = EZ > 1. Let f (z) = p(z) − zp (z), so that
f 0 (z) = −zp00 (z). Since zp00 (z) = k≥0 k(k − 1)pk z k−1 , and pk > 0 for some k ≥ 2, we have
f 0 (z) < 0 for all z > 0. Since f (0) = p(0) = p0 > 0 and f (1) = p(1)−p0 (1) < 0, we must have
0 < τ < 1; then f (z) > 0 for z ∈ (0, τ ) and f (z) < 0 for z ∈ (τ, 1). Let now r(z) = z/p(z).
Then, r0 (z) = f (z)/p(z)2 , so r(z) is increasing on (0, τ ) and decreasing on (τ, 1). Moreover,
r(0) = 0 and r(1) = 1, so ρ = r(τ ) > 1. Then, by Theorem 8, P[S = k] ∼ ck −3/2 ρ−k for a
constant c.
The conclusion is that, conditioned on the branching process dying out, the number of
light, red vertices is exponentially bounded. By the same argument one can show that the
16
number of heavy, red vertices is exponentially bounded. Since every vertex is adjacent to at
least one red edge, a blue vertex always generates at least one red vertex, and so the total
number of vertices is at most twice the number of red vertices generated. Hence, the number
of vertices generated by the multitype branching process is also exponentially bounded, and
since the probabilities πx,y,k,l were given by the actual structure of Gn,m , this holds also
for the components in the real graph process; in other words, if C(v) is the component
containing the vertex v,
ETm,k,l ≤ nP[|C(v)| = k] ≤ Knρ0k ,
for some constants K > 0 and 0 < ρ0 < 1.
Let T be an arbitrary tree on k vertices, whose edges are ordered and coloured red
and blue such that a edges are coloured red and b edges are coloured blue, where a +
b = k − 1, and such that all red edges appear before the blue edges in the ordering. Let
v1 w1 , v2 w2 , . . . , vm wm be the m edges added in the graph process, such that vi is first and
wi the second vertex chosen at every step. Let A be the event that exactly a vertices among
v1 , . . . , vH belong to A, and exactly b vertices among vH+1 , . . . , vm belong to A. Let us
assume that A holds, and let vi1 , . . . , vik−1 be the k−1 vertices chosen in A. Let xj ∈ {0, 1, 2}
be the number of vertices in {vij , wij } having minimum degree in Gn,ij −1 . Let yj be the
number of vertices of minimum degree in A. Let X = x1 · · · xk−1 and Y = y1 · · · yk−1 .
Conditioned on A, the probability that A forms a tree isomorphic to T preserving the edge
colouring, and such that the order in which the edges are added in the process matches the
ordering of the edges in T is equal to
xk−1 1
x1 1
···
y1 n − 1
yk−1 n − 1
n−k−1
n−1
m−k+1
=
X
Y
1−
k
n−1
m
(n − k − 1)1−k (24)
To see that this equation is correct, note that for the “correct” edge relative to T to be added
at the next step, one of the endpoints must be chosen as the first vertex. If we condition on
that vertex belonging to A, the probability that the vertex is one of the endpoints of that
edge is 2/yj if both endpoints have minimum degree, 1/yj if only one of the endpoints has
minimum degree, and 0 if none of the endpoints has minimum degree. The probability that
wij is the other endpoint of the graph is then 1/(n − 1). The last factor is the probability
that the remaining m − k + 1 edges in the graph lie completely outside of A.
Assume instead that A is a set of k vertices in Gn−k,m−k+1 . (The reason for this will be
explained below). Conditioned on A, the probability that A forms a tree isomorphic to T
becomes
m−2k+2
x1
1
x2
1
xk−1
1
n − 2k − 1
·
· ··· ·
y1 n − k − 1 y2 n − k − 1
yk−1 n − k − 1
n−k−1
m−k+1
X
k
=
1−
(n − 2k + 1)1−k
Y
n−k−1
which differs from (24) by a factor of (1 + O(k 2 /n)).
Let B be the event that A forms a tree of any kind in Gn,m . If B holds, there are
k − 1 edges inside A and no edges between A and V \ A, where V is the entire vertex set.
Thus, if we condition on B, the graph process Gn,m restricted to V \ A is identical to the
process Gn−k,m−k+1 . From this it follows that the expected number of pairs of trees of
type (k, l) is
E[Tm,k,l (Tm,k,l − 1)] = P[Cv = k]2 (1 + O(k 2 /n))n2 .
17
Hence
2
E[Var Tm,k,l ] = E[Tm,k,l
] − E[Tm,k,l ]2 = E[Tm,k,l (Tm,k,l − 1)] − E[Tm,k,l ]2 + E[Tm,k,l ]
= P[C(v) = k]2 (1 + O(k 2 /n))n2 − P[C(v) = k]2 n2 + nP[C(v) = k]
= O(k 2 n)P[C(v) = k]2 + nP[C(v) = k] ≤ K 00 nρ00k
for some constants K 00 and ρ00 with 0 < ρ00 < 1.
4.3
Small cyclic components
To complete the proof of Theorem 4, we have to show that the expected number of vertices
√
(n)
in cyclic components is o( n). Recall that Cm is the random variable denoting the number
of such vertices in Gn,m .
Let k− = B ln n for some constant B and let k+ = n1−ε for some ε with 0 < ε < 1/2. Let
A be the event that there is either a component in Gn,m with between k− and k+ vertices or
more than one component with at least k+ vertices. In [8] it is shown that PA = o(1); this
bound was made just as strong as was necessary to prove the main theorem of [8]. However,
it is immediate from the calculations in [8] that the probability can be made much smaller,
so for large enough B, we have, say, PA = o(n−1 ).
Let us call a component small if it has at most k− vertices. Let v and w be the two vertices
chosen at one stage in the graph process. The probability that v and w both belong to the
same component, having at most k− vertices, is bounded by k− /n. The expected number
of edges added in the graph process, such that both endpoints belong to the same small
component is therefore mk− /n. Thus, the expected number of vertices in small unicyclic
2
/n = O(ln2 n).
components is at most mk−
Since Cn,m clearly is at most n, we have
E[Cn,m ] = E[Cn,m | A]P[A] + E[Cn,m | A]P[A]
≤ O(ln2 n) + n · o(n−1 ) = O(ln2 n).
5
Other random graph processes
The previous section considered in detail the minimum-degree graph process, and used the
method explained in Section 3 to show that the giant component is asymptotically normally
distributed. The argument for this process is more complicated than it would be for several
other graph process it is natural to study, because of the “discontinuity” in the evolution of
the graph when the minimum degree increases from 0 to 1.
The most “important” necessary condition for this method to be applied is that the
differential equation method can be applied. The other conditions (the exponential bound
and the small cyclic components) are likely to hold in most typical random graph processes,
although they may be difficult to verify in some cases.
In the d-process, which is studied by Ruciński and Wormald [11, 12], a new edge is added
to the graph uniformly at random under the provision that the maximum degree remain at
most d. Suppose C is a component at some stage in the d-process. Then the probability that
the next edge to be added is incident to some vertex in C is proportional to the number of
vertices in C with degree strictly less than d. Thus, we need to keep track of the number of
vertices of degree d in every component. But this means that we also have to keep track of all
the other possible degrees. Thus, the classes of trees must be defined in terms of the entire
18
degree sequence, so we have {Tk,πk }k,πk , where πk is a partition of k. Although the number
of classes seems to be very large, perhaps making the actual computations prohibitive, the
number of classes of trees with k vertices is clearly O(k d ), so Theorem 2 can be applied.
To the author’s knowledge it has not been proven that there actually exist a phase
transition and a giant component in the d-process; however, since the evolution mechanism
is similar to the one in the standard GER
n,m model, one may reasonably conjecture that
there are, and from the above arguments it also seems safe to conjecture that whenever
there is a linear-sized giant component in the d-process, its order is asymptotically normally
distributed.
We also mention the Achlioptas process, which is studied by Spencer and Wormald [15].
In the general Achlioptas process, at every stage one is presented with two randomly chosen
edges, and one has to choose one of them to add to the graph, seeking either to promote
or delay the formation of a giant component. The problem is to find an optimal strategy
to either make sure that the giant component is formed as soon as possible, or as late
as possible. Our method probably does not work for the general problem, as it is likely
that there are strategies which actively prevent the giant component from being normally
distributed. However, Spencer and Wormald [15] suggest strategies where the decision about
which edge to pick depends solely on the orders of the components containing the endpoints
of the edges. This situation is ideal for the differential equation method. It is shown in [15]
that such processes exhibit a phase transition and a giant component. The point of the phase
transition depends, of course, on the exact strategy which is used; however, apart from that
the giant component behaves as in most other typical random graph processes, and with
the method in the present paper, it ought to be possible to show that it is asymptotically
normally distributed.
6
Proof of Theorem 1
Theorem 1 is a generalization of the main theorem in Seierstad [13]. The difference lies
in that we allow X 0 to be non-deterministic, provided that it converges to a multivariate
normal distribution; that is
X 0 − nµ0 d
√
→ N (0, Σ0 ).
(25)
n
√
In [13] X 0 has to be deterministic, or at least have deviations at most o( n) (although
this is unfortunately not stated very precisely in [13]). In this section we will show that
the theorem still holds when (25) is satisfied. We will not include every step in the proof
rigorously, as it is in large parts similar to the proof in [13], but will be content with including
the main steps in the argument which differ from the previous version. The proof is based
on a central limit theorem for near-martingales by McLeish [9]; the following is similar to
the version of that theorem which was proved in [13], except that we allow the inital state
to be nondeterministic.
Theorem 9. Let (Ωn , Fn , Pn ) be a sequence of probability spaces, and assume that a filtration Fn,0 ⊆ Fn,1 ⊆ · · · ⊆ Fn,mn ⊆ Fn is given. For n ≥ 1 and 0 ≤ m ≤ mn , let S n,m be an
Fn,m -measurable q-dimensional array of random variables, and let Σ0 and Σ∗ be symmetric
d
q × q-matrices. Suppose that S n,0 → N (0, Σ0 ). Assume that the following conditions are
satisfied.
(i) maxm k∆S n,m k has uniformly bounded second moment.
p
(ii) maxm k∆S n,m k → 0.
19
Pmn
p
(∆S n,m )2 → Σ∗ .
(iii) For all 1 ≤ i, j ≤ q, m=1
Pmn
p
(iv)
m=1 E[∆S n,m | Fm−1 ] → 0.
Pmn
2 p
(v)
m=1 E[∆S n,m | Fm−1 ] → 0.
d
Then S n,mn → N (0, Σ), where Σ = Σ0 + Σ∗ . Moreover, S n,0 and S n,m converge jointly.
Pm
Proof. Let S ∗n,m = S n,m − S n,0 = k=1 ∆S n,k . Then S ∗n,m has deterministic initial state.
d
By Theorem 3 in [13], if we condition on F0 , then S ∗n,m → N (0, Σ∗ ).
Assume first that q = 1, and let σ02 and σ ∗ 2 be the sole entries in Σ0 and Σ∗ , respectively.
∗
2 ∗2
Then the above can be expressed as E[eitS n,mn | F0 ] → e−t σ /2 for every t, where i is the
imaginary unit. Let a and b be arbitrary real numbers, and define T n = aS n,0 + bS n,mn .
Then, since Sn,0 is Fn,0 -measurable,
∗
∗
E[eitT n ] = E[eit(aS n,0 +bS n,m ) ] = E[eitaS n,0 E[eitbS n,m | F0 ]] → e−t
2
(a2 σ02 +b2 σ ∗ 2 )
.
Thus, S n,0 and S ∗n,mn tend jointly to jointly normal random variables, and in particular
d
S n,mn = S n,0 + S ∗n,mn → N (0, σ02 + σ ∗ 2 ). If q > 1, the theorem follows from the univariate
version by the same proof as that of Theorem 3 in [13].
We are now ready to prove Theorem 1. Let C = n−1/2 (X 0 − nµ0 ), and define X m =
√
d
X m − C n. By assumption C → N (0, Σ0 ) and X 0 = nµ0 , so X 0 is deterministic. Let
αm = α(m/n) and let Y m = X m − nαm . Clearly, ∆X m = ∆X m , so
E[∆X m | Fm−1 ] = F (n−1 X m−1 ) + o(n−1/2 )
= F (n−1 X m−1 + n−1/2 C) + o(n−1/2 )
= F (αm−1 + n−1 Y m−1 + n−1/2 C) + o(n−1/2 ).
Recall the definitions of J and T from Section 2. We let Tm = T (m/n), Am = J(αm ) and
Um = I − n−1 Am . Then (Lemma 1 in [13])
Tm = Tm−1 Um−1 + O(n−2 ),
so that
∆Tm = Tm−1 (Um−1 − I) = n−1 Tm−1 Am−1 .
(26)
q
Finally we define Z m = Tm Y m . We know from calculus that if a, y ∈ R , then
F (a + y) − F (a) = JF (a)y + O(kyk2 )
as y → 0. Thus
E[Um−1 Y m − Y m−1 | Fm−1 ]
= Um−1 (E[X m | Fm−1 ] − nαm ) − X m−1 + nαm−1
= (I − n−1 Am−1 )(X m−1 + F (αm−1 + n−1 Y m + n−1/2 C)
− nαm−1 − F (αm−1 )) − X m−1 + nαm−1 + o(n−1/2 )
= − n−1 Am−1 Y m−1 + JF (αm−1 )(n−1 Y m−1 + n−1/2 C) + o(n−1/2 )
= n−1/2 Am−1 C + o(n−1/2 ).
20
Hence
E[∆Z m | Fm−1 ] = E[Tm Y m − Tm−1 Y m−1 | Fm−1 ]
= Tm−1 (E[Um−1 Y m | Fm−1 ] − Y m−1 ) + O(n−2 )
= n−1/2 Tm−1 Am−1 C + o(n−1/2 ).
(27)
Let Ξ∗ (t) = Ξ(t) − Σ0 . Then Ξ∗ (t) is a solution of the integral in (3), satisfying Ξ∗ (0) = 0.
One can verify that
m
X
p
−1
n
(∆Z k )2 → Ξ∗ (m/n).
k=1
The rather lengthy calculations needed to prove this are identical to those in the proof of
Lemma 3 in [13]. Also similarly to [13], one can show that ∆Z m = O(β) and (∆Z m )2 =
O(β 2 ) a.s. We now define M m = n−1/2 Z m + Tm C. We will show that M m is almost a
d
martingale, in the sense that it satisfies the conditions of Theorem 9. We have M 0 = C →
N (0, Σ0 ) by assumption. It is easy to see that (i) and (ii) are satisfied. For (iii) we obtain
m
X
(∆M k )2 =
k=1
m
X
(n−1/2 ∆Z k + ∆Tk C)2
k=1
n−1
=
m
X
!
(∆Z k )2
p
+ o(1) → Ξ∗ (m/n).
k=1
For (iv), we find that
m
X
m
X
E[∆M k | Fm−1 ] =
k=1
(26,27)
=
k=1
m
X
E[n−1/2 ∆Z k + ∆Tk C | Fm−1 ]
n−1 Tm−1 Am−1 C − n−1 Tm−1 Am−1 C + o(n−1 )
k=1
= m · o(n−1 ) = o(1).
d
For (v) the calculations are similar. It now follows from Theorem 9 that M m → N (0, Ξ(t)),
which means that
Tm (X m − nαm )
d
√
+ Tm C → N (0, Ξ(t)).
n
Thus
√
Tm (X m − nαm )
Tm (X m − nαm + C n) d
√
√
=
→ N (0, Ξ(t)),
n
n
so
X m − nαm d
√
→ N (0, Σ(t)),
n
where Σ(t) = T (t)−1 Ξ(t)(T (t)0 )−1 .
References
[1] K. B. Athreya and P. E. Ney. Branching processes. Springer-Verlag, New York, 1972.
Die Grundlehren der mathematischen Wissenschaften, Band 196.
21
[2] D. Barraez, S. Boucheron, and W. Fernandez de la Vega. On the fluctuations of the
giant component. Combin. Probab. Comput., 9(4):287–304, 2000.
[3] M. Behrisch, A. Coja-Oghlan, and M. Kang. The order of the giant component of
random hypergraphs. submitted.
[4] P. Billingsley. Convergence of probability measures. Wiley Series in Probability and
Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, second edition, 1999.
[5] P. Erdős and A. Rényi. On the evolution of random graphs. Magyar Tud. Akad. Mat.
Kutató Int. Közl., 5:17–61, 1960.
[6] P. Flajolet and R. Sedgewick. Analytic combinatorics. Web edition (available from the
authors’ web sites). To be published in 2008 by Cambridge University Press.
[7] M. Kang, Y. Koh, S. Ree, and T. Luczak. The connectivity threshold for the min-degree
random graph process. Random Structures Algorithms, 29(1):105–120, 2006.
[8] M. Kang and T. G. Seierstad. Phase transition of the minimum degree random multigraph process. Random Structures Algorithms, 31(3):330–353, 2007.
[9] D. L. McLeish. Dependent central limit theorems and invariance principles. Ann.
Probability, 2:620–628, 1974.
[10] B. Pittel. On tree census and the giant component in sparse random graphs. Random
Structures Algorithms, 1(3):311–342, 1990.
[11] A. Ruciński and N. C. Wormald. Random graph processes with degree restrictions.
Comb. Probab. Comput., 1(2):169–180, 1992.
[12] A. Ruciński and N. C. Wormald. Connectedness of graphs generated by a random
d-process. J. Aust. Math. Soc., 72(1):67–85, 2002.
[13] T. G. Seierstad. A central limit theorem via differential equations. To appear in Ann.
Appl. Probab., available from the journal’s website.
[14] T. G. Seierstad. The phase transition in random graphs and random graph processes.
PhD thesis, Humboldt University Berlin, 2007.
[15] J. Spencer and N. C. Wormald. Birth control for giants. Combinatorica, 27(5):587–628,
2007.
[16] V. E. Stepanov. On the probability of the connectedness of a random graph Gm (t).
Theor. Probability Appl., 15:55–67, 1970.
[17] N. C. Wormald. Differential equations for random processes and random graphs. Ann.
Appl. Probab., 5(4):1217–1235, 1995.
[18] N. C. Wormald. The differential equation method for random graph processes and
greedy algorithms. In M. Karoński and H-J. Prömel, editor, Lectures on approximation
and randomized algorithms., pages 75–152. PWN, Warsaw, 1999.
22