The Size of the Giant Component of a Random Graph with a Given Degree Sequence

TLDR: The size of the giant component in the former case, and the structure of the graph formed by deleting that component is analyzed, which is basically that of a random graph with n′=n−∣C∣ vertices, and with λ′in′ of them of degree i.

Abstract: Given a sequence of nonnegative real numbers λ0, λ1, … that sum to 1, we consider a random graph having approximately λin vertices of degree i. In [12] the authors essentially show that if ∑i(i−2)λi>0 then the graph a.s. has a giant component, while if ∑i(i−2)λi<0 then a.s. all components in the graph are small. In this paper we analyse the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine e, λ′0, λ′1 … such that a.s. the giant component, C, has en+o(n) vertices, and the structure of the graph remaining after deleting C is basically that of a random graph with n′=n−∣C∣ vertices, and with λ′in′ of them of degree i.

Full text

THE SIZE OF THE GIANT COMPONENT
OF A RANDOM GRAPH WITH A GIVEN
DEGREE SEQUENCE
Michael Molloy
Department of Computer Science
UniversityofToronto
Toronto, Canada
Bruce Reed
Equip e Combinatoire
CNRS
Universite Pierre et Marie Curie
Paris, France
June 22, 2000
Abstract
Given a sequence of non-negativerealnumbers

0

1
:::
which
sumto1,we consider a random graph having approximately

i
n
ver-
tices of degree
i
. In 12] the authors essentially show that if
P
i
(
i
;
2)

i
>
0 then the graph a.s. has a giant component, while if
P
i
(
i
;
2)

i
<
0 then a.s. all comp onents in the graph are small. In
this pap er we analyze the size of the giant component in the former
case, and the structure of the graph formed by deleting that comp o-
nent. We determine
 
0
0

0
1
:::
such that a.s. the giant component,
C
,has
n
+o(
n
)vertices, and the structure of the graph remaining
after deleting
C
is basically that of a random graph with
n
0
=
n
;j
C
j
vertices, and with

0
i
n
0
of them of degree
i
.
1

1 Intro duction and Overview
Perhaps the most studied phenomenon in the eld of random graphs is the
behaviour of the size of the largest componentin
G
np
1
when
p
=
c=n
for
c
near 1. For
c<
1 the size of the largest comp onent is almost surely
2
(a.s.)
O(log
n
), for
c
= 1 the size of the largest comp onentisa.s. (
n
2
=
3
), and
for
c>
1 a.s. the size of the largest componentis(
n
) while the size of
the second largest comp onentisO(log
n
) (see 8], 7] or 9]). For
c>
1, this
largest comp onent is commonly referred to as the
giant component
and the
point
p
=1
=n
is referred to as the
critical point
or the
double jump threshold
.
For
c>
1, we can also determine the approximate size of the giantcom-
ponent,
C
,as well as the structure of the graph formed by deleting it. It's
size is a.s.

c
n
+o(
n
) where

c
is the unique solution to

+e
;
c
=1,and
the graph formed by deleting
C
is essentially equivalentto
G
n
0
p
=
d
c
=n
0
,where
n
0
=
n
;j
C
j
=(1
;

c
)
n
+o(
n
), and
d
c
=
c
(1
;

c
) (see 1] or 10 ]). (Note that
d
c
<
1.) The latter prop erty is referred to as the Discrete Duality Principle.
In 12], the authors showed that a similar phenomenon occurs among
random graphs with a xed degree sequence. Essentially,we considered
random graphs on
n
vertices with

i
n
+o(
n
)vertices of degree
i
, for some
xed sequence

0

1
:::
. Weintroduced the parameter
Q
=
P
i
(
i
;
2)

i
and showed that if
Q<
0thena.s. the size of the largest comp onentis
O(
!
2
log
n
), where
!
is the highest degree in the graph, and if
Q>
0, then
a.s. the size of the largest componentis(
n
), and the size of the second
largest comp onent is O(log
n
).
In this pap er we rene our arguments to determine the approximate size
of the giant comp onent in such a graph. We also nd an analogue to the
Discrete Duality Principle, showing that there is a sequence

0
0

0
1
:::
,such
that the graph remaining after deleting the giant component,
C
, is basically
equivalent to a random graph on
n
0
=
n
;j
C
j
vertices, with approximately

0
i
n
0
vertices of degree
i
for each
i
. Of course,
P
i
(
i
;
2)

0
i
<
0.
To b e expeditious, we will state our main theorems here, momentarily
postp oning the denition of a well-behaved sparse asymptotic degree se-
quence, whichwas introduced in 12].
1
G
np
is the random graph with
n
vertices where each edge appears indep endently with
probability
p
.
2
Wesay that a random event
E
n
holds
almost surely
if lim
n
!1
Pr
(
E
n
)=
1
.
2

Given a sequence of non-negative reals

0

1
:::
summing to one, weset
K
=
P
i

0
i
i
, and dene

:0

1]
!
R
as

(

)=
K
;
2

;
X
i

1
i
i

1
;
2

K

i
2

and we denote the smallest p ositive solution to

(

) = 0 (if such a solution
exists), by

D
.Now setting

D
= 1
;
X
i

1

i

1
;
2

D
K

i
2

0
i
=

i
(1
;

D
)

1
;
2

D
K

i
2
wehave
Theorem 1
Let
D
=
d
0
(
n
)
d
1
(
n
)
:::
be a wel l-behavedsparse asymptotic
degreesequence whereforeach
i

0
,
lim
n
!1
d
i
(
n
)
=n
=

i
and for which
there exists
>
0
such that for al l
n
and
i

n
1
=
4
;

,
d
i
(
n
)=0
.Suppose that
Q
(
D
)=
P
i
(
i
;
2)

i
>
0
.If
G
is a random graph with
n
vertices and degree
sequenc
e
D
n
then a.s. the giant component of
G
has size

D
n
+o(
n
)
.
Theorem 2
Let
D
beadegreesequencemeeting the conditions of Theorem
1. Let
G
bearandom graph with
n
vertices and degreesequence
D
n
. A.s.
the structure of the graph formed by deleting the largest component,
C
,from
G
is essential ly the same as that of a random graph on
n
0
=
n
;j
C
j
=(1
;

D
)
n
+o(
n
)
vertices, with degreesequence
D
0
, for some
D
0
=
d
0
0
(
n
)
d
0
1
(
n
)
:::
,
where
d
0
i
(
n
)=

0
i
n
+o(
n
)
.
Nowwe will recall the relevant denitions from 12]. Throughout this
paper, all asymptotics will b e taken as
n
tends to
1
and we only claim things
to b e true for suciently large
n
.By
A

B
we mean that lim
n
!1
A=B
=1.
Denition:
An
asymptotic degree sequence
is a sequenceofinteger-
valued functions
D
=
d
0
(
n
)
d
1
(
n
)
:::
such that
1.
d
i
(
n
)=0
for
i

n

3

2.
P
i

0
d
i
(
n
)=
n
.
Given an asymptotic degree sequence
D
,we set
D
n
to b e the degree
sequence
f
c
1
c
2
:::c
n
g
, where
c
j

c
j
+1
and
jf
j
:
c
j
=
i
gj
=
d
i
(
n
)foreach
i

0. Dene 
D
n
to b e the set of all graphs with vertex set 
n
] with degree
sequence
D
n
. A random graph on
n
vertices with degree sequence
D
is a
uniformly random member of 
D
n
.
Denition:
An asymptotic degreesequence
D
is
feasible
if

D
n
6
=

for
al l
n

1
.
Denition:
An asymptotic degreesequence
D
is
smo oth
if there exist
constants

i
such that
lim
n
!1
d
i
(
n
)
=n
=

i
.
Denition:
An asymptotic degreesequence
D
is
sparse
if
P
i

0
id
i
(
n
)
=n
=
K
+
o
(1)
for some constant
K
.
Denition:
Given a smooth asymptotic degreesequence,
D
,
Q
(
D
)=
P
i

1
i
(
i
;
2)

i
.
Denition:
An asymptotic degreesequence
D
is
well-b ehaved
if:
1.
D
is feasible and smooth.
2.
i
(
i
;
2)
d
i
(
n
)
=n
tends uniformly to
i
(
i
;
2)

i
 i.e. for al l
>
0
there
exists
N
such that for al l
n>N
and for al l
i

0
:
j
i
(
i
;
2)
d
i
(
n
)
n
;
i
(
i
;
2)

i
j
<:
3.
L
(
D
)= lim
n
!1
X
i

1
i
(
i
;
2)
d
i
(
n
)
=n
exists, and the sum approaches the limit uniformly, i.e.:
(a) If
L
(
D
)
is nite then for al l
>
0
, there exists
i

N
such that for
al l
n>N
:
j
i

X
i
=1
i
(
i
;
2)
d
i
(
n
)
=n
;
L
(
D
)
j
<:
(b) If
L
(
D
)
is innite then for al l
T>
0
, there exists
i

N
such that
for al l
n>N
:
i

X
i
=1
i
(
i
;
2)
d
i
(
n
)
=n > T :
4

We note that it is an easy exercise to showthatif
D
is well-behaved then:
L
(
D
)=
Q
(
D
)
:
Note that for a well-behaved asymptotic degree sequence
D
,if
Q
(
D
)is
nite then
D
is sparse. Note further that if
D
is sparse and well-behaved then
since for
i

>
1,
P
i>i

id
i
(
n
)
<
P
i>i

i
(
i
;
2)
d
i
(
n
), the sum lim
n
!1
P
i

1
id
i
(
n
)
=n
approaches its limit uniformly in the sense of condition 3 in the denition of
wel l-behaved
.
The main result of 12] is:
Theorem 3
Let
D
=
d
0
(
n
)
d
1
(
n
)
:::
be a wel l-behavedsp
arse asymptotic
degreesequence for which there exists
>
0
such that for al l
n
and
i>n
1
4
;

,
d
i
(
n
)=0
.Let
G
beagraph with
n
vertices,
d
i
(
n
)
of which have degree
i
,
chosen uniformly at random from amongst al l such graphs. Then:
(a) If
Q
(
D
)
>
0
then there exist constants

1

2
>
0
dependent on
D
such that
G
a.s. has a component with at least

1
n
vertices and

2
n
cycles. Furthermore, if
Q
(
D
)
is nite then
G
a.s. has exactly
one component of size greater than

log
n
for some constant

dependent on
D
.
(b) If
Q
(
D
)
<
0
and for some function
0

!
(
n
)

n
1
8
;

,
d
i
(
n
)=0
for al l
i

!
(
n
)
, then for some constant
R
dependent on
Q
(
D
)
,
G
a.s. has no componentwithatleast
R!
(
n
)
2
log
n
vertices, and a.s.
has fewer than
2
R!
(
n
)
2
log
n
cycles. Also, a.s. no component of
G
has more than one cycle.
Consistent with the model
G
np
,we call the component referred to in
Theorem 3(a) the
giant component
.
Toprove Theorem 3, weworked with the conguration mo del introduced
in this form by Bollobas6] and motivated in part by the work of Bender and
Caneld4]. This mo del arose in a somewhat dierentforminthe work of
Bekessy, Bekessy and Komlos3]and Wormald13, 14 ]. A random congura-
tion with
n
vertices and a xed degree sequence is formed by taking a set
L
containing deg(
v
) distinct copies of eachvertex
v
,and cho osing a random
matching of the elements of
L
.Each conguration represents an underlying
multigraph whose edges are dened by the pairs in the matching. Weoften
5

Frequently asked questions

Q1. what is the probability of a random graph?

An asymptotic degree sequence is a sequence of integer valued functions D d n d n such thatdi n for i nPi di n nGiven an asymptotic degree sequence D the authors set Dn to be the degree sequence fc c cng where cj cj and jfj cj igj di n for each i De ne Dn to be the set of all graphs with vertex set n with degree sequence Dn A random graph on n vertices with degree sequence D is a uniformly random member of DnDe nition 

Q2. what is the key to the proof of the theorem?

If a random con guration with degree sequence D a s has a property P then a random graph with degree sequence D a s has PThe key to the proof of Theorem is the manner in which the authors exposed the con guration Given D the authors expose a random con guration F on n vertices with degree sequence D as follows 

Q3. what is the simplest way to prove the gn p?

The copies of partially exposed vertices which are not in exposed pairs are openForm a set L consisting of i distinct copies of each of the di n vertices which have degree iRepeat until L is emptya 

Q4. What is the smallest positive zero of Gn?

In fact with probability D the giant component is the rst component exposedUpon completion of the exposure of the giant component the con gura tion induced by the unexposed vertices is a uniformly random con guration with di j vertices of degree i for each i where j is the number of exposed pairs By Theorem this con guration a s has nD n verticesino n of which have degree i Recall that G a s has exactly one component of size greater than log nso it should not be surprising that Pi i i i as the authors will now see For DK X i i iKiK Xii iki KAK Xii iDki Xii iDkiAK D D X i i i i K D D Q DFurthermore the inequality is strict for D and so Q D as otherwise D could not be the smallest positive zero ofThe Model Gn pThe authors close by noting that some previously known results about Gn p c n are special cases of Theorems andSelect 

Q5. What is the probability that the largest component of a function lies in a cyclic?

For j and for each j such that XJ for all blog nc J j blog nc i j di j blog nc For any other j the authors de nei ji j with probability i i j M j i j otherwiseNow for any xed i by applying Theorem with Yj i j C m n and f s z izK s the authors see that with probability at least o nfor every i d ne Zi n o nwhereZi diKiis the unique solution toZi di nZ i iZiKSince their degree sequence has maximum degree o n a s holds for every iNote that Xj M j Pi idi j 

Q6. How do the authors prove the gn p theorem?

Expose a pair of F by rst choosing any member of L and then choosing its partner at random Remove them from Lb Repeat until there are no partially exposed verticesChoose an open copy of a partially exposed vertex and pair it with another randomly chosen member of L Remove them both from LAll random choices are made uniformly