A Study on the Amount of Random Graph Groupies

0.14 MB·English

by  Daodi Lu

A Study on the Amount of Random Graph
3
Groupies
1
0
2
Daodi Lu ∗
n
a
J January 15, 2013

3
1

]
O Abstract
C
In 1980, Ajtai, Komlos and Szemer´edi defined “groupie”: Let G=
h. (V,E)beasimplegraph, V =n, E =e. Foravertexv V,letr(v)
| | | | ∈
t denote the sum of the degrees of the vertices adjacent to v. We say
ma v ∈V isagroupie,if dreg(v(v)) ≥ ne.Inthispaper,weprovethatinrandom
graph B(n,p), 0 < p < 1, the proportion of groupies converges in
[
probabilitytowardsΦ(1) 0.8413asnapproachesinfinity,whereΦ(x)

1 is the distribution functi≈on of standard normal distribution N(0,1).
v
We also discuss the asymptotic behavior of the proportion of groupies
7
in complete bipartite graph B(n ,n ,p).
4 1 2
7
2
1 Introduction
.
1
0
3 The definition of groupie was first given in [1]. In that paper, Ajtai, Komlos
1
and Szemer´edi used the fact that every nonempty simple graph has at least
:
v one groupie to give an upper bound for Ramsey number R(3,k).
i
X
Definition 1 Suppose G= (V,E) isa nonempty simple graph, with V = n
r | |
a and E = e. For a vertex v of G, denote by r(v) the sum of the degrees of
| |
neighbors of v. The vertex v is called a groupie, if the average degree of the
neighbors of v is not less than the average degree of all vertices in G, i.e.
r(v) e
. (1)
deg(v) ≥ n
For the case that v is isolated, v is a groupie if and only if all vertices in G
are isolated.
∗DepartmentofMathematics,CaliforniaInstituteofTechnology,Pasadena,CA91125,
[e-mail]
1
The references [4], [10], [9], and [8] discuss the properties of groupies in
simple graphs. Mackey [9] proved that thereare at least two groupies in any
simple graphs with at least two vertices. The original definition of groupie
was revised in [6] and [11]. In these papers, the term “groupie” is defined as
avertex whosedegreeisnotlessthantheaverage degreeofitsneighborsina
simplegraph. FernandezdelaVega andTuza[6]proved thattheproportion
ofgroupieingraphB(n,p)isalmostalwaysverynearto 1 asn . Shang
2 → ∞
[11] investigated the amount of groupies in bipartite graph B(n ,n ,p), and
1 2
proved that the proportion of groupie in B(n ,n ,p) is almost always very
1 2
near to 1 when n ,n and n = n . However, he did not show
2 1 2 → ∞ 1 2
whether the proportion of groupies converges in probability when n ,n
1 2
go to infinity together in general case. In our paper, the term “groupie”
is used as Definition 1. In Section 2, we will investigate the proportion of
groupiedefinedin Definition 1inarandomgraphB(n,p). As thenumberof
vertices n , the proportion of groupie converges in probability towards
→ ∞
Φ(1) 0.8413, where Φ(x) is the cumulative distribution function of the
≈
standard normal distribution N(0,1). The main result in this section is:
Theorem 1 Suppose G= B(n,p), 0< p < 1, is a complete random simple
graph on n vertices. Let N(n) be the number of groupies in G. Then for
any ǫ > 0, we have
N(n)
P Φ(1) > ǫ 0, as n , (2)
(cid:18)(cid:12) n − (cid:12) (cid:19) → → ∞
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where Φ(x) is the cumulative distribution function of the standard normal
distribution N(0,1).
In Section 3, we will discuss the asymptotic behavior of the proportion
of groupie in a random bipartite graphs B(n ,n ,p) as n ,n and
1 2 1 2
→ ∞
the ratio n1 α, where α is a fixed nonnegative number. If n n
n2 → | 1 − 2| →
as n ,n , the proportion of groupies in B(n ,n ,p) converges in
1 2 1 2
∞ → ∞
probability towards max(1,α), while when α = 1 and n n = c where c is
1+α 1− 2
a fixed integer, the proportion of groupies converges in probability towards
1(Φ(1+ pc )+Φ(1 pc )). This limit is monotone decreasing with the
2 2(1−p) −2(1−p)
absolute value of c. When c = 0, the limit is Φ(1) 0.8413. As c is large,
≈ | |
it isnear to 1, whichcoincides withthelimit whenα= 1 and n n
2 | 1− 2|→ ∞
as n ,n .
1 2
→ ∞
2
2 In The Complete Random Graph B(n,p)
In random graph G= B(n,p), denote the n vertices to bev ,v ,...,v . For
1 2 n
k = 1,2,...,n, let A be the event “The vertex v is a groupie”. Let X be
k k k
the characteristic function of A , i.e. X = 1 .
k k Ak
Weclaimthatwecanexcludethecasethatthereexistsanisolatedvertex
in graph G, because as n ,
→ ∞
P(there exists an isolated vertex in B(n,p)) n (1 p)n−1 0.
≤ · − →
We first prove:
Lemma 2 The limit lim E[X ] Φ(1) is equal to 0.
n→∞ 1
−
Note that E[X ] = P(A ), i.e. E[X ] is the probability that the vertex
1 1 1
v is a groupie. Supposei is the degree of v . In a graph G = B(n,p), let V
1 1 1
bethesetofivertices whichareadjacenttov ; V thesetofn 1 ivertices
1 2
− −
which are not adjacent to v ; e the number of edges whose two vertices are
1 1
all in the vertex set V ; e the number of edges whose two vertices are all in
1 2
the vertex set V ; e the number of edges with one vertex in the vertex set
2 3
V , and the other vertex in V . Then the event A is equivalent to
1 2 1
2(i+e +e +e ) i+2e +e
1 2 3 1 3
,
n ≤ i
i.e.
2(n i)e +(n 2i)(e +i) 2ie 0.
1 3 2
− − − ≥
Note that the conditional expectation
E[2(n i)e +(n 2i)(e +i) 2ie ;i]
1 3 2
− − −
=ip((n i)(i 1)+(n 2i)(n 1 i) (n 1 i)(n 2 i))+(n 2i)i
− − − − − − − − − − −
=i((n 2)p+(n 2i)),
− −
and the conditional variance
Var[2(n i)e +(n 2i)(e +i) 2ie ;i]
1 3 2
− − −
i(i 1)
= (2(n i))2 − +(n 2i)2i(n 1 i)
(cid:18) − 2 − − −
(n 1 i)(n 2 i)
+(2i)2 − − − − (p p2)
2 (cid:19) −
,σ2.
3
Since i ∼ B(n 1,p), for any ǫ > 0, we can pick N N, such that
0 0
− ∈
for all n > N , P(i pn n0.5Ω(n)) < ǫ , where Ω(n) approaches in-
0 0
| − | ≥
finity as n , with the speed slower than any power of n. Denote by
→ ∞
F (x), F (x), F (x) the cumulative distribution function of the binomial
1 2 3
random variables 2(n−i)e1, −2ie2, (n−2i)e3, and let Φ (x), Φ (x), Φ (x) be
σ σ σ 1 2 3
the cumulative distribution function of the normal distribution in central
limit theorem corresponding to these binomial random variables. Suppose
X,X ,...,X areindependentidenticallydistributedd-dimensionalrandom
1 n
vectors, and X = (X(1),...,X(d)). In addition, suppose the third moment
of all components of X exist, and the first moment E[X] = 0. Let
X +...+X
1 n
Y = ,
n
F(x ,...,x ) the cumulative distribution function of Y, and Φ(x ,...,x )
1 d 1 d
the cumulative distribution function of the d-dimensional normal distribu-
tion Y that has the same first moment and covariance matrix as Y. By
the Berry-Esseen Theorem [3][7][2][5], there exist a constant C(d) that only
e
depends on the dimension d, so that
C(d) d E[X(i) 3]
sup F(x) Φ(x) | | .
x∈Rd| − | ≤ n12 Xi=1 (E[(X(i))2])23
By the1-dimensional Berry-Esseen theorem[3][7], thereexists aconstant
C that does not depend on n,i,p, so that
C(p2+(1 p)2) 1
sup F (x) Φ (x) < − , j = 1,2,3,
j j
x∈R| − | p(1 p) Ej
−
p p
whereE denote thenumberof randomedges which influencee , after given
j j
the degree i of vertex v , for example, E = i .
1 1 2
If i pn < n0.5Ω(n), we have 1 < C(cid:0)n(cid:1)−1, then
| − | √Ej 1
sup F (x) Φ (x) < C n−1,
j j 2
x∈R| − | ·
where C and C are constants which depend on p.
1 2
We claim an easy proposition, which is useful to generalize results in
Berry-Esseen theorem for i.i.d. random vectors.
Proposition 1 Suppose X ,X ,...,X , and Y ,Y ,...,Y are independent
1 2 n 1 2 n
d-dimensional random vectors, d 1, with cumulative distribution functions
≥
F ,F ,...,F , and G ,G ,...,G . Let
1 2 n 1 2 n
X = X +...+X ,
1 n
4
Y = Y +...+Y ,
1 n
and F,G the cumulative distribution functions of X,Y. If
sup F (x) G (x) ǫ ,
k k k
x∈Rd| − | ≤
where 1 k n, then
≤ ≤
n
sup F(x) G(x) ǫ .
i
x∈Rd| − | ≤ Xi=1
The proof of Proposition 1 can be given by induction on the number n
of summands in X and Y.
We go back to the proof of Lemma 2. By Proposition 1, there exist a
constant C which only depends on p, so that
3
i(n 2)p+(n 2i)
P(A ; i pn < n0.5Ω(n)) Φ − − < C n−1.
1 3
(cid:12) | − | − (cid:18) σ (cid:19)(cid:12) ·
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Since Φ(x) is continuous,
i(n 2)p+(n 2i)
lim Φ − − = Φ(1).
n→∞ (cid:18) σ (cid:19)
In addition, since P(i pn n0.5Ω(n)) < ǫ ,
0
| − |≥
P(A ) P(A ; i pn < n0.5Ω(n)) < ǫ .
1 1 0
| − | − | |
Since the choice of ǫ is arbitrary, we have
0
lim P(A ) Φ(1) = 0.
1
n→∞ −
Lemma 2 is proved. (cid:3)
We will prove another lemma.
Lemma 3 The limit lim Cov[X ,X ] is equal to 0.
n→∞ 1 2
We have changed the definition of the vertex sets V ,V in this part: Let V
1 2 1
be the set of vertices which are adjacent to v , but not adjacent to v , and
1 2
its size V = i ; V the set of vertices which are adjacent to both v and
1 1 2 1
| |
v , and its size V = i ; V the set of vertices which are adjacent to v , but
2 2 2 3 2
| |
5
not adjacent to v , and its size V = i ; V the set of vertices which are not
1 3 3 4
| |
adjacent to either v or v , and its size V = i = n 2 i i i . Then
1 2 4 4 1 2 3
| | − − − −
E[X X ]
1 2
=P(A A v ∼ v )+P(A A v ≁ v )
1 2 1 2 1 2 1 2
∧ ∧ ∧ ∧
=p P(A A v ∼ v )+(1 p) P(A A v ≁v ).
1 2 1 2 1 2 1 2
· ∧ | − · ∧ ∧
Denote e ,1 j k 4, be the number of edges whose two vertices are in
jk
≤ ≤ ≤
V and V , respectively.
j k
If v is adjacent to v , the event A A is equivalent to the following
1 2 1 2
∧
inequalities:
B , 2(n (i +i +1))(e +e +e +i )
1 1 2 11 22 12 2
−
 +(n 2(i +i +1))(e +e +e +e +(i +i +i +1))
1 2 13 14 23 24 1 2 3
 −

 2(i1 +i2+1)(e33 +e34 +e44) 0
 − ≥






B , 2(n (i +i +1))(e +e +e +i )
2 3 2 33 22 23 2
−

 +(n−2(i3+i2+1))(e13 +e34+e12+e24+(i1 +i2+i3+1))

 −2(i3 +i2+1)(e11 +e14 +e44)≥ 0.

Since the e ’s are independent when the value of i ,i ,i is fixed, then
jk 1 2 3
the conditional expectation and variance of B and B are given by
1 2
E[B ;i ,i ,i ]= (n (i +i +1))((n 2)p+(n 2(i +i +1))),
1 1 2 3 1 2 1 2
− − −
E[B ;i ,i ,i ]= (n (i +i +1))((n 2)p+(n 2(i +i +1))),
2 1 2 3 3 2 3 2
− − −
Var[B ;i ,i ,i ]
1 1 2 3
(i +i +1)(i +i )
= (2(n (i +i +1)))2 1 2 1 2
1 2
(cid:20) − 2
+(n 2(i +i +1))2(i +i +1)(n 1 (i +i +1))
1 2 1 2 1 2
− − −
(n 1 (i +i +1))(n 2 (i +i +1))
+(2(i +i +1))2 − − 1 2 − − 1 2 (p p2),
1 2
2 (cid:21) −
Var[B ;i ,i ,i ]
2 1 2 3
(i +i +1)(i +i )
= (2(n (i +i +1)))2 3 2 3 2
3 2
(cid:20) − 2
+(n 2(i +i +1))2(i +i +1)(n 1 (i +i +1))
3 2 3 2 3 2
− − −
(n 1 (i +i +1))(n 2 (i +i +1))
+(2(i +i +1))2 − − 3 2 − − 3 2 (p p2).
2 3
2 (cid:21) −
6
In addition, the conditional covariance of B and B is given by
1 2
Cov[B ,B ;i ,i ,i ]
1 2 1 2 3
i
1
= 4(n (i +i +1))(i +i +1)
1 2 2 3
(cid:20)− − (cid:18)2(cid:19)
i
2
+4(n (i +i +1))(n (i +i +1))
1 2 2 3
− − (cid:18)2(cid:19)
i
3
4(n (i +i +1))(i +i +1)
2 3 1 2
− − (cid:18)2(cid:19)
n 2 i i i
1 2 3
+4(i +i +1)(i +i +1) − − − −
1 2 2 3
(cid:18) 2 (cid:19)
+2(n (i +i +1))(n 2(i +i +1))i i
1 2 2 3 1 2
− −
+(n 2(i +i +1))(n 2(i +i +1))i i
1 2 2 3 1 3
− −
2(n 2(i +i +1))(i +i +1)i (n 2 i i i )
1 2 2 3 1 1 2 3
− − − − − −
+2(n (i +i +1))(n 2(i +i +1))i i
2 3 1 2 2 3
− −
+(n 2(i +i +1))(n 2(i +i +1))i (n 2 i i i )
1 2 2 3 2 1 2 3
− − − − − −
2(i +i +1)(n 2(i +i +1))i (n 2 i i i ) (p p2).
1 2 2 3 3 1 2 3
− − − − − − (cid:21) −
Treat B = (B ,B ) as a 2-dimensional random vector. Denote by
1 2
F(x ,x ) the cumulative distribution function of B, N = (N ,N ) the
1 2 n2 1 2
normal distribution with the same expectation and covariance matrix of B,
n2
and Φ(x ,x ) the cumulative distribution function of N.
1 2
Note that the contribution of e , 1 j k 4, to B ,B are linear,
jk 1 2
≤ ≤ ≤
by Berry-Esseen theorem of independentidentically distributed sequence on
Rd proved by Bergstro¨m [2][5], and Proposition 1, If we have
Condition 1 i np(1 p) n0.5Ω(n), i np2 n0.5Ω(n), i np(1
1 2 3
| − − | ≤ | − |≤ | − −
p) n0.5Ω(n),
| ≤
then there exists a constant C which only depends on p, so that
3
sup F(x ,x ) Φ(x ,x ) < C n−1.
1 2 1 2 3
| − | ·
When n , and Condition 1 holds, we have
→ ∞
Var[B ;i ,i ,i ] = n4Var[N ;i ,i ,i ] = O(n4), j = 1,2.
j 1 2 3 j 1 2 3
In addition, the covariance
Cov[B ,B ;i ,i ,i ] = n4Cov[N ,N ;i ,i ,i ] = O(n3.5Ω(n)) = o(n4).
1 2 1 2 3 1 2 1 2 3
7
For any bounded region R in R2
lim sup P(N x )P(N x ) P(N x ,N x ) = 0.
1 1 2 2 1 1 2 2
n→∞,δ1→0(x1,x2)∈R| ≤ ≤ − ≤ ≤ |
Under Condition 1, Var[N ;i ,i ,i ] (j = 1,2) are bounded, in addition,
j 1 2 3
when n , the probability that Condition 1 holds approaches 1. Thus,
→ ∞
lim P(A ,A v ∼ v ) P(A v ∼ v )P(A v ∼ v )= 0.
1 2 1 2 1 1 2 2 1 2
n→∞ | − | |
When v is not adjacent to v , by similar argument, we have
1 2
lim P(A ,A v ≁ v ) P(A v ≁ v )P(A v ≁ v )= 0.
1 2 1 2 1 1 2 2 1 2
n→∞ | − | |
Thus the equality lim Cov[X ,X ]= 0 holds. Lemma 3 is proved.
n→∞ 1 2
We go back to the final proof of Theorem 1, for any ǫ > 0,
N(n)
P Φ(1) > ǫ
(cid:18)(cid:12) n − (cid:12) (cid:19)
(cid:12) (cid:12)
(cid:12)N(n) (cid:12) ǫ ǫ
(cid:12) (cid:12)
P E[X ] > +P E[X ] Φ(1) > .
1 1
≤ (cid:18)(cid:12) n − (cid:12) 2(cid:19) (cid:18)| − | 2(cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
By Lemma 2, w(cid:12)henn , th(cid:12)esecond term in the right sideof theinequal-
→ ∞
ity above approaches 0. For the first term, by Chebyshev’s Inequality,
N(n) ǫ
P E[X ] >
1
(cid:18)(cid:12) n − (cid:12) 2(cid:19)
(cid:12) (cid:12)
4Va(cid:12)r[X +...+X(cid:12)]
(cid:12) 1 n(cid:12) .
≤ n2ǫ2
In addition,
Var[X +...+X ]
1 n
n
= Var[X ]+2 Cov[X ,X ]
1 i j
Xi=1 Xi<j
<n+n(n 1)Cov[X ,X ]= o(n2),
1 2
−
so the first term also approaches 0 as n . We complete the proof of
→ ∞
Theorem 1.
8
3 In The Complete Bipartite Random Graph B(n ,n ,p)
1 2
In the complete bipartite random graph G = B(n ,n ,p), where 0 < p < 1
1 2
is fixed, when n and n approaches infinity, the asymptotic behavior of the
1 2
proportion of groupies in G depends on the ratio n1. Suppose this ratio n1
n2 n2
has a limit α 0as n ,n . Without loss of generality, we may assume
1 2
≥ → ∞
that α 1. If n n , the proportion of groupies in G converges in
1 2
≤ | − | → ∞
probability towards 1 .
1+α
Theorem 4 Suppose G = B(n ,n ,p), 0 < p < 1, is a complete bipartite
1 2
random graph with n = n + n vertices. Let N(n ,n ) be the number of
1 2 1 2
groupies in G. Then for any ǫ > 0, we have
N(n ,n ) 1
1 2
P > ǫ 0,
(cid:18)(cid:12) n − 1+α(cid:12) (cid:19) →
(cid:12) (cid:12)
(cid:12) (cid:12)
as n ,n , n1 α(cid:12) 1, and n n =(cid:12) c(n) .
1 2 → ∞ n2 → ≤ | 1− 2| → ∞
Denote the two parts P ,P of vertices in G by v ,v ,...,v , and
1 2 1,1 1,2 1,n1
v ,v ,...,v . Let A be the event that the vertex v is a groupie,
2,1 2,2 2,n2 j,k j,k
where j = 1,2, and 1 k n ; E the characteristic function of the event
j j,k
≤ ≤
A , i.e., E = 1 .
j,k j,k Aj,k
We claim that for vertices v in the part P , the probability that v is
1,k 1 1,k
a groupie approaches 0, and for vertices v in the part P , the probability
2,k 2
that v is a groupie approaches 1.
2,k
Lemma 5 As n ,n , if n n , then
1 2 1 2
→ ∞ − → −∞
limE[X ]= 0,
1,1
limE[X ] = 1.
2,1
Note that E[X ] = P(A ) = P(A ), for any j = 1,2, and k =
j,1 j,1 j,k
1,2,...,n . Wemayfirstexcludethecasethatvertices v orv isisolated
1 1,1 2,1
because its probability (1 p)n2 or (1 p)n1 approaches 0 as n ,n .
1 2
− − → ∞
We then estimate the value of E[X ]. Suppose the vertex v has degree
1,1 1,1
i. Let V be the set of i vertices that are adjacent to v , and V be the
1 1,1 2
other n i vertices in the set P that are not adjacent to v . Let e be
2 2 1,1 j
−
the number of edges with one endpoint in the set V , where j = 1,2, and
j
another endpoint other than v . Then the event that v is a groupie is
1,1 1,1
equivalent to
e +i 2(e +e +i)
1 1 2
,
i ≥ n +n
1 2
9
i.e.,
(n +n 2i)e 2ie +(n +n 2i)i 0.
1 2 1 2 1 2
− − − ≥
Note that e is subject to the binomial distribution B(i(n 1),p), and
1 1
−
e is subject to the binomial distribution B((n i)(n 1),p). Then we
2 2 1
− −
have
E[(n +n 2i)e 2ie +(n +n 2i)i;i]
1 2 1 2 1 2
− − −
=(n +n 2i)i(n 1)p 2i(n i)(n 1)p+(n +n 2i)i
1 2 1 2 1 1 2
− − − − − −
=i((n 1)p(n n )+(n +n 2i));
1 1 2 1 2
− − −
and
Var[(n +n 2i)e 2ie +(n +n 2i)i;i]
1 2 1 2 1 2
− − −
=(n +n 2i)2i(n 1)p(1 p) (2i)2(n i)(n 1)p(1 p)
1 2 1 2 1
− − − − − − −
=i(n 1)p(1 p)(n2+n2+2n n 4in ).
1− − 1 2 1 2− 1
Note that the degree i of v is subject to the binomial distribution
1,1
B(n ,p). Thus for any ǫ > 0, when n approaches infinity, we have P(i
2 0 2
| −
n p n0.5Ω(n )) < ǫ .
2 | ≥ 2 2 0
Under the condition that i n p < n0.5Ω(n ), we have
| − 2 | 2 2
µ , E[(n +n 2i)e 2ie +(n +n 2i)i;i] = O(n2c(n)),
1 2 1 2 1 2
− − −
σ2 , Var[(n +n 2i)e 2ie +(n +n 2i)i;i] = O(n4).
1 2 1 2 1 2
− − −
Note that (n +n 2i)e 2ie +(n +n 2i)i is subject to the linear
1 2 1 2 1 2
− − −
combination of binomial random variables. In addition, as n ,n , by
1 2
→ ∞
the Berry-Esseen Theorem, there exists a constant C which only depends
on p, such that
µ
P(A ) Φ < Cn−1.
1,1
(cid:12) − (cid:18)σ(cid:19)(cid:12)
(cid:12) (cid:12)
As n1,n2 → ∞, we have(cid:12)(cid:12) µσ → −∞, then (cid:12)(cid:12)
µ
Φ 0.
(cid:18)σ(cid:19) →
In addition, by P(i pn n0.5Ω(n )) < ǫ ,
| − 2|≥ 2 2 0
P(A ) P(A ;P(i pn n0.5Ω(n )) < ǫ ) < ǫ .
| 1,1 − 1,1 | − 2| ≥ 2 2 0 | 0
For the arbitrary selection of ǫ > 0,
0
10