Statistics & Probability Letters 58 (2002) 363–368
Gaussian tail for empirical distributions of MST
on random graphs
Sungchul Leea;∗;1 , Zhonggen Sua; b;2
b
a
Department of Mathematics, Yonsei University, Seoul 120-749, South Korea
Department of Mathematics, Zhejiang University, Hangzhou 310028, P.R. China
Received November 2001; received in revised form March 2002
Abstract
Consider the complete graph Kn on n vertices and the n-cube graph Qn on 2n vertices. Suppose independent
uniform random edge weights are assigned to each edges in Kn and Qn and let T(Kn ) and T(Qn ) denote
the unique minimal spanning trees on Kn and Qn , respectively. In this paper we obtain the Gaussian tail
c 2002 Elsevier Science B.V. All
for the number of edges of T(Kn ) and T(Qn ) with weight at most t=n. rights reserved.
MSC: primary 60D05; 60F05; secondary 60K35; 05C05; 90C27
Keywords: Empirical distribution; Gaussian tail; Minimal spanning tree
1. Introduction and statement of main results
If G is a connected graph with aweight w(e) assigned to each edge e, a spanning tree S of G
has associated total weight L(S) = e∈S w(e). A minimal spanning tree (MST) of G is a spanning
tree T such that L(T) 6 L(S) for any spanning tree S. Minimal spanning trees of certain graphs
have been studied by a number of authors. Typically, one is interested in the total weight L(T) of
the MST T, the vertex degrees of T, or the empirical distribution of L(T). Among the most studied
graphs are the complete graph on n vertices and the n-cube graph.
∗
Corresponding author. Tel.: 0082-2365-7966; fax: 0082-2365-2579.
E-mail addresses: sungchul@.yonsei.ac.kr (S. Lee), zgsu@mail.hz.zj.cn (Z. Su).
1
This work was supported by the BK21 project of the Department of Mathematics, Yonsei University, the
interdisciplinary research program of KOSEF 1999-2-103-001-5, and Com ¿ 2MaC in POSTECH.
2
This work was supported by the BK21 project of the Department of Mathematics, Yonsei University, and by NSFC
10071072.
c 2002 Elsevier Science B.V. All rights reserved.
0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 2 ) 0 0 1 4 4 - X
S. Lee, Z. Su / Statistics & Probability Letters 58 (2002) 363–368
364
This paper is mainly devoted to the study on the asymptotic behaviour of empirical distributions
of the edge weights of the MST of the complete graph and the n-cube graph.
Let Gn be a connected graph with vertex set Vn = {1; 2; : : : ; n} and edge set En = {e}. We assume
that the degree of each vertex i ∈ Vn is dn so that the number of edges in En is ndn =2. More
specically, we consider the following two connected graphs. Let Kn be the complete graph with
vertex set Vn = {1; 2; : : : ; n} and edge set En = {(i; j): 1 6 i ¡ j 6 n}, and let Qn be the n-cube graph
with vertex set {0; 1}n and nearest neighbor edges, i.e., two vertices are taken to be adjacent if they
dier in exactly one coordinate. The edge weights U (e) are dened for each edge e of Gn , Kn ,
and Qn , and taken to be independent uniform random variables on the unit interval [0; 1]. Denote
the MST of Gn , Kn , and Qn by T(Gn ), T(Kn ), and T(Qn ), respectively. Since we assign dierent
weights for dierent edges with probability 1, by Lemma 5 of Lee (1997) we see that T(Gn ),
T(Kn ), and T(Qn ) are uniquely determined.
From now on, we assume that T(Gn ), T(Kn ), and T(Qn ) are unique.
The one remarkable result concerning the total weight of T(Kn ) is due to Frieze (1985), who
gave the exact asymptotic expected value of L(T(Kn )), i.e., as n → ∞
EL(T(Kn )) → (3):
In the case of the n-cube, Penrose (1998) found that the same limit is also valid for (n=2n )L(T(Qn )).
There are also interesting results regarding the vertex degrees in T(Kn ) and T(Qn ). For these results
see Aldous (1990), Penrose (1998), and Kim and Lee (2001).
Now, let’s consider the empirical distributions. For 0 ¡ f(n; t) ¡ 1, we dene
N (Gn ; t) =
1(U (e) 6 f(n; t)):
e∈T(Gn )
For Kn and Qn we consider a more specic form of the empirical distributions. For t ¿ 0, we dene
t
t
; N (Qn ; t) =
1 U (e) 6
1 U (e) 6
N (Kn ; t) =
n
n
e∈T(Kn )
e∈T(Qn )
and
F(Kn ; t) =
N (Kn ; t)
;
n−1
F(Qn ; t) =
N (Qn ; t)
:
2n − 1
As a consequence of a deep result on the asymptotic empirical distribution of edge lengths of Aldous
(1990), one can easily see that
1 t
2
F(Kn ; t) →
(1 − (s)) ds in probability;
2 0
where (t) is the survival probability for a Galton–Watson branching process from a single ancestor
with Poisson(t) ospring. In the case of the n-cube, Penrose (1998) showed that the same limit is
also valid for F(Qn ; t).
In this paper, as a rst step toward the central limit theorems we obtain the Gaussian tail bound
for N (Kn ; t) and N (Qn ; t). The central limit theorems for N (Kn ; t) and N (Qn ; t) are largely open.
Our main results are as follows.
S. Lee, Z. Su / Statistics & Probability Letters 58 (2002) 363–368
365
Theorem 1. For any strictly positive but nite constant C1 such that ex 6 1 + 1:5x; 0 6 x 6 C1 ;
and for any 0 6 6 (2C1 )1=2 ndn f(n; t)
2
:
P(|N (Gn ; t) − EN (Gn ; t)| ¿ ) 6 2 exp −
4ndn f(n; t)
Corollary 1. For any strictly positive but nite constant C1 such that ex 6 1 + 1:5x; 0 6 x 6 C1 ;
and for any 0 6 6 (8C1 n)1=2 t
√
1 2
P(|N (Kn ; t) − EN (Kn ; t)| ¿ n) 6 2 exp − :
4t
Corollary 2. For any strictly positive but nite constant C1 such that ex 6 1 + 1:5x; 0 6 x 6 C1 ;
and for any 0 6 6 (C1 2n+1 )1=2 t
√
1 2
n
P(|N (Qn ; t) − EN (Qn ; t)| ¿ 2 ) 6 2 exp − :
4t
As a simple corollary to Corollaries 1 and 2, by the Borel–Cantelli lemma we have the following.
We leave this as an easy exercise to the reader.
Corollary 3. For t ¿ 0; as n → ∞
1 t
2
F(Kn ; t) →
(1 − (s)) ds
2 0
and
1
F(Qn ; t) →
2
t
2
(1 − (s)) ds
a:s:
a:s:
0
2. Gaussian tail
In this section, we introduce a martingale inequality. We also introduce an algorithm for the
construction of a minimal spanning tree. Using these two tools we obtain Theorem 1. From Theorem
1 we obtain the Gaussian tail bound for N (Qn ; t) and N (Kn ; t). So, here we mainly concentrate on
Theorem 1.
Let’s start with a martingale inequality. Here is the idea behind the inequality. Recall that Kn is
the complete graph on n vertices with vertex set Vn and edge set En . Also recall that {U (e): e ∈ En }
are iid uniform on [0; 1]. For an easy presentation, we rename {U (e): e ∈ En } as {Uk : 1 6 k 6 m}
where m = ( n2 ). Let for i = 1; : : : ; m,
t
1 U (e) 6
N = N (Kn ; t) = (n − 1)F(Kn ; t) =
n
e∈T(Kn )
S. Lee, Z. Su / Statistics & Probability Letters 58 (2002) 363–368
366
and let
di = E(N |U1 ; : : : ; Ui ) − E(N |U1 ; : : : ; Ui−1 )
= : : : (N (U1 ; : : : ; Ui ; Ui+1
; : : : ; Um ) − N (U1 ; : : : ; Ui−1 ; Ui ; : : : ; Um )) dUi : : : dUm
(2.1)
; : : : ; Um ) − N (U1 ; : : : ; Ui−1 ; Ui ; : : : ; Um );
i = N (U1 ; : : : ; Ui ; Ui+1
(2.2)
and
where {Uk : 1 6 k 6 m} is an independent copy of {Uk : 1 6 k 6 m}. In our study i has values 1,
0, and −1, and P(i = 1) = P(i = −1) is small. In this case with some extra technical conditions
the following martingale inequality is quite powerful.
Lemma 1. For an arbitrary random variable X ∈ L1 (; F; P) and for an arbitrary ltration {∅; }=
F0 ⊂ F1 ⊂ · · · ⊂ Fm = F. We let di ; 1 6 i 6 m; be the martingale dierence; that is di =
E(X |Fi ) − E(X |Fi−1 ). Suppose that di = E(i |Fi ) for some i ∈ L1 (; F; P). In addition; assume
that for a xed c ¿ 0; i takes only c; 0; and −c; and that for a xed p and for 1 6 i 6 m;
P(i = c|Fi−1 ) 6 p
a:s:
(2.3)
Then, for any strictly positive but nite constant C1 such that ex 6 1 + 1:5x, 0 6 x 6 C1 , and for
any 0 6 6 (8C1 )1=2 mcp,
2
:
(2.4)
P(|X − EX | ¿ ) 6 2 exp −
8mc2 p
Remark. If one uses Azuma’s inequality in the situation considered in the above lemma; one has
2
P(|X − EX | ¿ ) 6 2 exp −
:
(2.5)
2mc2
If p is small; then (2.4) beats (2.5) and actually this is the case in our study. For Azuma’s inequality
and its application see Steele (1997) and Yukich (1998).
Proof. We just estimate the moment generating function Ees(X −EX ) and apply the Markov inequality
to get (2.4). One can also get the similar inequality by Lemma 1.6 of Ledoux and Talagrand (1991)
with b2 = mpc2 . We leave these to the interested reader.
Now we recall the add and delete algorithm for the construction of an MST which one can nd
in Kim and Lee (2001). Here is the situation. Let G = (V; E; w) be a connected weighted graph with
|V | ¡ ∞, |E| ¡ ∞ where w : E → [0; ∞) is a weight function dened on the edge set E. Let T
be an MST on G. Now, suppose only one edge e changes its weight from w(e) to w (e), that is
suppose we are facing a new connected weighted graph G = (V; E; w ) where
w(e ) for e = e;
w (e ) =
w (e) for e = e:
S. Lee, Z. Su / Statistics & Probability Letters 58 (2002) 363–368
367
In order to construct an MST T on G we use the following add and delete algorithm:
Case 1: If e ∈ T and if w (e) 6 w(e), we just let T = T.
Case 2: If e ∈ T and if w (e) ¿ w(e), we delete e from T. Then two connected components are
generated. Choose f such that (T \{e}) ∪ {f} is connected and
w (f) = min{w (f ): (T \{e}) ∪ {f } is connected}:
Now, we let T = (T \{e}) ∪ {f}.
Case 3: If e ∈ T and if w (e) ¿ w(e), we just let T = T.
Case 4: If e ∈ T and if w (e) ¡ w(e), then add e to T, i.e., form T ∪ {e}. T ∪ {e} contains
a unique circuit C. Choose an edge f ∈ C such that w (f) = max{w (e ): e ∈ C}. Delete f from
T ∪ {e}, i.e., we let T = (T ∪ {e}) \{f}.
Lemma 2. The spanning tree T is a minimal spanning tree of G .
Proof. This is Lemma 2.2 of Kim and Lee (2001).
We need the following property of an MST.
Lemma 3. Let G = (V; E; w) be a connected weighted graph with |V | ¡ ∞ and |E| ¡ ∞. If there
exists a path = (e1 ; : : : ; en ) in G from v1 ∈ V to v2 ∈ V such that w(ei ) 6 ; 1 6 i 6 n; then in
any MST T on G there exists a path = (e1 ; : : : ; em ) in T from v1 to v2 such that w(ej ) 6 ;
1 6 j 6 m.
Proof. This is Lemma 2 of Kesten and Lee (1996).
Now we are ready for the main argument of this paper.
Proof of Theorem 1. Let’s recall di and i from (2.1) and (2.2); that is di is some kind of integration
of i where
i = N (U1 ; : : : ; Ui ; Ui+1
; : : : ; Um ) − N (U1 ; : : : ; Ui−1 ; Ui ; : : : ; Um ):
; : : : ; U ) and the old congLet T and T be MST under the new conguration (U1 ; : : : ; Ui ; Ui+1
m
uration (U1 ; : : : ; Ui−1 ; Ui ; : : : ; Um ); respectively. We also let ej be the edge where the weight Uj or
Uj is attached to. We claim that i has values 1; 0; and −1; and that P(i = 1) = P(i = −1) is
small. Furthermore; it satises technical conditions in Lemma 1 so that in our study we can use the
martingale inequality in Lemma 1.
First, one can easily see that by Lemma 2, i takes only 1, 0, and −1. Suppose that Ui ¿ f(n; t)
and Ui ¿ Ui . There are two cases we have to consider; the rst case is that ei ∈ T and the second is
that ei ∈ T. If ei ∈ T, then by Lemma 2 we rst delete ei from T. Then, we add the smallest edge
; : : : ; U )
f which makes (T \{ei }) ∪ {f} connected. Under the new conguration (U1 ; : : : ; Ui ; Ui+1
m
the weight attached to the edge f is either U (f) or U (f). For simplicity we call this weight U . By
Lemma 3 the weight U is greater than Ui . Therefore, under the new conguration 1(U 6 f(n; t))=0.
Since under the old conguration, 1(Ui 6 f(n; t)) = 0, we have i = 0. If ei ∈ T, then by Lemma 2,
S. Lee, Z. Su / Statistics & Probability Letters 58 (2002) 363–368
368
T = T and of course i = 0. By the similar arguments we see that i = 1 only when Ui ¿ f(n; t)
and Ui 6 f(n; t). Therefore with the ltration Fi−1 = (U1 ; : : : ; Ui−1 )
P(i = 1|Fi−1 ) 6 P(Ui 6 f(n; t)) = f(n; t) a:s:
Now, we apply Lemma 1 for m = ndn =2, c = 1, p = f(n; t), to obtain Theorem 1.
√
Proof of Corollaries 1 and 2. We apply Theorem 1 for = n; ndn =2 = ( n2 ); f(n; t) = t=n; to
√
obtain Corollary 1. To obtain Corollary 2; we apply Theorem 1 for = 2n ; ndn =2 = 2n−1 n;
f(n; t) = t=n.
Acknowledgements
The authors would like to thank David Aldous for helpful discussions regarding the branching
process argument of Aldous (1990) and the referee for his thoughtful comments on an earlier version.
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