poisson convergence in the n-cube

Math. Xachr. 131 (1987) 49-57

Poisson Convergence in the n-Cube

By KARL WEBER of Rostock

(Receive3 September 11, 1985)

Abstract. We consider two types of random subgraphs of the n-cube Q, obtained by independent deletion the vertices (together with all edges incident with them) or the edges of Q,,, respectively, with a prescribed probability q = 1 - p . For these two probabilistic models we determine some values of the probability p for which the number of (isolated) L-dimensional subcubes or the number of vertices of a given degree k , respectively, has asymptotically a Poisson or a Normal distribntion. The technique which will he wed is that of Poisson convergence introduced by BARBOUR [I] (see also [a]).

1. Introduction

The n-cube Q, is the graph consisting of the 2" vertices (a l , . . .) a,), a; E {O, 1) and the n271-1 edges between vertices differing in exactly one coordinate. A spanning subgraph g of Q, has the same vertex set as Q,. An induced subgraph f of Q, with the vertex set A Q, contains exactly those edges of Q, that join two vertices in A. (Note that by Q, or f are not only denoted the graphs but also its vertex sets, g stands also for the edge set of 9.) Choosing the edges of g (the vertices of f ) a t random, independently of each other with the same probability p , we arrive at a random spanning ( induced) subgraph whose probabilities are defined as Prob (9) = plglpna"-'--lgl ( P rob (f) = plflq?"-lfl where q = 1 - p . We say almost all g (or f ) have a given property if the probability that g (or f) has this property tends to 1 as n + cu. In the sequel for these two types of random graph9 in the n-cube the limit distribu-

tions of certain random variables are investigated : Jk(f) (Ik(f)) = number of k-dimensional subcubes of Q , being also (isolated) subgraphs

of f (briefly called k-cubes and isolated k-cubes of f, respectively), Fk(f) = number of vertices with degree k in f.')

For random spanning subgraphs the corresponding random variables are written with bar: J;, I;, V;.

More precisely, assuming k = 0, 1,2, . . . is fixed, we determine some values of the probability p for which these random variables have asymptotically a Poisson or a Normal distribution, respectively, as n -+ 00.

) >

l) For the sake of convenience the index n is often omitted.

4 Math. Nachr., Bd. 191 -

50 JIeth. Nachr. 131 (1987)

Note that the subcubes of Q, are very important induced subgraphs of Q,. They play a fundamental role in the theory of minimization of Boolean functions.

Henceforth all limits, asymptotics etc. are considered as n -+ co. For two sequences oc = a(n) and f i = P(n) we write a 5 j3 if OL 5 j3(1 + o(l)) , OL ,< f i if a = O(&, some- times ci < f i for K = o(P), (y. - f i if a/fi -+ 1 and OL x j3 if (y. ,< and 5 a. Everywhere p = p(n) denotes a sequence tending to infinity arbitrarily slowly, x is an arbitrary real, whereas c is a positive real constant and k stands for an arbitrarily fixed non- negative integer unless otherwise specified. The binary logarithm is denoted by log. The distance e means the customary Hamming distance in the cube.

2. Poisson Convergence

Let (X,,), n = 1,2, .. . be a sequence of random variables. As usual, we say X, is asymptotically distributed as the random variable X (notation: X, -+ X) if (X,) con- verges in distribution to X . Let P o ( A ) and N(0, 1) denote random variables which are POISSON distributed with expectation A and standard normally distributed, respectively. For showing that X , -+Po ( A ) or X, *f N(0, l), respectively, the method of moments is often used. Let E(X, ) , denote the .r-th factorial moments of X,, r = 1, 2, . . ., and 2 is a positive real constant. Then

(2.1) E(X, ) , + A r , r = 1, 2, ... niplies that X , ccf Po ( A ) . If

(2 .2) EX', 3 & J xre-2a/2 dx, r = 1, 2 , . . . -m

then X, N(0, 1). The standard technique to show (2.1) is the following. Represent X, as a sum of identically distributed 0 - 1 random variables: X , = Xn, i , and then

E(X,) , = C E(X,,iLXn,;, . . . X,,{J is estimated, where the latter sum is taken over all ordered r-tuples of pairwise different X,,i.

To establish (2.2) for 2, = ( X , -- E X , ) / f E X , instead of X , ERDOS and RENYI [3] introduced the following technique which was used more carefully by SCHURGER [ 71 and since then by many authors [4, 5, 6, 81. One shows

1

-

where I , = EX,,. The sum on the right-hand side is the appropriately normalized r-th

central moment of Po ( I , ) , and it tends to (1//2n) j" xre-2*/2 dx, the r-th moment of

It is clear that D 2 X , N EXn (where D2X, is written for the variance of X,) is a necessary condition for this technique. But furthermore (2.3) requires E(X,) , = A: + o( 1) uniformly in r as n -+ 00. That leads in general to very hard restrictions on the probability p of the underlying probabilistic model, and often it is difficult to manage. Therefore BARBOUR [ 11 used an alternative approach (see also [4]) : The random variable

00

N ( 0 , l), provided that A, -+ co as n -+ 00. --m

Weber, Poisson Convergence 51

X, on the non-negative integers is called Poisson convergent if the so-called total variation distance between X, and Po (A,)

(2.4) sup IProb ( X , E &I) - Prob (Po (I ,) E ")I + 0 'MZN

as n -+ m, where A, = E X , and iV = (0, 1, 2, ...}. (Henceforth we use the short notation d(X,, Po ( j . , ) } for the total variation distance

between X, and Po (A,), i.e. for the left-hand side of (2.4).) Kote that (2.4) implies that X,, c3 P o ( A ) if A, -+ A and r?,+N(O, 1) if i,, + m a s n -+ CO. The latter approximation follow-s by the well-knownfact that (Po (A,) - A,) /V/ .n -3 N ( 0 , 1) if 1, + 00.

Following [l] we can derive two upper bounds for d ( X , , PO ( I , ) ) , where A, = EX,.

Let X, = 2 Xn,i, where X,,i are identically distributed zero-one random variables

with Prob ( X , , j = 1) = P, i = 1, 2, ..., Z(n). Then in = Z(n) P. F i r s t m e t h o d : Suppose that the conditional probabilities Prob (9, = j 1 X, , i = 1)

do not depend on i, and let Y , be a non-negative integer valued random variable with Prob ( Y , = j - 1) = Prob (X, = j I X,,l = l), j = 1, 2, .. ., Z(n). Furthermore define x+ = (!.I + 2 ) / 2 . Then

l ( n )

i = l

(2 .5) rqx,, Po (A,)) I 2 ( E ( X , - Yn)+ + E( Y,, - XJ-1.

S e c o n d m e t h o d : D e f i n e N ; = ( j : Xn,;andX,,iareindependent] andiFi = 11, 2, ..., I ( , % ) } \ iVi. Assume INiI = lNll, and F Prob (X,, i = 1 I Xn,i = 1) gives also the same value for all i = 1, 2, . . ., Z(n). Then j E N ~ \ { i J

If P = Prob (X,,i = 1) 5 Prob ( X , , j = 1 I Xn. l = 1) for all j E of (2.6) we can also use the upper bound

\ {l} then instead

d(X,, Po (A,)) 5 2P + 4 2 Prob (X,, i = 1 1 Xn,l = 1). j€iT,\(l]

(2-7)

P r o o f s of (2 .5) and (2.6): Following [l], for each A > 0 and i1.I 5 N , the function a =

Prob (Po ( I ) E LM n Cm) - Prob (Po (A) E 31) Prob (Po ( A ) E Cm) a(m + 1) = , m 2 1 ,

Prob (Po (A) = m)

where C , = (0, 1, . . ., m} has the properties

(2.8)

for any non-negative integer-valued random variable X and

Aa = sup la(m + 1) - a(m)i 5 -.

N -+ R defined by a(0) = 0 ;

Prob ( X E iM) - Prob (Po ( A ) E fW) = E{Aa(X + 1) - X a ( X ) }

2 mEN 11 (2.9)

Now let the conditions for the first method be fulfilled. Then by (2.8)

Prob ( X , 6 M) - Prob (Po (A,) E H) = E{A,a(X, j- I) - X n a ( X , ) } .

4"

52 Math. Nachr. 131 (1987)

= 1, 1 l ( n )

E { a ( X n + 1)) - C Prob (Xn, i = 1) E(a(Xn) I Xn,i = 1) An i=l

= 2 2 Prob ( X n , j = 1 1 Xn,l = 1). j € L V , \ l l I


3. Isolated Subcuhes

Let I k ( f ) denote the number of isolated k-cubes of f and put p, = (nk/2'2n/2')-1 J %(pa) k l n n x -1, p , = 1 - 2-4 1 - - + - + o (i)). Then - 1 - 2 - 2 - k

assuming p < p, or p 2 p , almost ali f do not contain an isolated k-cube but if pr < p 5 p2 then EIk 3 co and there is an Ek = Ek(n) --f 0 such that Prob (11, - EIkl < ckEIk) 3 1 (cf. [S]). Now we shall characterize the distribution of I k more exactly provided that p, 5 p I p4.

( Zkn (-) n ( 2kn 2% k l n n + y

1 - - -

Theorem 3.1. Let p , 5 p 5 p.,. Then is POISSON convergent i f any of the following two conditions is satisfied:

( i ) k 2 2 ,

Corollury3.2. Let p , < p 5 p , . Then f k + N ( O , I ) if ( i ) or ( i i ) of Theorem3.1 is satisfied. If p - cp, then I k 4 Po (2.) with 1 = c2"/k! zk, whereas if p = p , then I k -+ Po (A), where 1 = e z ( l - 2-2-')Zk/k!.

Proof of Theorem 3.1: In order to bound d ( I k , Po ( E I k ) ) the first method is used.

(ii) k E (0, I} and p< l /n o r p > l / r ~ . ~ )

Denote the k-cubes of &, by K,, K,, . . ., KT, T = W k , define the 0 - 1 random T

variables X,,;(f) = 1 iff Ki is an isolated subcnbe of f and put X, = C X n , i . Then

X,: = Ik, Prob (X,, i = 1) = pz'q(n-k)2" = P and 1, = E X , = TP. Let f 1 denote the subgraph of f obtained from f by deleting the vertices 14 with &K,) 5 1 and all edges incident with them. Define Y , as the number of isolated k-cubes in f'. Then the equality Prob ( Y , = j - 1) = Prob (X,, = j X,,,l = 1) is obvious, and we can use

(2.5). Since there are no more than ( 1 + (n - k)) 2k k-cubes Kj with e(K,, Rf) 5 1

1=1

we get (3 E ( X , - Y,)+ 5 (1) (1 + n - k) 2kf'.

There exist exactly (" i-: ') (' ') (:) 2 k - 8 k-cubes Ki in&, such that ,o(K1, JKj) = 2

and the vertices g CxKi with &z; K,) = 2'induce an s-cube. The probability that such a k-cube K j is isolated in f1 but not in f is Pq-2'*'( 1 - (1 - P)~ '+ ' ) Pq-2"'. Thus

and consequently by (2 .5)

z, Note that the result for k f {O, I} given in [8] is not correct for p (except when p - l/n and k = 0).

54 >lath. Nschr. 181 (1957)

for fixed k = O,* 1 ,2 , . . . and the given range of probability. From this upper bound our theorem follows immediately considering that 2k > k + 1 for k 2 2 .

Note that the method of ERDOY and RENYI (see Section 2) yields f k -+N(O, 1 ) for p , < p 5 only for its application requires besides EI, -+ ca that (EI,)?

c]

\ I I

We have an analogous result for random spanningsubgraphs. P u t pi = (n1/3k-'2n;k2k-' )-I ( k 2 l), p2, p , and p, are defined as above. (An inequality of the form p i < p has to be understood as p 5 p2 for k = 0, i.e. we ignore p ; when i t is not defined.)

p ,

Theorem 3.3. Assume p i 5 p 5 p, . Then I: is POISSON converyent if an2 of the following three conditions is satisfied:

(i) k 2 2 , (iii) k = 0 and p > l /n .

(ii) k = 1 und p < l / n or p > l / n ,

Corollary 3.4. Asmme pi < p 5 p 2 . Then 1; -+ N(0, 1) i/ (i), (ii) or (iii) of Theorem 3.3 is satisfied. If p - cpi then I ; --+ Po (I.) with 1 = ck2'-'/k! 2k ( k 2 l), whereas if p = p4 then 1; -+ Po (i), where 2, = es(1 - 2-2'k)k2k''/k!.

Since the proof of Theorem 3.3 follows the same line as that of the preceding theoretn we omit it.

Note that 16 for p < l /n and p -4 c/'tz (c + 2 ) and I : for p - c/n ( c =+ 1) can not be POISSON convergent as for these probabilities the necessary condition D21L -El;. is not fulfilled.

Finally let us remark that also the second method may be applied to establish POISSON convergence of I , and 1;. But it provides a slightly weaker result for k E {0,1,2} or k E (0, 11, respectively.

4. Subcubes

Now we are concerned with the nutnber Jk(f ) of k-cubes of f . Clearly EJk = 2n-kp2'

and EJ, -+ 0 for p < p , what implies Prob (Jk = 0) + 1. On the other hand EJk --f cc

(cf. [B]). Finally if p - cp, then EJ, - c2"/k! 2k. Let us determine some values of the probability p for which Jk is POISSON convergent.

(3 if p > p, and there exist &k = EL.('^) -+ 0 such that Prob ( I J , - EJk] < E k E J k ) -+ 1

Theorem 4.1. Suppose p1 5 p . Then Jk is Poisson Convergent i f uny of the following two conditions is satisfied:

(i) k z l and p<<n-k'(z '- l) , (ii) k = O and p + O .

Corollary 4.2. Suppose p1 < p . Then j k -+ N(0, 1) i f (i) or (ii) of Theorem 4.1 i s satisfied. If p N cp, then Jk 4 P o ( I ) , where 1 = c Z k / k ! 2,.

Proof of Theorem 4.1. Now the second method is used. Let again K,, K,, ..., K,,

T = (l) 2n-k, be the k-cubes in Q,,. X n , i are the indicator random variablesof the events

K , S f and X, is the sum of all the LY,,,~, i = 1, 2, . . ., T . Then x, = J k , Prob (x,,i = 1)


= p2' = P and I , = TP Obviously X , , i and Xn,j are independent iff Ki n K j = 0. The values INi\ and Prob ( X , , j = 1 1 X,,i = 1) (summed over j E wi \ (i}) are independent of i and P < Prob ( X , , j = 1 j Xn,l = 1) for j E \ (1). Thus by (2.7) it is not difficult to show that the conditions nkp2*-I + 0 (k 1 1) and p -+ 0 (k = 0), respectively, are sufficient for POISSON convergence.

Remark. Of course, J , is binomially distributed with the parameters 2" and p . There- fore by MOIVRE-LAPLACE the random variable ( J , - E J , ) / ~ ~ , -+ N(0, 1) for p q P --f 00. But J , is POISSON convergent only if 2-n < p < 1 since D2J, = pq2n N p2n = E J , does hold for these probabilities only.

For the number J;(g) of k-cubes of g (k >= 1 as Jh(g) = 2" for all y) we have a similar result.

Theorein -1.3. Let p i 5 p . Then JL is Poisson convergent if any of the jollowiny two conditions is satisfied;

( i ) k 2 - 2 and p < n - ( k - l ) ' ( h z k - ' - l ) , (ii) k = 1 and p + 0 .

Corollary 4.4. Let p i < p . Then Ji. -3 N ( 0 , 1) i f (i) or (ii) o/ Theorem 4.3 i s sutis/ied. If p - cpi then JL -+ Po ( A ) , where i, = ck2'-'/k!

The proof of Theorem 4.3 is siniilar to that of the preceding one. Easy calculations show that d(J i , Po (EJJ)) 5 nk-lpk8*" (k 2 2) and d(J: , Po (EJ:)) 5 p .

Xote that J i is binomially distributed with parameters n 2,-l and p , and consequently its standardization tends to N ( 0 , 1) in distribution provided that pqn2" + co. But again J { is POISSOX convergent for p --f 0 only.

( k 2 1).

C

5 . Vertex Degrees

The number of vertices of degree k in g is denoted by J7i(g). Furthermore piit

+ 0 (i). 1 k1n.n x 1 k I n n 2 2n n 2 2n n

' and p7 =-+--- p i = (n2ni9-l (k 2 l), p , = - + - - - Then E V J = 2% (:I p*pn-k co for p ; << p p , (ignore the lower bound for k = 0).

Theorem 5.1. Assume p i 5 p I_ pi. Then V ; is POISSON convergent if any of the following two conditions is satisfied:

(i) k 2 2 and p < n - ( k + l ) ' k or p 2 Inn + k l n l n n + pl

n Inn + k In In n + pl

n (ii) k E {0, 1) and p 2

Corollary 5.2. Let pg 4 p $pe. Then vi -+ N(0 , 1) i f condition (i) or (ii) of Theorem 5.1 is satisfied. If p N cpg then V;--tPo ( A ) with I. = ck/k! (k >= l), whereas if p = p 7 then VJ --f P o ( A ) , where A = eZz/k! .

Recall that we already mentioned a better result for Vh(= 16) in Section 3 (see Theorem 3.3 and Corollary 3.4).

Proof of Theorem 5.1. The second method is employed in order to show the POISSON convergence claimed in the theorem. Denote the vertices of Q, by gl, a,, ..., g2" and

56 Math. Nachr. 131 (1987)

2"

define X,,i(g) = 1 if the degree of ai in g is k, S,,i(g) = 0 otherwise and X, = 2 Xn, i .

Then X , = V;, Prob (X,,,i = 1) = pkqn-k = P and A,, = EX,, = ZnP. Now Xn, i and

Xn, are independent iff ,o(a;, a j ) 2 2. Obviously the necessary symmetry conditions are fulfilled too so that we may use (2.8). We get P Irli = P(n + 1) x nk+lpkqn-k and

i = l t) ' Prob (XnLj = 1 and Xn,l = 1)

P C Prob (X,,j = 1 I X,,,l = 1) = n

j€.Y,\{ll

=: (pn)k-] q -? + n k f l p k p k

(the sum appears for k 2 1 only). Thus by (2.6) d(X, , Po ( in)) -+ 0 if any of the conditions (i) and (ii) of Theorem 5.1 does hold.

Note that for k 2 1 and pf < p ;g p , we have D2V; -El:; (unless p =: l /n ) so that the conditions ( i ) and ( i i ) in Theorem 5.1 are probably technical ones only. Eut never- theless our result goes far beyond thah of PA4LKA and RUCIBSKI [6 ] . They proved POISSOX

convergence of Vg for p i 5 p 5 p;lfo(l) and for - - o(1) 5 p 5 p;, respectively. An

analogous theorem holds for the number v k ( f ) of vertices o f degree k in f . The proof can easily be adapted from that of the previous theorem.

1 2

Theorem 5.3. Put p , = (n1/(k+1)2n/(kf1))-1 and let p 5 5 p 5 p i . Then vk is P O I S S O S

convergent if and of the following two conditions is satisfied:

Inn + (k + 1) In Inn + cp

n (i) k 2 2 and p < n - ( k + 2 ) / ( k + 1 ) or p 2 ,

In n + ( k + 1) In Inn + q~ n

(ii) k E (0, l} and p 2

Corohry 5.4. Let p 5 <I, 5 p6 . l'hen 8, += N(0, 1) if (i) or (ii) of Theorem 5.3 is satisfied. If p - cp, then vk 1'0 ( A ) with I = ckcl /k! , whereas if p = pi then V, += Po (A), where I = e22/2k!.

For V , ( = I,) we proved already a better result in Section 3 (see Theorem 3.1 and Corollary 3.2).

References

A. D. BARBOGR, Poisson convergence and random graphs. Math. Roc . Camb. Phil. Soc. 92 (1982) 349-359 B. BOLLOB~S, Vertices of given degree in a random graph. Journal of Graph Theory 6 (1982)

P. ERDOS, A. RENYI, On the evoluti.on of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5 (1960) 17-61

147-155


[4] 31. KAROBSKI, Balanced Subgraphs of Large Random Graphs. Poznan 1984 [.?I Z. PALXA, On the number of vertices of given degree in a random graph. Journal of Graph

[6] Z. PALKA, A. RUCI~TSRI, Vertex-degrees in a random subgraph of a regular graph. Preprint

[7] K. SCHURGIER, Limit theorem for complete subgraphs of random graphs. Periodica Math.

[S] K. WEBER, Snbcubes of random Boolean functions. Elektron. Informationsverarb. u. Kyber-

[ O ] -, Subcube coverings of random spenning subgraphs of the n-cube. XliLth. Xiichr. 120 (1985)

Theory 8 (1984) 167-170

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net. 19 (1983) 7/8, 365-374

327-345

Ingenieurhochschule f u r Seejuhrt DDR - Warnemi i~de 2530

poisson convergence in the n-cube

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