On the connectedness of random hypergraphs

Series I: COMMENTATIONES MATHEMATICAE XXV (1985) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXV (1985)

Wo jc ie c h Ko r d e c k i (Wroclaw)

On the connectedness of random hypergraphs

1. Introduction

Problems of connectedness of random graphs have been considered at first by Erdos and Renyi in [1] and by Gilbert in [3]. Next, those problems have been investigated by Erdos and Renyi in their fundamental paper [2]

and many others.

Wide, but not complete bibliography until 1974 is given in author’s paper [5], and until 1980, is given in Karonski’s survey article [4], which covers 200 papers.

The aim of this paper is to give a generalization of basic theorem concerning the connectedness of random graph, to random hypergraphs.

At first, we define a random hypergraph and present some well-known combinatorial and algebraic results. Next, in Section 3 we obtain some exact formulas for probability of connectedness and other parameters in finite case.

Sections 4 and 5 contain limit theorems in general case and in the case of the so-called power measure. In Section 6 we present several examples.

This paper is a revised version of my Ph. D. thesis at Technical University in Wroclaw. 1 wish to thank my thesis advisor, Professor S. Gladysz, for his advice, many valuable conversations and constant encouragement. I also wish to thank Professor L. Jesmanowicz for his very critical remarks.

2. Preliminaries

Combinatorial algebra. The construction and properties of combinatorial algebra were given by Stepanov [10].

Let Q denote a denumerable set; A, B , ... c f l . The sequence of sets П

(Alt . .. , An), with A{ n A j ^ 0 for i Ф j, and ( J At — A <= Q we call the n-

i = 1

partition of set A. The partition (A lt . .. , An), where all A{ are non-empty, is called the exact n-partition of set A. All exact n-partitions which differ only in the order we can include in one class. Each of these classes we call the unordered n-partition of set A. Families of all n-partitions, exact n-partitions and unordered и-partitions we denote by (A)„, [Л]„ and {A}„, respectively.

Let us notice that if n > \A\, then [Л]„ = \A)n = 0 . Now, we shall consider the family 3F of all real functions F, G, ... defined on the family of all finite subsets included in Ü. In J 27, we can introduce three operations:

multiplication by a real number, addition and convolution {F * G)(A) = X F (A l )G (A 2).

(A) 2

The family with these operations is a commutative algebra with zero G (A) = 0 and unity,

HA) = ¹ 0

for A = 0 , for А Ф 0 .

The family c; of all functions F e & which satisfy F (0 ) = 0 is an ideal in 3F. The converse element (with respect to convolution) exists iff F $ & о-

The projection F -> F = F —F ( 0 ) 1 is the projection of algebra J*7 to its ideal & 0 .

If F j , . . . , F ne then

( F i * . . . * F n)( A )= X ^ ( ^ . . . F ^ J .

W n

If F j = ... = F H = F, the left-hand side of the above formula denotes the nth power of the element F, n > 0. Let us denote this power by F *". By definition we take F * ° = I.

If F e . f 0, then

F * n(A) = nl X F (A 1) . . . F ( A n).

If \A\ < n, then F * n(A) = 0 for F e J^ 0 -

Let К [x ] denote the algebra of formal power series of variable x with real coefficients. Elements of this algebra we shall denote by /, g, ft, ... ; for example,

/(x) = X a* x<£- k= 0

Let us determine some element F e and let us state the projection of algebra К [x ] to algebra 3? by the formula

00 M l

f : / - / * ( f ) = I ^{ak F*k = I akF * k.}

k = О к = 0

/ (x) = exp x = k= 0z ^xk/k\.

Let

Then for F e ,F 0 and А Ф 0 we have

ехр *Т (Л ) = J I F { A , ) . . . F { A k).

fc=i и}к

Pr o p o s i t i o n 1. I f F et F0, then the element G — ^{e x p} * F is reversible in !F and F = In* ( 1 4- G), where

oo ( _ l f - a

and

In (1 T x ) = X

F ( A ) = X ( - l f - ' ( k - l ) £ G {A x) . . . G { A k), □

k= 1 И Ik

The function L e , F is called additive if for every pair of finite and disjoint sets A, В we have L(A и B) = L(A) + L(B). The additive function is determined by its values on one-element s’ets: L(A) = £ Aa, where

aeA

Aa = L ({a}).

A non-negative additive function is a measure on Q. The measure is elemeniary if takes value 1 on the exactly one one-element set \a], and 0 on the other ones. We denote this measure by La.

ô 3F

Let L be an additive function. The correspondence — : F -► -^r of the

dL dL

algebra ,F into itself we define by the formula dF

dL(A) = L (A )F (A ).

This projection we shall call differentiation with respect to the measure L.

The operator д/ô L has some properties of usual differentiation. In particular, d(F * G) „ ÔG ^ dF

—--- - = F * --- 1- G * — .

dL dL dL

The proofs which were omitted and other properties are in Stepanov’s paper [10].

Let us denote by Qs the family of all two-elements at least and s- elements at most subsets of Q. Analogously, As denotes the family of all such subsets of set A, where \A\ ^ 2. If \A\ < 2, then As = 0 . Let (A x, . .. , A„)s

= (Û -4<)S- Û л?.

i = 1 i = 1

Now, we shall define a measure of special type on 0 s.

Let L b e a measure on Q, L {{a }) — Aa, a e Q , L(A ) = ]T Aa, A c: Q. The

aeA

measure Ls is defined by the following formula:

Ц (В ) = I y „ ,

veB

where B c Q3, and yv is defined as

У» I

P 1 + - + r m = S

Г,-# о

. . я ?

! ’

where t? = {al5 aM}, |y| = m. This measure is called the power measure generated by L. Since

we have

( L ^ = I _{i = 1 r i + ... + r „} = s ' 1 • •••'«•77—77

W ) = Z I

X1 . . . Xn ,

/1 /и

Л«1 • • • Ч

= 2 | a j , . . . V Cj4 ri + ... + rm = * Гх ! . . . Гт ! Œ ^ ) s

__ aeA aeA

s! 5.1

From this we have the following Proposition 2.

m

5! 5!

Ц.(А’) = [1 5 (A )- £ A»]/s!

aeA

and fo r disjoint sets A, В:

Z ,[ (A u B )s] = [ f (S) L l (A )n ~ k(B )~ £ A;]/s!,

k = 0 '* / a eA u B

(I) L ,(A , В) = У Q l‘ (A )Z T *(B ). □

Hypergraphs. Definitions and terminology from the domain of hypergraph theory correspond to those given by Zykov in his survey article [ I I ] . However, we shall restrict them to hypergraphs without multiple and isolated edges.

A hypergraph Я is a pair Я = (X , R ), where X is a finite set of vertices and R is some family of non-empty subsets of set X. The elements E e R are called edges. We shall assume that for every E a R : \E\ ^ 2. Isolated vertex is a vertex which is not contained in any edge.

A subhypergraph H' of hypergraph Я = (X , R) is any hypergraph H'

= (X', R x ) such that X' с X and

R x. = 2 x' n R .

We shall say “subgraph of hypergraph” instead „subhypergraph of hyper graph” and we shall write H' — (X', R) instead of H' = (X\ R x.).

A hypergraph is connected if for any partitions of X into non-empty sets

X x, X 2 there exists an edge E e R such that X x c\E Ф 0 and X 2 n E Ф 0 . Every maximal connected subgraph of hypergraph is called a component.

We assume that hypergraph with one vertex is connected. It is clear that any hypergraph has a unique representation as the sum of its components.

By a random hypergraph we mean a random variable taking values in the family of all hypergraphs (X , R ), where X is a fixed finite set of vertices and the events of the form { E e R } are independent. We shall use the following terminology and notation:

“the edge E = {X j , . . . , arises with probability p = pj j ” or “the edge belongs to hypegraph with probability p” if E e R ,

“the edge E vanish with probability c f if E ф R, q = 1 — p.

- ii

We often write q = e * for probability of disappearance of edge (x,- , . .. , x ,J and t > 0 is a real number. In this way we have for every edge the real number Я,- f > 0. This number is called intensivity.

We shall assume that a finite set of vertices A is a subset of the denumerable set Q. Next, we define a measure L o n 0 s in such a way that L ( { a x, . . . , ak}) is the intensivity of the edge [ a x, . . . , ak] e A . If Ls is a power measure generated by L, then Т ({а4}) = Я, is the intensivity of the vertex at.

These random hypergraphs are denoted by J f L{A\t) in these two cases. In the case s = 2 (random graphs) we shall write g L(A\t).

It is clear that the following propositions are true:

Proposition 3. I f A, B a Q are disjoint, then random hypergraphs Ж Ь(А\1) and Ж Ь{ В|t) are independent. □

Proposition 4. I f A{c Q are pair-disjoint, then probability o f disappearance o f edges join ed vertices o f sets At is equal to e \ q

3, Finite distributions

Let P L(A\t) denote the probability of connectedness of random hypergraph J^ L{A\t) and PL{A 1, . .. , An\t) denote the probability that J4?L(A\t) is decomposed into components J f L(Ai\t). Since every hypergraph has the unique decomposition to components, we have

00

(2) I I P iWi... Л И ) » 1-

л = 1 {A}n

The hypergraphs Ж Ь{А{ \1) are components of Ж Ь{А | f) iff:

I. There are no edges joining different Ah

II. Every Ж Ь(А( I f) is connected. The probability of the event is

П П ^ т(Д 10- Therefore

«=1

PL{A 1, . . . , A n\t) = e~ t^ П P L{At \ t ) ^ . i = i

6 — Prace Matematyczne 25.2

From this and from (2) we have

00 n

Y I ПРьШ1)е'иА‘' = е'«А\

л= 1 {A}„ i= 1

If we introduce

F L(A\t) = PL(A\t)e'UA‘\

then we obtain the equation

(3) exp* F L( • |f) = etL.

From Proposition 1 we obtain a solution of this equation:

(4) P L(A\t) = e ~ ' ^ Y ( - Г ‘ ( » - 1 ) ! I exp [t Y L(A$\.

k = l {A)„ i = l

Now, let L denote a measure on Q. Then for the power measure on Qs generated by L, we have from (3)

(5) exp* F L ( • I f) = etLS,s',

where

F L(A\t) = P L( A \ t ) ^ l ‘ and from (4)

00 П

(6) PL(A\t) = e - L™ ‘ Y ( —1)"— 1 (о — 1)! Y exp[t I )/*•']•

л= 1 {А}„ i= 1

Next, we shall differentiate equation (3) with respect to the measure L a on Ü, where L a is an elementary measure concentrated in a. Hence

(?) ^{Y pLu}11|о^,ци‘И2Л = 1.

aeA i c A A 2 = A\AX

Analogously, differentiating (5) (for the power measure) with respect to L', we have from (1) for L' = L:

(⁸)

(a^a2)

L(Al> P L(A t \t)e ₌ 1,

and for L' = L„

(9) x

aeA j A 2 = A\Ai

-t s I 1

P L{A x\t)e i=lW = 1.

Formulas (7), (8) and (9) are linear recurrent relations for PL{-\t), because PL({a}\t) = 1 for each a e Q .

In the next part of this section we shall consider only random hypergraphs j f L(A\t) with power measures.

In the same way as Stepanov [10] we obtain following formulas:

(10) (L(A)-k.)PL(A\t)

= £ Ц А 2) ( е 1°~ 'Ш 2 т ~"' - l ) e

( A ) 2

s — 1

*,5i РьШ 1 ) Рь(А2\1),

(11) PL(A\t)

А^эа А2э1>

x P b i A ^ P ^ A ^ t ) , where Xa — L ({a}).

Formulas (10) and (11) are non-linear recurrent relations for P L(A\t).

It is easy to see that the function etLS,s' satisfies the equation

(12) detLS,sl 1 8s e LS/sl

dt s! ÔU

There is a well-known formula (see Riordan [9]) for nth derivative of ex{t):

xlr) V- nl f X(t)\kl dnx(t)/dtn Y ", n! J

where the summation ranges over all integrable and non-negative solutions of the equation k x -I- l k 2 + ... +nk„ = n.

After substituting this to (12) we have after simple computation (13) e^suvs\

dt

1 s_ ^J Ц.

- - - Г , , ^ , П П L‘ (Ail) P L(A„\t)e'LHAil'IS' 11.... ^ n M C i ! ) ' * 1 j l .... V M M

S — 1

or

(14) ^tLs(A)/s! (A I t) _ y, 1

e dt^ - kl>-'ks ]-J L 7=1---xf i i= 1

1 ki

x I 1 П П li(Ail)P L(Aij\t)eL,iAiim-

(A l t ...,A s - x) ( A ^ , . . . ^ ) i= 1 j = 1

Formulas (13) and (14) are differentiable and recurrent formulas for ы т .

If in (13) we shall take s = 2, then we shall obtain the simple formula dPL^ U) = i £ L(A1) L( A2IPl(A1 \t)PLiA2 \ t)e ‘UA]}UA2>.

a l ( A V A 2 )

The above formulas were given in the same way by Stepanov [10] in the case s = 2 and for power measure.

Next, we shall give formulas for the expectation and variance of number of components in random hypergraph J f L(A\t), where L will be some measure on 0 s.

Let YL(A\t) denote the number of components in a random hypergraph J f L(A\t). Let X a = l/Mjl if and only if the vertex a belongs to component on A 1. Then

Clearly

YL(Alt)= £ X „

aeA

E X a Z ^{1 P L(A\t)e}-tU(Al ,A2)s) where L ((A l9 A2)s) = L {A S) ~ L(A \)~ L(A S2), and

(15) EYL (A\t)= X ?l(A i\t)e~tU(Ai’A2)S).

Ai <=A

Next, we shall calculate D2 YL(A\t). At first we shall find E Y 2(A\t) = E ( £ X„)2 = E £ X I + 2E £ X .X „ .

aeA aeA {a,b}c.A

We shall calculate both summands

and

Next

E X 2 = Z (777) Р Л Л \ 1)е-т Л 1'А2П aeAi cA \l^ll

I E X l = Y. { Г7-7 ) I t ) e ' m A l-Ain

aeA A1 c

z ¹

A 1'A 2 <=a Ш \а2\

A 1e\A 2 = 0 a eA i> b eA 2

P L(A 1\t)P[.(,Aî \t)e~‘m Al’A2’Ab,S> +

A1

-tmA^Atf*)

and

2 £ E X .X > = I P LiA 1\t)PL(,A2\ t)e-m / , " A2'A3)S' +

{a,b }^ A A },A 2 C A

A i n A 2 ~ @

+ £ PL(A1_lt)Pl(A1lt)e~,u,J,,-A3».

A^ <=A

Hence we obtain

( 1 6 ) D2 YL(A\t) = EYL(A\t) — (EYL(A\t))2 +

+ I PL(A,\t)PL(A 2\t)e~'LnAl’A2’A3>!',

A A 2 *—А Ai n A 2 = 0

4. Limiting distribution - power measure

The contents of this section generalize the theorem of Stepanov [10] to random hypergraphs.

Let us denote for A, B a 12, t > 0:

p(A , B\t) = ^ ex p [ —tA0Ls_ 1 (^4)/(s —1)!], p(A \ t) = p {A , A\t)

aeB

and for A n B = 0 :

L (A , B) = ( 8) й ( А ) П - 1(В)/8\,

i = 1 W

where L(A) denotes a measure on Q (see Section 2).

Th e o r e m 5 . I f fo r the sequence A ⁼ A(n) and t ⁼ tn^{> 0}

(17) p{A\t) = 0 (1 )

and if there exists a function L 0 = L ffiA ,,, tn) such that

(18) tLs0 = o ( 1),

(19) t ü - ' W L o ( )з

(20) max p{A , A'\t) = o (l),

A' < L (A ’) < L 0

and, moreover, fo r s > 2

(21) max tXa L 0 U ~ 2(A) = o (l),

aeA

and n(A\tfs) = o ( l ) or l im ii(A \t/s) = c > 0 , then

P L{A\t)e^iAU) = l + o ( l ) ,

where PL(A\t) denotes the probability o f connectedness o f J f L(A\t).

R e m a rk . In the following we shall write A, t, L 0 instead of A(n\ tn, L(ô}.

P r o o f o f T h e o r e m 5. In this proof we shall assume that

(22) tLs~ l (A )L 0 ^ oo.

If

lim ji(A\t/s) ^ c > 0, (22) follows from (19). The case

fi(A\t/s) = o (l) satisfies assumptions of Theorem 6.

From (20), it follows that

(23) rLs_1 (A)m in2a -» oo.

aeA

Next, it follows that

tU (A) -* oo, and therefore from (18) we have

(24) L 0 = o(L(A )).

In this proof we shall use formula (8). The sum in that formula shall be parted into three parts: 1 Ь I 2, £ 3, where the summation in Г 3 ranges over Ai such that L (A i) ^ L 0, in Z 2 such that L 0 < L (A i) ^ L(A) — L 0 and in I x such that L {A i) > L(A) — L 0. We shall prove that I 3 — o(l), I 2 = 'o(l)- Then Zi = l + o ( l ) .

At first, we shall estimate Г 3. Since P L(A1|t) ^ 1 and L (A X) ^ L 0, we have

£ 3 ^ 7 7 7 7 X exP ^{C “} tL(Ai, Ai)l^• Ш ) A^A

Next

L (A X, A 2) = L (A i) [ I T 1 (A)/(s— 1)! + a + ( — 1 Г 1 Ls~1 (+ x)], where for s > 2 we shall denote

« =

(У

By assumption (21) we obtain the estimation

£ exp [ —rL(Ai) [Ls_1(A)/(s —l ) ! + a + o (l) ]].

Next

L exp [ —(Ц Л ,) [Ls - 1 (/4)/(s —l ) !+ a + o (l)}]

= n ( l + e x p [ - i ^ s_10 4 )/ (s -l)!+ « + <?(!)]) = е ^ )=о(1) = 0 (1 ).

aeA

Using the fact L 0/L (A ) = o (l) we obtain = o (l). Now we estimate I 2

^ 2 ^ X exp l - t L { A u Л 2У].

A± <=A Lq^ L (A 1) < U A ) - L 0

Since L (A 1) + L (A 2) = L(A), then L {A X) ^ L{A)/2 or Ц А 2) ^ Ц А )/2.

From this we obtain

* 2 ^ 2 £ [ - t’l ( ^ ' ( Л О С Ь ^ - Ц Л , ) ] 5-'/*!]-

A l <=A i= i \ I/

L 0 ^ L ( A l )< L (A )l 2

Let us consider the function

S — 1

/ ( * ) = I = U - ^ - ^ L - x f .

It is easy to see that this function has the maximum at the point

* = L/2. Hence it is f (x) ^ gx + h in [L 0, L/2], where

9 =

Lso -2 (L / 2 )s + ( L - L 0)s L/2 — L 0

L/Z — L 0

Substituting x = L(yl1), L = L(A ) we obtain ехР [ - ' Е * Q ^ I ^ H I - L t / U r V s ! ]

^ exp( — fh/s!)exp( — tg L (A x)!s\) Let us estimate the exponent in the first factor:

h > £ ( / l ) - ( £ 0 (L(/l)-Z.„)s)(l + o (l)) = §.

£ e\ p( — tg Ц А ^ /s'.) ^ exp £ exp(-<gA„/v!)

A i <=A aeA

Next

and

exp( — f/?/s!)exp £ exp ( - t g À J s l )

aeA

= exp( — tf/sl) I - 7 7 £ exp { - t g À J s l ) . Ф aeA

From formula (24) we have

tgkjs\ » ( r i r 1 И)Я„/5 !)(1 + o (l)).

From (19) we have

I 7 7 Z e x p ( -r ^ 2e/s!) = o (l)

Ф aeA

and from conditions (23) and (24), e~ tfilsl = o (l), and I 2 — o (l). From this we obtain

(25) £ exp [ —r L M ,, /!,)] = o (l).

A i <=A Lq < L ( Aj) ^ L ( A ) - Lq

We have the following definition. We shall say that a partition {Ax, A 2) of a set A is a partition of the hypergraph Ж ь {А\г) if there do not exist any edges joining A x and A 2.

We have the following

Lemma 6. Under the assumptions o f Theorem 5, if the random hyper graph J f L(A{t) is disconnected, then with probability tending to 1 there exists the component j f L (A'\t) such that L(A') < L 0.

P r o o f. The probability that (A l , A2) is a partition of J^ L(A\t) is equal to exp [ —L (T 1? T 2)].

The probability that J^ L{A\t) possesses the partition (Аг , A 2) such that L 0 < L (T j) ^ L(A ) — L 0 (in this case also L 0 < L (A 2) ^ L(A) —L 0) is not greater than £ ex p [ — tL {A 1, Л 2)] and from (25) tends to

Aj cA

Lq < L ( A j ) < Ц А ) — Lq

zero. □

Next parts of the proof are exactly the same as the Stepanov’s ones (see [ 10]) and we omit them.

Theorem 7. T he thesis o f Theorem 5 remains true, i f condition (21) will be replaced by

(26) tL 0 U ~ 2(A) — o (l).

The proof is similar to that of Theorem 5. □ Theorem 8. I f /

(27) p(A\t/s) = o(\),

then

P l {A = l + o (l).

P ro o f. If condition (27) holds, then p(A\t) = o ( l) and conditions (17) and (20) for any function L 0 hold. If we take

L 0 = [(tLs_1(+))1_s2 min Aa] 1/s2,

a s A

then

(min Xa \ (s2 _ sys j

--- = o (l),

М Л ) ) 1 ^ m i n ^ ( s 2 - s - D / S

аеЛ

because from (27) we obtain that tH ~ l (A)minXa - + o o . Moreover, tL 0 I T 1 {A) = (fLs“ 1 (Л) min Aa)1/s2,

aeA

then conditions (18) and (22) hold. But conditions (21) and (26) may be not true. The proof is similar to that of Theorem 5. However, we shall use the following estimation of Г 3:

I ° exp [ tL {A u A 2y\

<=A L(A)

U A ^ ^ Lq

I T/ A, exP [ tL {A x)U 1 (A)/s!]

A j <=A L (A)

< 7 7 7 7 exP Z e x p [ - d flLs 1 (v4)/s!] = o (l). □ Ц Л )

5. Limiting distribution — general case

The contents of this section generalizes a theorem of Kovalenko [7] to hypergraphs.

Let us consider the random hypergraphs Ж (n) with set of vertices N

= N n = {1, 2, n} and with the probability of arising of an edge {h> - ik}, к < s, equal to pfl...ik = pÇ..v

Let us denote

4 M ) = U П П ««i-ir*

r = l k = 1 { i , i <=Л {‘ k + l ’ - > ' r )<=N\A

Qiv = min q M ) , l ^ v ^ n ,

\A\ = v + l

for any A <= N covering some fixed vertex ie A . Obviously, a , = <?i(n\!/}) = f l I

r - l i n , —,ir )cN \ {i}

is the probability that i is an isolated vertex.

The set of isolated vertices is denoted by К and a = Q i n = P { i e K ] , A = £ a , » = w ,

i = 1

(1, where i e K , j 0, where i $ K . Theorem 9. I f

(28) max Qi = o (l) ,

(29) A = 0 ( 1),

I h

l^k^n/2 K-

/ « о \k (30)

t e l ) - 1- " - ' " " 1

fo r every v, then with probability tending to 1 fo r n-> оо , the random hypergraph Ж {п) has p + 1 com ponents;

P (p = k) — e~ x~Xk----> 0 k\

fo r every к and n -> oo.

Co r o l l a r y 10. T he hypergraph Ж (n) has under conditions (28)-(30) one

“giant" component and some isolated vertices with probability tending to 1. □ P r o o f o f T h e o r e m 9. Erdos and Renyi obtained in [2] the following lemma.

Lem m a 11. L et e 1, . . . , e n be random variables on the same probability space, taking only values 0 and 1. I f A > 0 and

then

E I

1 i < ... <Σ ^ n * •

P( f , e,=j)

i = 1 O(l),

fo r j = 0, 1, ... □

From assumptions (28)-(30) we obtain the following asymptotic estimations:

(31)

from (30) we obtain

(32)

I a, - 6^-77 = o(i),

1 <i'i <... <ik^n K‘

Qi^k ôijli ^{Q h} ••• Q ik = o(l)/k\ ,

where о ( 1) is tending to 0 uniformly with respect to к ^ n il. Hence

(33)

l l

Q n Q ik x k

Q i xk Q ikk k! ⁰(1).

These properties are used in proofs of the following two lemmas:

Lem m a 12. Under conditions (28)-(30):

Р (ц = k) — e kXk k\ ^{o ( l ) .}

P ro o f. From

we have

(34)

v = 1

f t , - G*

f t , ft

f t , * ' f tikk

Comparing this formula with formulas (31)—(33) and Lemma 11 we have the thesis. П

Proof of this lemma is similar to that in Kovalenko’s paper [7].

We shall say that a set A c N is closed if there are no edges joining vertices from A with vertices from the set N\A. Moreover, we shall say that a set B c= N is simple if В is a set of isolated vertices or N\B is a set of isolated vertices.

Lem m a 13. Probability that there exists in Ж {п) a closed and non-simple set is tending to 0 fo r n -> oo.

P r o o f.

(34) P ( { iu . . . , ik] is closed)

S — 1 r

-П n П

r = 1 V = 1 Ü 1 ... = ... 'fc }

U v + l ’ - J r + ^ ^ H W l ’ - ’ik)

s - 1 r

E

A\ П П П

« 'e { » l...* r ) r = l k = 1 { j v . . . , j v - 1 }* = {* ' 1 >

U '» . —

O'l»—»Jr}cW\{0

> Q h - Q , t - On the other hand

(35) ...» ik} is closed)

< П

s — 1

П ^{0 Q ij\ —j r}

«kî r = 1 O 'l- . .,jr }^ N \ {i1 ,...,ik )

s — 1

п ^П П Q i j l - J r

^ i e f i l , ->*'k)r= 1 tfi.—Jr}< = N \ $i... *k>

П П П

Г= 1 »=1 Ü1... »fc}

U'r+l — -Л }с№\{»1 — -»'к}

Q h Q ik

^ — - ... ^—

Q i t k O ikk

It is obvious that

. . . , ik} is simple) ^ P ( { i u . .. , ifc} <= K) and

P ( { ii, . . . , I*} is closed and non-simple)

= P ({i’i, . . . , ik} is closed) — P ({il5 . . . , ik} is simple).

By this and (34)-{35) we obtain

P ({ iu . . . , ik] is closed and non-simple) < ~ ~ . . . 7^ - - 6 ,, • •• Q,-.

Q i 1k V Îijk

beginning with some n.

If {il5 . . . , ik} is simple (closed), then {ik+1, . . . , i„} is also simple (closed).

Hence we may consider only the case к ^ n/2. Then

P(exist /с-element closed and non-simple set)

P ( { ii, . . . , ik} is closed and non-simple) = ^{o ( l )} k\ ’ 1 <ij <... <ik^n

where o (l) is uniformly tending to 0, with respect to к ^ nf2. □ Proof of Theorem 9 is a simple consequence of these lemmas. □ 6. Examples

Let us denote by Jf„ (t) the random hypergraph defined by the power measure which was generated by the measure L(A) = \A\ = n and by P„(t) — the probability of connectedness.

From (6) we obtain for

Ml к

Р М = е - ^ т £ ( —l)1- 1** —1)! £ exp [t £ К Г / *!].

k = 1 M)k ⁱ = 1

from where

I ki~l «

( - 1 Г * ( £ * , - 1 ) ! (36) P„(t) = n! £ - - - ;^{--- —}- - - -

‘‘..П b w f

exp [t £ 1'4,/s!].

i = 1

Formula (9) implies

n - 1 (37)

i= 1

П — 1 P.U ) = 1 - £ ("

k = 1

A non-linear recurrence formula for random hypergraph Ж п{г) is obtained from (11). Let a e A l , |>41| = к, b e A 2, \Л2\ = n — к. The number of

/n — 2 different partitions of the set A such that a e A 1 and b e A 2 is equal to

Jk — 1 Thus

(38) p „ ( l ) = £ (" ^(e«"-*»',s- 1», - l ) < ! - ,("s- ‘s- (" - ‘ rt'l!P|t( t) P .-t(t) . The recurrence-differential equation for Pn(t) we obtain from (13).

dP n{t) - 1

(39)

dt £ n —

*>.... *«-> П *(!(»!)*'

«= 1

n\

s— 1 к

s - 1 ^*j

(”1....".-P <"ч....V f [ f ] Щ ! П П "IjP.,М)е%

i=l j = 1 1

,/s!

i = 1 J= 1 X

where successive summations ranges: the first over all solutions of the equation k 1 + 2 k 2 + ... + { s —l ) k s- 1 = s, the second over all permutations of solutions of the equation n1 + ... + n s_i = n, the third over all permutations of solutions of the equation n( l + ... +Щк, = Щ-

Fromulas (36) and (37) have been obtained by Gilbert [3]. Formulas (38) and (39) for s = 2 have been obtained by Stepanov [10].

Formulas (15) and (16) are

(40) EY„(t) = Д

(41) D2 Yn(t) = EY„(t) — (EY„(t))2 +

n — 1 n — i + E I 7

nl

i = l j= i i l j' .{ n - i- j) \:— ;— P , (t) P j (t) e - ^" ■iS - jS - (n - 1- j)S]/s' for Jf„(t).

For s — 2 formulas (40) and (41) were obtained by Naus and Rabinowitz

[8].

We shall write

Am = (1, . . . , m ), L (A m) = m, Xt = l . Then

Hm(t) = rne~,mS \

We shall suppose that nm(t) > c > 0. Conditions (17)—(21) hold for (s — 1)! lnm-f 0 (1 )

and from (18) we have

L 0 = m(s_y/71n“m, 1/s < a < 1, and for

we have

(s — 1)! 1пт + х + о(1)

! = H r *

p „ ( 0 = ( i + o (i))<; - e “ ' .

For s = 2 this result is well known (see Erdos and Renyi [1]).

Let now А,- = i~a, a ^ 0. In this case we shall denote the random hypergraph by Jfa,„(0> an<3 the probability of its connectedness by Pat„(t). If a = 0 then ^ а>„(0 = Thus we shall assume that a > 0. It is known

that

m

i ■-

' m1 *

--- (-0(1), 0 < a < 1, 1 —a

In m + C + 0 (m _1), a = 1, . La + 0 ( m - ( « - % a > 1, where C = 0 .5 7 7 ... is Euler’s constant.

It is easy to see that if

L 0 = m1/(s_ ^/ln m,

then (17)-(20) hold. Condition (26) is true only for a > 2(s~ 1)/s and (21) is true only for a < 1/s.

Hence, if

(42) tE~nl(s — 1)! = ma(lnm — lnlnm + x — lna + o(l)), then

(43) Pa,m ( t ) = ( l+ o ( \ ) ) e e for a < 1/s or a > 2s~ 1/s.

For s = 2 formulas (42) and (43) are valid for every a < 0.

Finally, let us notice that for a > 1 it is t -> 0 and for a < 1/s it is t -» oo.

Similar results (however, without non-linear and differential-recurrence formulas) may be obtained for the measure L o n 0 s with L(A S) = 'Hi

. 5 , The suitable result for n -> oo in this case may be obtained from Theorem 9.

These results and limit behaviours for n ->■ oo and t = const can be obtained directly (see Kordecki [6]).

References

[1] P. E rd ôs, A. R en yi, On random graphs I, Publ. Math. (Debrecen) 6 (1959), 290-297.

[2] —, —, On the evolution o f random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960), 17-61.

[3] E. N. G ilb e r t, Random graphs, Ann. Math. Statis. 30 (1959), 1141-1144.

[4] M. K a ro n ski, A reviev o f random graphs, J. Graph Theory 6 (1982), 349-389.

[5] W. K o rd e c k i, Elements o f the theory o f random graphs, Sci. Papers of the Inst. Math.

Wroclaw Techn. Univ. 15 (1976) (in Polish).

[6] —, On the connectedness o f random s-graphs, Disc. Math. 4 (1980), 51-58.

[7] I. N. K o v a le n k o , On the theory o f random graphs, Kibernetika 4 (1971), 1-4 (in Russian).

[8] J. I. N aus, L. R a b in o w itz , The expectation and variance o f number o f components in random linear graphs, Ann. Prob. 3 (1975), 159-161.

[9] J. R io rd a n , Combinatorial Identities, New York 1968.

[10] V. E. S te p a n o v , Combinatorial algebra and random graphs, Theory Prob. Appl. 14 (1969), 373-399 (393-420; in Russian).

[11] A. A. Z ykov, Hypergraphs, Usp. Mat. Nauk 39 (1974), 89-154 (in Russian).