On the growth of components with non fixed excesses

On the gro wth of comp onen ts with non fixed excesse s Anne-Elisa b et h Baert a , Vlady Rav eloma nana b and Lo y s Thi monier a a L aRIA CNRS EA 208 3, Universit ´ e de Pic ar die Jules-V erne 33, R ue du Moulin-Neu f 80000 Am iens, F r anc e b LIPN CNRS UMR 70 30, Universit ´ e de Paris-Nor d. 99, av. J.B. Clement 93430 Vil letaneuse, F r anc e Abstract Denote b y an l -comp onent a connected graph with l edges more than v ertices. W e pro ve th at the exp ected n umber of creations of ( l + 1)-compon ent, b y means of adding a new edge to an l -comp onen t in a r andomly gro wing graph with n v ertices, tends to 1 as l , n tends to ∞ bu t w ith l = o ( n 1 / 4 ). W e also sho w, un der th e same conditions on l and n , th at the exp ecte d n umb er of v ertices th at eve r b elong to an l -comp onent is ∼ (1 2 l ) 1 / 3 n 2 / 3 . Key wor ds: Random graphs; asymptotic en umeration; W righ t’s co efficien ts. 1 In tr o duction W e consider here simple lab elled gr a phs, i.e., g raphs with lab elled v ertices, without self-lo ops and m ultiple edges. A ra ndom graph is a pair ( G , P ) where G is a family of graphs a nd P is a probability distribution o ver G . T opics on random graphs provide a large and particularly activ e b o dy of researc h. F or excellen t bo o ks on thes e fields, see Bollobas [3] or Janson [8]. In this pap er, w e consider the c ontinuous time ra ndom graph mo del { G ( n, t ) } t whic h consis ts on assigning a random v ariable, T e , to eac h edge e of the complete graph K n . The  n 2  v aria bles T e are indep enden t with a common con tin uous distribution and the edge set of { G ( n, t ) } t is constructed with all edge e suc h that T e ≤ t . Throughout this pap er, a ( k , k + l ) gra ph is o ne ha ving k v ertices and k + l edges. its exc ess is k . A ( k , k + l ) c onne cte d graph is said an l -c omp onent . Preprint su bmitted to Elsevier 21 No ve m b er 2018 These terms are due, as far as w e kno w, resp ectiv ely to Janson in [6] and to Janson et al in [7]. T o obtain the results presen ted here, metho ds of the probabilistic mo del { G ( n, t ) } t , studied in [6], are com bined with asymptotic en umeration metho ds (ha ving the “coun ting fla v our”) dev elopp ed b y W righ t in [10 ] and b y Bender et al. in [1,2]. The problems w e consider are in essence com bina t orial prob- lems. Often, com binatorics and probabilit y theory are closely related and t he approac hes giv en here furnish efficien t uses of metho ds of asymptotic analysis to get extreme c haracteristics of random lab elled graphs. F ollowing Janson in [6 ], w e define b y α ( l ; k ), the expected n umber of times that a new edge is added to an l - component of order k wh ic h b ecomes an ( l + 1)-comp onen t. This transition is denoted l → l + 1. It has b een pro ve d b y Janson et al. in [7] tha t with probability tending to 1, there is exactly one suc h transition, see Theorem 16 and its pro of . Our purp ose in this paper is t o giv e a differen t approach to obtain this result, when l → ∞ with n . More precisely , we prop ose an alternativ e method and direct calculatio ns of the difficult results giv en in [7], connecting these results to those of Bender et al. in [1,2]. T o do this, w e ev aluate α l = P n k =1 α ( l ; k ), the expected n umber of transitions l → l + 1, and sho w that, whenev er l → ∞ with n but l = o ( n 1 / 4 ), α l ∼ 1. Moreo v er, let V l b e the num b er of v ertices that ev er b elong to an l -comp onen t and V l max the order of the largest l -comp onen t that ev er a ppears. W e prov e that, whenev er l ≡ l ( n ) → ∞ with n but l = o ( n 1 / 4 ), V l ∼ (12 l ) 1 / 3 n 2 / 3 and V l max = O ( l 1 / 3 n 2 / 3 ). In particular, these results improv e the w ork of Janson and confirm his predictions (cf. [6, Remark 8]). 2 The Exp ected N um b er of Creations of ( l + 1) -excess Graphs When adding a n edge in a rando mly gr owing graph, there is a p ossibilit y that it joins t w o v ertices of t he same comp onen t, increasing its excess. Let α ( l ; k ) b e the exp ected num b er of times that a new edge is added to an l -comp onen t of order k . D enote b y c ( k , k + l ) the num b er of connected ( k , k + l )-graph then w e hav e the follo wing lemma. Lemma 1 F or al l l ≥ − 1 and k ≥ 1 , we have α ( l ; k ) = ( n ) k ( k + l )! k ! c ( k , k + l ) ( k 2 − 3 k − 2 l ) 2 ( nk − k 2 / 2 − 3 k / 2 − l − 1)! ( nk − k 2 / 2 − k / 2)! (1) and for al l l , k and n such that l = O ( k 2 / 3 ) , 1 ≤ k ≤ n 2 α ( l ; k ) = 1 2 ρ l k (3 l +1) / 2 n l +1 exp − k 3 24 n 2 + lk 2 8 n 2 + lk 2 n ! × 1 + O k n + k 4 n 3 + 1 k ! + O l 3 k 2 + l 1 / 2 k 1 / 2 + ( l + 1) 1 / 16 k 9 / 50 !! (2) wher e ρ l = 1 2 s 3 π  e 12 l  l 2 (1 + O (1 /l )) (3) ✸ Before pro ving lemma 1, let us recall the extension, due to Bender, Canfield and McKa y [2], of W righ t’s formula for c ( k , k + l ), the asymptotic n umber of connected sparsely edged graphs [10]. Theorem 2 (Bender-Canfield-McKa y 1992) Ther e exists se quenc e r i of c o nstants such that for e ach fixe d ǫ > 0 and in te ger m > 1 /ǫ , the numb er c ( k , k + l ) of c onne cte d sp arsely e dge d gr aphs satisfies c ( k , k + l ) = s 3 π w l 2  e 12 l  l/ 2 k k +(3 l − 1) / 2 × exp   m − 2 X i =1 r i l i +1 k i + O   l m k m − 1 + s l k + ( l + 1) 1 / 16 k 9 / 50     (4) uniformly for l = O ( k 1 − ǫ ) . The first few values of the c on stant r i ar e r 1 = − 1 2 , r 2 = 701 2100 , r 3 = − 263 1050 , r 4 = 538 859 2 695 00 0 . ✸ Note that, the factor w l in (4) is giv en, for l > 0, b y w l = π Γ( l ) Γ(3 l / 2) d l s 8 3 27 l 8 e ! l/ 2 (5) where d l = 1 / (2 π ) + O ( 1 /l ) and w 0 = π / √ 6 (see [1,10]). W e also remark here that in lemma 1, w e restrict our a tten tion to v alues of l such that l = O ( k 2 / 3 ) whic h will b e sho wn to b e sufficien t to obtain the result in theorem 4. Pro of of lemma 1. The pro of give n here are ba sed on t he w orks of Ja nson in [5,6]. Ho w ev er, the main difference comes from the f a ct that our para meter, represen ting the excess of the sparse comp onen ts l , is no more fixed as in [6 ]. When a new edge is added to an l -comp onen t of order k , there a re  n k  c ( k , k + l ) manners to c ho ose an l -comp onen t and  k 2  − k − l wa ys to c ho ose the new edge. F urthermore, the probabilit y that suc h p ossible comp onen t is one 3 of { G ( n, t ) } t is t k + l (1 − t ) ( n − k ) k + ( k 2 ) − k − l and with the conditional probabilit y dt (1 − t ) that a g iv en edge is added during the inte rv al ( t, t + d t ) and not earlier, in tegrating ov er all times, w e obtain α ( l ; k ) = n k ! c ( k , k + l ) k 2 − 3 k − 2 l 2 ! Z 1 0 t k + l (1 − t ) ( n − k ) k + ( k 2 ) − k − l − 1 dt (6) whic h ev aluation leads to (1). F o r 1 ≤ k ≤ n a nd l = O ( k 2 / 3 ), the v alue of the in tegral in (6) is ( nk − k 2 / 2 − 3 k / 2 − l − 1)! ( nk − k 2 / 2 − k / 2)! = k − k − l − 1 ( n − k / 2 ) − k − l − 1 1 + O ( k n ) ! . (7) F urthermore, ( n ) k ( n − k / 2 ) k = exp − k 3 24 n 2 !  1 + O ( k / n + k 4 /n 3 )  (8) and ob viously k 2 ! − k − l = k 2 2 (1 + O (1 /k )) . (9) Th us, comb ining (7),(8) and (9) in (1), w e infer that α ( l ; k ) = 1 2 ( k + l )! k ! c ( k , k + l ) exp  − k 3 24 n 2  ( n − k / 2) l +1 k k + l − 1  1 + O (1 / k + k /n + k 4 /n 3 )  . (10) Using T a ylor expansions ln ( k + l )! k l k ! ! = l 2 2 k + O ( l 3 /k 2 ) + O ( l /k ) (11) whic h is sufficien t for our presen t purp ose ( l = O ( k 2 / 3 )). Also, w e get ( n − k / 2 ) l +1 n l +1 = exp − lk 2 n − lk 2 8 n 2 !  1 + O ( k / n + lk 3 /n 3 )  . (12) Note that k 4 n 3 dominates the term lk 3 n 3 in (12). Then, using the asymptotic form ula f o r the num b er c ( k , k + l ) giv en b y theorem 2 in (10 ) , w e obtain (2). ✷ The form giv en b y equation (2), in lemma 1, suggests us to consider the asymptotic b eha viour of n X k =1 k a exp − k 3 24 n 2 + lk 2 8 n 2 + lk 2 n ! where a = 3 l +1 2 , l ≡ l ( n ) a s n → ∞ . 4 Lemma 3 As n → ∞ , l ≡ l ( n ) = o ( n 1 / 4 ) and a = 3 l +1 2 , w e have n X k =1 k a exp − k 3 24 n 2 + lk 2 8 n 2 + lk 2 n ! ∼ 2 a +1 3 ( a − 2) / 3 Γ  a + 1 3  n 2( a +1) / 3 . (13) ✸ Pro of. W e start estimating the summation b y an integral using, for e.g., the classical Euler-Maclaurin metho d for asymptotics estimates o f summations (see [4]), n X k =1 k a exp − k 3 24 n 2 + lk 2 8 n 2 + lk 2 n ! ∼ Z n 0 t a exp − t 3 24 n 2 + lt 2 8 n 2 + lt 2 n ! dt (14) If w e denote by I n the in tegr a l, w e ha v e after substituting t = 2 n 2 / 3 e z : I n ∼ 2 a +1 n 2( a +1) 3 Z + ∞ −∞ exp( h ( z )) dz (15) where h ( z ) = − 1 3 e 3 z + l 2 n 2 / 3 e 2 z + l n 1 / 3 e z + ( a + 1) z . (16) W e ha ve h ′ ( z ) = − e 3 z + l n 2 / 3 e 2 z + l n 1 / 3 e z + ( a + 1 ) (17) and h ′′ ( z ) = − 3 e 3 z + 2 l n 2 / 3 e 2 z + l n 1 / 3 e z . (18) Let z 0 b e the solutio n of h ′ ( z ) = 0. z 0 is lo cated near 1 3 ln( a + 1) b ecause a = (3 l + 1) / 2 is large. Note tha t z 0 can b e obtained solving the cubic equation h ′ ( z ) = 0. Straigh tfor w ard calculations leads to the follo wing estimate z 0 = 1 3 ln  3 2 ( l + 1)  + O l 1 / 3 n 1 / 3 ! = 1 3 ln ( a + 1) + O l 1 / 3 n 1 / 3 ! (19) and h ( z 0 ) = a + 1 3 ln ( a + 1) − a + 1 3 + O ( l 4 / 3 n 1 / 3 ) . (20) W e ha ve also h ′′ ( z 0 ) = −  3( a + 1) + 2 l /n 1 / 3 e z 0 + l /n 2 / 3  and more generally h ( m ) ( z 0 ) = − 3 m − 1 ( a + 1) + A m l n 1 / 3 e z 0 + B m l n 2 / 3 e 2 z 0 . (21) Th us, Z + ∞ −∞ e h ( z ) dz = e h ( z 0 ) Z + ∞ −∞ exp h ′′ ( z 0 ) ( z − z 0 ) 2 2 + P ( z − z 0 ) ! dz (22) 5 where P is a p o wer series of the form P ( x ) = ( a + 1) P i ≥ 3 p i x i and h ′′ ( z 0 ) < 0. A t this stag e, one can consider exp  h ′′ ( z 0 ) ( z − z 0 ) 2 2  as the main factor of the in tegrand. W e refer here to the b o ok of De Bruijn [4 , § 4.4 and § 6.8] for more discussions abo ut a sym ptotic estimates on integrals of the form give n b y (22) and w e infer that Z + ∞ −∞ e h ( z ) dz ∼ s − 2 π h ′′ ( z 0 ) exp ( h ( z 0 )) . (23) Using the Stirling formu la for Gamma function, i.e., Γ( t + 1) ∼ √ 2 π t t t e t and the fact that z 0 is lo cated near 1 3 ln( a + 1) , h ( z 0 ) ∼ ( a +1) 3 (ln( a + 1) − 1) and h ′′ ( z 0 ) ∼ − 3( a + 1), w e can see t ha t (23) leads to (13) whic h is a lso the form ula obtained b y Janson in [6]. ✷ T o estimate α l , due to (2), it is conv enien t to compare the magnitudes of n X k =1 k (3 l +1) / 2 n l +1 exp − k 3 24 n 2 + lk 2 8 n 2 + lk 2 n ! (24) and of n X k =1 k 4 n 3 k (3 l +1) / 2 n l +1 exp − k 3 24 n 2 + lk 2 8 n 2 + lk 2 n ! . (25) Also w e need to compare (24) t o the other “error terms” contained in the “big-ohs” of (2). Using the asymptotic v alue given b y lemma 3, w e eas- ily obtain the estimates of the t wo quan tities and we c ompute resp ectiv ely 2 (3 l +3) / 2 3 ( l − 1) / 2 Γ(( l + 1) / 2) for (24) and 2 (3 l +11) / 2 3 l/ 2+5 / 6 Γ(( l + 1) / 2 + 4 / 3) /n 1 / 3 for (25). Th us, the term “ O ( k 4 /n 3 )” in (2) can b e neglected if l = o ( n 1 / 4 ) otherwise t he quantit y represen ted b y (25) is not smal l compared to that r ep- resen ted by (24 ). Similarly , straig h tforw ar d calculations using (3) sho w that the terms k /n , 1 /k , l 3 /k 2 , l 1 / 2 /k 1 / 2 and ( l + 1) 1 / 16 /k 9 / 50 can also b e neglected. Using Stirling form ula fo r Ga mma function, lemmas 1 and 3, w e hav e α l ∼ ρ l 2 2 3 l +3 2 3 l − 1 2 Γ( l + 1 2 ) . (26) After nice cancellations, it results that: Theorem 4 In a r andomly gr owing gr aph of n vertic es, if l , n → ∞ but l = o ( n 1 / 4 ) , the exp e cte d numb er of tr ansitions l → l + 1 , for al l l -c omp onents, is α l ∼ 1 . ✸ Note that in [7, p 301–306, § 16– 1 8], the authors a lready prov ed, by en t ir ely differen t methods, that the most probable ev olution of a ra ndom graph, when regarding the excess of connected comp onen t, is to pass dir ectly from 1- comp onen t to 2-comp onen t , f rom 2- component to 3-comp onen t, and so on. 6 Similarly , as an immediate consequence o f calculations ab o ve and [6, Theorem 9], w e hav e: Corollary 5 As n → ∞ an d l = o ( n 1 / 4 ) , the exp e cte d numb er of vertic es that ever b elong to an l -c omp onent is E V l ∼ (12 l ) 1 / 3 n 2 / 3 and the exp e cte d o r der of the la r gest l -c omp onent that ever app e ars i s E V l max = O ( l 1 / 3 n 2 / 3 ) . ✸ Note that these results answ er the last remark in [6]. 3 Conclusion W e briefly p o in t out a remark concerning the restrictions on l in constrained graphs problems, i.e., the creation and g ro wth of comp onen ts with prefixed configurations. 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