DEGREE DISTRIBUTION IN THE LOWER LEVELS OF THE UNIFORM RECURSIVE TREE

Transcription

1 Aales Uiv. Sci. Budapest., Sect. Comp DEGREE DISTRIBUTION IN THE LOWER LEVELS OF THE UNIFORM RECURSIVE TREE Áges Backhausz ad Tamás F. Móri Budapest, Hugary Commuicated by Imre Kátai Received November 23, 2011; accepted December 15, 2011 Abstract. I this ote we cosider the kth level of the uiform radom recursive tree after steps, ad prove that the proportio of odes with degree greater tha t log coverges to 1 t k almost surely, as, for every t 0, 1. I additio, we show that the umber of degree d odes i the first level is asymptotically Poisso distributed with mea 1; moreover, they are asymptotically idepedet for d 1, 2, Itroductio Let us cosider the followig radom graph model. We start from a sigle ode labelled with 0. At the th step we choose a vertex at radom, with equal probability, ad idepedetly of the past. The a ew ode, vertex, is added to the graph, ad it is coected to the chose vertex. I this way a radom tree, the so called uiform recursive tree, is built. Key words ad phrases: Radom recursive tree, permutatio, degree distributio, Poisso distributio, method of momets Mathematics Subject Classificatio: Primary 05C80, Secodary 60C05, 60F CR Categories ad Descriptors: G.2.2. The Europea Uio ad the Europea Social Fud have provided fiacial support to the project uder the grat agreemet o. TÁMOP /B-09/KMR

2 54 Á. Backhausz ad T.F. Móri This model has a log ad rich history. Apparetly, the first publicatio where the uiform recursive tree appeared was [11]. Sice the a huge umber of papers have explored the properties of this simple combiatorial structure. Recursive trees serve as probabilistic models for system geeratio, spread of cotamiatio of orgaisms, pyramid scheme, stemma costructio of philology, Iteret iterface map, stochastic growth of etworks, ad may other areas of applicatio, see [6] for refereces. For a survey of probabilistic properties of uiform recursive trees see [5] or [9]. Amog others, it is kow that this radom tree has a asymptotic degree distributio, amely, the proportio of odes with degree d coverges, as, to 2 d almost surely. Aother importat quatity is the maximal degree, which is kow to be asymptotically equal to log 2 [4]. Cosiderig our graph a rooted tree, we ca defie the levels of the tree i the usual way: level s the set L k of the vertices that are of distace k from vertex 0, the root. It is ot hard to fid the a.s. asymptotics of the size of level k after step ; it is L k E L k log k, k 1, 2,.... k! Recursive trees o odes 0, 1,..., 1 ca be trasformed ito permutatios σ σ 1, σ 2,..., σ i the followig recursive way. Start from the idetity permutatio σ 1, 2,...,. The, takig the odes 1, 2,..., 1 oe after aother, update the permutatio by swappig σ i+1 ad σ i+1 j if ode i was coected to ode j < i at the time it was added to the tree. It is easy to see that i this way a oe-to-oe correspodece is set betwee trees ad permutatios, ad the uiform recursive tree is trasformed ito a uiform radom permutatio. Aother popular recursive tree model is the so called plae orieted recursive tree. It was origially proposed by Szymański [10], but it got i the focus of research after the semial paper of Barabási ad Albert [3]. A o-orieted versio of it starts from a sigle edge, ad at each step a ew vertex is added to the graph. The ew vertex is the coected to oe of the old odes at radom; the other edpoit of the ew edge is chose from the existig vertices with probability proportioal to the istateeous degree of the ode preferetial attachmet. This ca also be doe i such a way that we select a edge at radom with equal probability, the choose oe of its edpoits. I this tree 4 the proportio of degree d odes coverges to dd+1d+2 with probability 1. Katoa has show [7] that the same degree distributio ca be observed if oe is cofied to ay of the largest levels. O the other had, if we oly cosider a fixed level, the asymptotic degree distributio still exists, but it becomes differet [8]. This pheomeo has bee observed i other radom graphs, too. A geeral result of that kid has bee published recetly [2].

3 Degree distributio i the lower levels of the uiform recursive tree 55 I the preset ote we will ivestigate the lower levels of the uiform recursive tree. We will show that, ulike i may scale free recursive tree models, o asymptotic degree distributio emerges. Istead, for almost all odes i the lower levels the degree sequece grows to ifiity at the same rate as the overall maximum of degrees does. We also ivestigate the umber of degree d vertices i the first level for d 1, 2,..., ad show that they are asymptotically i.i.d. Poisso with mea Nodes of high degree i the lower levels Let deg i deote the degree of ode i after step i. Further, let Z,k t deote the proportio of odes i level k with degree greater tha t log. Formally, Z,k t 1 {i : i L k, deg L k i > t log }. The mai result of this sectio is the followig theorem. Theorem 2.1. For k 1, 2,... ad 0 < t < 1 lim Z,kt 1 t k a.s. For the proof we eed some auxiliary lemmas, iterestig i their ow right. Let the umber of steps be fixed, ad 1 < i <. Firstly, we are iterested i X deg i 1. Lemma 2.1. Let 0 < ε < t < 1. The for every i > 1 t+ε we have PX > t log exp ε2 2t log. Proof. X I i+1 + I i I, where I j 1, if vertex i gets a ew edge at step, ad 0 otherwise. These idicators are clearly idepedet ad EI j 1/j, hece EX 1 i Let us abbreviate it by s. Clearly, log i + 1 s log i.

4 56 Á. Backhausz ad T.F. Móri Let a > s, the by [1, Theorem A.1.12] we have where β a/s. Hece PX a e β 1 β β s, PX a e a s s a e a s 1 a s a a a a s exp a s a + 1 a s 2 a s exp. a 2 a 2a Now, set a t log. The s t ε log, ad t log s2 PX t log exp 2t log exp ε2 2t log. Lemma 2.2. Let 0 < t < 1, ad 0 < ε < 1 t. The for every i 1 t ε 1 we have PX t log exp ε2 2t + ε log. Proof. implies that This time s > log i+1 t + ε log, thus [1, Theorem A.1.13] s t log 2 PX t log exp. 2s Notice that the expoet i the right-had side, as a fuctio of s, is decreasig for s > t log. Therefore s ca be replaced by t + ε log, ad the proof is complete. Proof of Theorem 2.1. Sice deg i is approximately equal to log i, it follows that deg i t log is approximately equivalet to i 1 t. Based o Lemmas 2.1 ad 2.2 we ca quatify this heuristic reasoig. Let 0 < ε < mi{t, 1 t}, ad a a 1 t ε 1, b b 1 t+ε. The by Lemma 2.2 P i L k such that i a, deg i 1 + t log a P i L k, deg i 1 + t log a P i L k P deg i 1 + t log EL k exp ε2 2t + ε log.

5 Degree distributio i the lower levels of the uiform recursive tree 57 Similarly, by Lemma 2.1, P i L k such that i > b, deg i > 1 + t log P i L k, deg i > 1 + t log Itroduce the evets ib+1 ib+1 EL k exp P i L k P deg i > 1 + t log ε2 2t log. A { L a k {i L k : deg i > 1 + t log } L b k }. The the probability of their complemets ca be estimated as follows. P A 2E L k exp ε2 2t log. Note that L a k 1 t ε k L k, ad L b k 1 t + ε k L k, a.s. Let c > 2t + εε 2, the 1 A P c <, hece by the Borel Catelli lemma it follows almost surely that A c occurs for every large eough. Cosequetly, This implies 1 t ε k L ck 1 + o1 {i L ck : deg ci > 1 + t log c } 1 t + ε k L ck 1 + o1. lim if Z c,kt 1 t ε k ad lim sup Z c,kt 1 t + ε k for every positive ε, hece Theorem 2.1 is prove alog the subsequece c. To the idices i betwee we ca apply the followig esimatio. For c N + 1 c with sufficietly large we have 1 { Z N,k t i L +1 ck : deg L ck +1 ci t log } c L+1 ck L ck Z +1c,k t log. log + 1

6 58 Á. Backhausz ad T.F. Móri Here the first term teds to 1, while the secod term s asymptotic behaviour is just the same as that of Z +1c,kt. Hece Z N,k t 1 + o1 1 t k. Similarly, Z N,k t L ck L+1 ck Z c,k t 1 + o1 Z c,kt 1 + o1 1 t k. log + 1 log This completes the proof. 3. Nodes of small degree i the first level Lookig at the picture Theorem 2.1 shows us o the degree distributio oe ca aturally ask how may poits of fixed degree remai i the lower levels at all. I this respect the first level ad the other oes behave differetly. It is easy to see that degree 1 odes i level 1 correspod to the fixed poits of the radom permutatio described i the Itroductio. Hece their umber has a Poisso limit distributio with parameter 1 without ay ormalizatio. More geerally, let X[, d] {i L 1 : deg i d} ; this is the umber of odes with degree d i the first level after steps. The mai result of this sectio is the followig limit theorem. Theorem 3.1. X[, 1], X[, 2],... are asymptotically i.i.d. Poisso with mea 1, as. Proof. We will apply the method of momets i the followig form. For ay real umber a ad oegative iteger k let us defie a 0 1, ad a k aa 1 a k + 1, k 1, 2,.... I order to verify the limitig joit distributio i Theorem 3.1 it suffices to show that d 3.1 lim E X[, i] holds for every d 1, 2,..., ad oegative itegers k 1,..., k d. This ca easily be see from the followig expasio of the joit probability geeratig 1

7 Degree distributio i the lower levels of the uiform recursive tree 59 fuctio of the radom variables X[, 1],..., X[, d]. E d z X[,i] i k 10 d E X[, i] k d 0 d z i 1 ki.! I the proof we shall rely o the followig obvious idetities a + 1 k a k k a k 1, a [ ] a 1 k b + 1 l a k b l l ak+1 b l 1 k a k b l, a k 1 k k+1. ak Let us start from the coditioal expectatio of the quatity uder cosideratio with respect to the sigma-field geerated by the past of the process. d d 3.5 E X[ + 1, i] F d X[, i] + S j, where i the rightmost sum j equals 0, 1,..., d, accordig to whether the ew vertex at step + 1 is coected to the root j 0, or to a degree j ode i level 1. This happes with coditioal probability 1 X[, 1] X[, d],,...,, respectively. That is, j0 S 0 1 d [ X[, X[, i] 1] + 1 k 1 X[, 1] ] k 1, i2 ad for 1 j d 1 X[, j] S j X[, i] i {j,j+1} [ X[, j] 1 kj X[, j + 1] + 1 kj+1 X[, j] ] kj X[, j + 1]. kj+1 Fially, S d X[, d] d 1 [ X[, X[, i] k d] 1 i k X[, d] ]. d k d Let us apply 3.2 to S 0 with k k 1, 3.3 to S j with k k j, l k j+1

8 60 Á. Backhausz ad T.F. Móri 1 j d 1, ad 3.3 to S d with k k d, l 0, to obtai S j k j+1 S 0 k 1 i {j,j+1} d X[, i] X[, 1] i2 X[, i] X[, j] S d k d d X[, i] k j+1. k 1 1, X[, j + 1] k j k j+1 1 d X[, i], I it ca happe that some of the k j s are zero, ad, though a 1 has ot bee defied, it always gets a zero multiplier, thus the expressios do have sese. Let us plug ito 3.5. d E X[ + 1, i] d X[, i] Itroducig d 1 + j1 F 1 1 d k j k j+1 j1 i {j,j+1} + k 1 d X[, i] X[, 1] i2 X[, i] X[, j] d E, k 1,..., k d E X[, i] we have the followig recursio. E + 1, k 1,..., k d d 1 + j1 k j+1 X[, j + 1], K k k d, k k j K E, k 1,..., k d + k 1 E, k 1 1, k 2,..., k d + k j+1 E, k 1,..., k j + 1, k j+1 1,..., k d,

9 Degree distributio i the lower levels of the uiform recursive tree 61 or equivaletly, 3.9 K E + 1, k 1,..., k d 1 K E, k 1,..., k d + d + 1 K 1 k j Ek 1,..., k j 1 + 1, k j 1,..., k d. j1 Based o 3.9, the proof ca be completed by iductio o the expoet vectors k 1,..., k. We say that k k 1, k 2,..., k d is majorized by l l 1, l 2,..., l d, if k d l d, k d 1 +k d l d 1 +l d,..., k 1 + +k d l 1 + +l d. This is a total order o N d. Now, 3.1 clearly holds for k 1, 0,..., 0, sice EX[, 1] 1 for every 1, 2,..., which is obvious cosiderig the fixed poits of a radom permutatio. I every term of the sum o the right had side of 3.9 the argumet of E is majorized by k k 1,..., k d, hece the iductio hypothesis ca be applied to them. We get that K E + 1, k 1 K E, k + 1 K 1 K 1 + o1, from which 3.4 gives that 1 K E, k 1 K, that is, as eeded. lim E, k 1,..., k d 1, Turig to higher levels oe fids the situatio chaged. Fixig a degree d we fid, roughly speakig, that each ode i level k 1 has a Poisso umber of degree d childre i level k a freshly added ode is cosidered as the child of the old ode it is coected to. Now, strog-law-of-large-umbers-type heuristics imply that the umber of odes with degree d i level k 2 is approximately equal to L k 1, that is, their proportio is L k 1 L k 1 k log. Aother iterestig problem worth of dealig with is the umber of odes with uusually high degree. I every fixed level Theorem 2.1 implies that the proportio of odes with degree higher tha log is asymptotically egligible, but they must exist, sice the maximal degree is approximately log 2 log 2 e log. How may of them are there? We are plaig to retur to this issue i a separate paper.

10 62 Á. Backhausz ad T.F. Móri Refereces [1] Alo, N. ad J.H. Specer, The Probabilistic Method, 2d ed., Wiley, New York, [2] Backhausz, Á. ad T.F. Móri, Local degree distributio i scale free radom graphs, Electro. J. Probab., , [3] Barabási, A.-L. ad R. Albert, Emergece of scalig i radom etworks, Sciece, , [4] Devroye, L. ad J. Lu, The strog covergece of maximal degrees i uiform radom recursive trees ad dags, Radom Structures Algorithms, , [5] Drmota, M., Radom Trees, Spriger, Wie, [6] Fuchs, M., H.-K. Hwag ad R. Neiiger, Profiles of radom trees: Limit theorems for radom recursive trees ad biary search trees, Algorithmica , [7] Katoa, Zs., Levels of a scale-free tree, Radom Structures Algorithms, , [8] Móri, T.F., A surprisig property of the Barabási Albert radom tree, Studia Sci. Math. Hugar., , [9] Smythe, R.T. ad H.M. Mahmoud, A survey of recursive trees, Theory Probab. Math. Statist., , [10] Szymański, J., O a ouiform radom recursive tree, A. Discrete Math., , [11] Tapia, M.A. ad B.R. Myers, Geeratio of cocave ode-weighted trees, IEEE Tras. Circuit Theory, CT , Á. Backhausz ad T.F. Móri Departmet of Probability Theory ad Statistics Faculty of Sciece Eötvös Lorád Uiversity H-1117 Budapest, Pázmáy P. sétáy 1/C Hugary ages@cs.elte.hu moritamas@ludes.elte.hu