The Degree Sequence of Random Apollonian Networks

THE DEGREE SEQUENCE OF RANDOM APOLLONIA N NETW ORKS CHARALAMPOS E. TSOURAKAKIS Abstract. W e analyze the asymptotic behavior of the degree sequence of Rando m Ap ollonian Net works [11]. F or previo us weaker results see [10, 11]. 1. Intr oduction 1.1. Results. Random Ap ollonian Net w orks (RANs) is a p opular mo del of planar graphs with p o w er law properties [1 1], see also Section 1.3. In this note w e analyze the degree sequence of RANs. F or earlier, w eak er results see [10, 11]. Our main result is T heorem 1. Theorem 1. L et Z k ( t ) denote the numb er of vertic es of de gr e e k at time t , k ≥ 3 . F or any k ≥ 3 ther e exists a c onstant b k (dep ending on k ) such that for t sufficiently lar ge | E [ Z k ( t )] − b k t | ≤ K, wher e K = 3 . 6 . F urthermor e let λ > 0 . F o r any k ≥ 3 Pr [ | Z k ( t ) − E [ Z k ( t )] | ≥ λ ] ≤ 2 e − λ 2 72 t . 1.2. Related W ork. Bollob´ as, R iordan, Sp encer and T usn´ ady [3 ] prov ed rigorously the p ow er la w distribution o f t he Barab´ asi-Alb ert mo del [2]. Random Ap ollonian Netw orks w ere introduced in [11]. Their degree sequenc e was analyzed inaccurately in [11] (see commen t in [10]) and sub- sequen tly in [10] using ph ysicist’s metho do lo gy . Co op er & Uehara [4] and Gao [8] analyzed the degree distribution of random k trees, a closely related mo del to RANs. In RANs –in con trast to random k trees– the random k clique chos en at eac h step has nev er previously b een selected. F or example in the t w o dimens ional case an y c hosen triangular face is b eing sub divided into three new triangular faces b y connecting the incoming v ertex to the v ertices of the b oundary . Darrasse and Soria analyzed the degree distribution of random Ap ollonia n net work structures in [6]. 1.3. Mo del. F or con venie nce, w e summarize the RAN mo del here, see also [11]. A RAN is gener- ated b y starting with a triangular face and doing t he fo llo wing un til the net w ork reac hes the desired size: pick a triangular face uniformly at rando m, insert a v ertex inside the sampled face and connect it to the vertice s o f the b oundary . 1.4. Prerequisites. In Section 2 w e in v ok e the following lemma. Lemma 1 (Lemma 3.1, [5 ]) . Supp ose that a se q uenc e { a t } sa tisfies the r e curr enc e a t +1 = (1 − b t t + t 1 ) a t + c t Key wor ds and phr ases. Degree Distribution, Random Ap ollo nian Net works. Suppo rted b y NSF grant ccf10 1311 0. 1 for t ≥ t 0 . F urthermor e supp ose lim t → + ∞ b t = b > 0 and lim t → + ∞ c t = c . Then lim t → + ∞ a t t exists and lim t → + ∞ a t t = c 1 + b . W e also use in Section 2 the Azuma-Ho effding inequalit y 2, see [1, 9]. Lemma 2 (Azuma-Ho effding inequalit y) . L et ( X t ) n t =0 b e a martingale se quenc e with | X t +1 − X t | ≤ c for t = 0 , . . . , n − 1 . Also, let λ > 0 . Th en: Pr [ | X n − X 0 | ≥ λ ] ≤ 2 exp  − λ 2 2 c 2 n  2. Proo f of Theorem 1 F or simplicit y , let N k ( t ) = E [ Z k ( t )], k ≥ 3. Also, let d v ( t ) denote the degree of v ertex v at time t and let 1 ( d v ( t ) = k ) b e an indicator v ariable whic h equals 1 if d v ( t ) = k , otherwise 0. Then, f or an y k ≥ 3 w e can express the exp ected num b er N k ( t ) of vertice s of degree k as a sum of exp ectations of indicator v ariables: N k ( t ) = X v E [ 1 ( d v ( t ) = k )] . (1) W e distinguish t w o cases in the followin g. • Case 1 k = 3: Observ e that a v ertex of degree 3 is created only b y an insertion of a new v ertex. The exp ectation N 3 ( t ) satisfies the following recurrence 1 N 3 ( t + 1) = N 3 ( t ) + 1 − 3 N 3 ( t ) 2 t + 1 . (2) The basis fo r Recurrence (2) is N 3 (1) = 4. W e prov e the following lemma whic h sho ws that lim t → + ∞ N 3 ( t ) t = 2 5 . Lemma 3. N 3 ( t ) sa tisfies the fol lowing ine quality: | N 3 ( t ) − 2 5 t | ≤ K, wher e K = 3 . 6 (3) Pr o of. W e use induction. Assume tha t N 3 ( t ) = 2 5 t + e 3 ( t ). W e wis h to pro v e that for all t , | e 3 ( t ) | ≤ 3 . 6. The result tr ivially holds fo r t = 1. Assume the result holds f or some t . W e shall sho w it holds for t + 1. 1 The three initial vertices participate in one les s face than their degree . How ever, this leaves the a symptotic analysis unc hanged. 2 N 3 ( t + 1) = N 3 ( t ) + 1 − 3 N 3 ( t ) 2 t + 1 ⇒ e 3 ( t + 1) = e 3 ( t ) + 3 5 − 6 t + 15 e 3 ( t ) 10 t + 5 = e 3 ( t ) + 3 5(2 t + 1) − 3 e 3 ( t ) 2 t + 1 ⇒ | e 3 ( t + 1) | ≤ K (1 − 3 2 t + 1 ) + 3 5(2 t + 1) ≤ K Hence b y induction, Inequalit y (3) holds fo r all t ≥ 1.  • Case 2 k ≥ 4: F or k ≥ 4 the follo wing Equation holds for each indicator v ariable 1 ( d v ( t ) = k ): E [ 1 ( d v ( t + 1) = k )] = E [ 1 ( d v ( t ) = k )] (1 − k 2 t + 1 ) + E [ 1 ( d v ( t ) = k − 1)] k − 1 2 t + 1 (4) Therefore, subs tituting in Equation (1) the expression from Equation 4 w e obtain for k ≥ 4 N k ( t + 1) = N k ( t )(1 − k 2 t + 1 ) + N k − 1 ( t ) k − 1 2 t + 1 . (5) No w, w e use induction to sho w that lim t → + ∞ N k ( t ) t for k ≥ 4 exists . Lemma 4. F or k ≥ 3 lim t → + ∞ N k ( t ) t exists. L et b k = lim t → + ∞ N k ( t ) t . T h e n, b 3 = 2 5 , b 4 = 1 5 , b 5 = 4 35 and for k ≥ 6 b k = 24 k ( k +1)( k +2) . F urthermor e, for al l k ≥ 3 | N k ( t ) − b k t | ≤ K, wher e K = 3 . 6 . (6) Pr o of. F or k = 3 the result holds by Lemma 3. W e use induction. Rewrite Recurrence (5) as: N k ( t + 1) = (1 − b t t + t 1 ) N k ( t ) + c t where b t = k / 2, t 1 = 1 / 2 , c t = N k − 1 ( t ) k − 1 2 t +1 . Clearly , lim t → + ∞ b t = k / 2 > 0 and b y the inductiv e h yp othesis lim t → + ∞ c t = lim t → + ∞ b k − 1 t k − 1 2 t +1 = b k − 1 ( k − 1) / 2. Hence, b y in vokin g Lemma 1 we obta in b k = lim t → + ∞ N k ( t ) t = ( k − 1) b k − 1 / 2 1 + k / 2 = b k − 1 k − 1 k + 2 . Therefore b 3 = 2 5 , b 4 = 1 5 , b 5 = 4 35 for an y k ≥ 6, b k = 24 k ( k +1)( k +2) whic h sho ws a p o w er la w degree distribution with exp o nen t 3. Finally consider the pro of of Inequalit y (6). The case k = 3 w a s pro v ed in the Lemma 3. Assume the result holds for some k ≥ 3, i.e., | e k ( t ) | ≤ K where K = 3 . 6. W e will show it holds for k + 1 to o. Let e k ( t ) = N k ( t ) − b k t . Substituting in Recurrence (5) and using the fact that b k − 1 ( k − 1 ) = b k ( k + 2) w e obtain the following: 3 e k ( t + 1) = e k ( t ) + k − 1 2 t + 1 e k − 1 ( t ) − k 2 t + 1 e k ( t ) ⇒ | e k ( t + 1) | ≤ | (1 − k 2 t + 1 ) e k ( t ) | + | k − 1 2 t + 1 e k − 1 ( t ) | ≤ K (1 − 1 2 t + 1 ) ≤ K Hence b y induction, Inequalit y (6) holds fo r all k ≥ 3.  It’s worth p o inting out that Lemma 4 ag r ees with [7] where it w as show n that the maxim um degree is Θ ( √ t ). Finally , the Lemma 5 prov es the concen tr a tion of Z k ( t ) around its expected v alue for k ≥ 3. This lemma a pplies the Azuma-Ho effding inequality 2 and completes the pro of of Theorem 1. Lemma 5. L et λ > 0 . F or any k ≥ 3 Pr [ | Z k ( t ) − E [ Z k ( t )] | ≥ λ ] ≤ 2 e − λ 2 72 t . (7) Pr o of. Let (Ω , F , P ) b e the probabilit y space induced b y the construction of a Random Ap ollo nian Net work (see Section 1.3) after t insertions. Fix k ( k ≥ 3) and let ( X i ) i ∈{ 0 , 1 ,...,t } b e the martingale sequence defined b y X i = E [ Z k ( t ) |F i ], where F 0 = { ∅ , Ω } and F i is the σ - algebra generated b y the RAN pro cess after i steps. Notice X 0 = E [ Z k ( t ) |{∅ , Ω } ] = N k ( t ), X t = Z k ( t ). W e show that | X i +1 − X i | ≤ 6 for i = 0 , .., t − 1. Let P j = ( Y 1 , . . . , Y j − 1 , Y j ), P ′ j = ( Y 1 , . . . , Y j − 1 , Y ′ j ) b e tw o sequence s of face c hoices differing only a t time j . Also, let ¯ P , ¯ P ′ con tin ue from P j , P ′ j un t il t . W e call the faces Y j , Y ′ j sp ecial with resp ect t o ¯ P , ¯ P ′ . W e define a measure preserving map ¯ P 7→ ¯ P ′ in the following w a y: for ev ery c hoice of a non-sp ecial face in pro cess ¯ P at time l we mak e the same face c hoice in ¯ P ′ at time l . F or ev ery choice of a face inside the sp ecial face Y j in pro cess ¯ P w e mak e an isomorphic (w.r.t., e.g., clockw ise order and depth) c hoice of a fa ce inside the sp ecial face Y ′ j in pro cess ¯ P ′ . Sinc e the num b er of v ertices of degree k can c ha ng e by at most 6, i.e., the (at most) 6 vertice s in v olv ed in the tw o faces Y j , Y ′ j the follo wing holds: | E [ Z k ( t ) | P ] − E [ Z k ( t ) | P ′ ] | ≤ 6 . F urthermore, this holds for any P j , P ′ j . W e deduce that X i − 1 is a w eigh ted mean of v alues, whose pairwise differences a r e all at most 6. Th us, the distance of the mean X i − 1 is at most 6 f rom eac h of these v alues. Hence, for any one step refinemen t | X i +1 − X i | ≤ 6 ∀ i ∈ { 0 , . . . , t − 1 } . By a pplying the Azuma-Ho effding inequalit y a s stated in Lemma 2 w e obtain Pr [ | Z k ( t ) − E [ Z k ( t )] | ≥ λ ] ≤ 2 e − λ 2 72 t . (8)  A corollary immediately obtained b y the previous lemma b y setting λ = √ t lo g t is the fo llo wing: Corollary 1. Pr  | Z k ( t ) − E [ Z k ( t )] | ≥ √ t lo g t  = o (1) . A ckno wledgements The author w ould like to thank D eepak Bal and Alan F rieze for helpful commen ts on the man uscript. 4 Reference s [1] Azuma, K.: Weighte d sums of c ertain dep endent varia bles T ohoku Math J 3, pp. 357- 3 67, 19 67 [2] Barab´ asi, A., Alb ert, R.: Emer genc e of Sc aling in Ra ndom Networks Science 28 6, pp. 509-512, 1 999 [3] Bollob´ as, B., Rio rdan, O., Sp encer, J., T usn´ ady , G.: The De gr e e Se quenc e of a S c ale F r e e R andom Gr aph Pr o c ess Random Struct. Algorithms, 18(3), pp. 279-2 90, 200 1 [4] Co op er, C., Uehara, R.: S c ale F r e e Pr op erties of ra ndom k -t r e es Mathematics in Computer Science, 3(4), pp. 489–4 96, 20 1 0 [5] Ch ung Graham, F., Lu, L.: Complex Gr aphs and Networks American Mathematical So ciety , No. 107 (200 6) [6] Darrasse , A., So r ia, M.: De gr e e distribut ion of r andom Ap ol lonian network stru ctur es and Boltzmann sampling 2007 Conference on Analysis of Algorithms, AofA 07, DMTCS Pro ceedings. [7] F r ie z e, A., Tsourak a kis, C .E .: High De gr e e V ertic es, Eig envalues and Diameter of R andom Ap ol lonian Networks Av ailable at Arxiv http ://arx iv.org/abs/1104.5259 [8] Gao, Y.: The de gr e e distribution of r andom k-tr e es Theoretica l Computer Science, 410(8-10), 2009. [9] Hoe ffding, W.: Pr ob ability ine qualities for s u mes of b ounde d r andom variable s J Amer Statist Ass o c 5 8, pp. 13-30 , 196 3 [10] W u, Z.-X., Xu, X.-J., W a ng, Y.- H.: Comment on “Maximal planar networks with lar ge clustering c o efficient and p ower-law de gr e e distribution ” P h ysica l Rev iew, E 73, 058101 (2006) [11] Zhou, T., Y an, G., W ang, B.H: Maximal planar networks with lar ge clustering c o efficient and p ower-law de gr e e distribution Physical Review, E 71 , 046141 (2005) Dep ar tment of Ma thema tical S ciences, Carnegie Mellon U niversity, 5000 Forbe s A v., 15213, Pittsburgh, P A, U. S.A E-mail addr ess : ctso urak@ math.c mu.edu 5