The Degree Distribution of Random Birth-and-Death Network with Network Size Decline

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1 The Degree Dstrbuton of Random Brth-and-Death etwork wth etwork Sze Declne Xaojun Zhang *, Hulan Yang School of Mathematcal Scences, Unversty of Electronc Scence and Technology of Chna, Chengdu 673, P.R. Chna Abstract In ths aer, we rovde a general method to obtan the exact solutons of the degree dstrbutons for RBD wth network sze declne. Frst by stochastc rocess rules, the steady state transformaton euatons and steady state degree dstrbuton euatons are gven n the case of m 3, 0<</, then the average degree of network wth n nodes s ntroduced to calculate the degree dstrbuton. Esecally, takng m=3 as an examle, we exlan the detaled solvng rocess, n whch comuter smulaton s used to verfy our degree dstrbuton solutons. In addton, the tal characterstcs of the degree dstrbuton are dscussed. Our fndngs suggest that the degree dstrbutons wll exhbt Posson tal roerty for the declnng RBD. Key words: random brth-and-death network (RBD), stochastc rocesses, Markov chan, generatng functon, degree dstrbuton * Corresondng author: Xaojun Zhang Emal address: sczhxj@uestc.edu.cn

2 The Degree Dstrbuton of Random Brth-and-Death etwork wth etwork Sze Declne. Introducton In the real world, there exst many brth-and-death networks such as the World Wde Web [- 3], the communcaton networks [4-6], the frend relatonsh networks [7-0] and the food chan networks [-3], n whch nodes may enter or ext at any tme. For ths knd of evolvng networks, the degree dstrbuton s always one of the most mortant statstcal roertes. Several methods have been roosed to calculate ther degree dstrbutons lke the frst-order artal-dfferental euaton method by Saldana [4], the mean-feld aroach [5] by Slater et al. [6], the rate euaton aroach by Sarshar and Roychowdhury [7] Moore et al. [8], Garca-Domngo et al. [9] Ben- am and Kravsky [0]. Whle these aroaches am at the networks whose sze s growng or reman unchanged at each tme ste kee, Zhang et al. [] ut forward the stochastc rocess rules (SPR) based Markov chan method to solve evolvng networks n whch the network sze may ncrease or decrease at each tme ste. Furthermore, a random brth-and-death network model (RBD) s consdered [], n whch at each tme ste, a new node s added nto the network wth robablty (0< <) wth connecton to m old nodes unformly, or an exstng node s deleted from the network wth the robablty =-. For network sze declne (0<</), snce m s a crtcal arameter for the degree dstrbutons, soluton methods may vary for dfferent m. Whle Zhang et al. [] only consder a secal case m=,, n ths aer, a general aroach s roosed for solvng the degree dstrbutons of RBD wth network sze declne n case of m 3. Takng m=3 as an examle, we rovde the exact solutons of the degree dstrbutons. Our fndngs also ndcate that the tal of the degree dstrbuton for the declnng RBD s subject to Posson tal. Ths aer s structured as follows: Secton gves the RBD model wth 0<</ and ts steady-state euatons; Secton 3 rovdes the method of solvng degree dstrbuton of RBD wth 0<</; Secton 4 further dscusses the tal characterstcs of the degree dstrbuton and Secton 5 concludes the aer.

3 . Steady State Euatons of RBD wth etwork Sze Declne. RBD model Consder the RBD model []: () () The ntal network s a comlete grah wth m+ nodes, where m s a ostve nteger; At each unt of tme, add a new node to the network wth robablty (0<</) and connect t wth m old nodes unformly, or randomly delete a node from the network wth robablty =-. ote: (a) Here we assume the low-bound of the network sze n0, that s, f the number of nodes n the network s n 0 at tme t, then at tme t, we only add a new node to the network wth robablty and connect t to the old node n the network. (b) If at tme t, a new node s added to the network and the network sze s less than m, then the new node s connected to all old nodes.. Steady state transformaton euatons Usng SPR [], we use nk, to descrbe the state of node v, where n s the number of nodes n the network that contans v, and k s the degree of node v. Let K t denote the state of node v at tme t, the stochastc rocess K t, t 0 s an ergodc aerodc homogeneous Markov chan wth the state sace E n, k, n,0 k n. Let P t be the robablty dstrbuton of K t,.e., nk, P t P K t n k () The state transformaton euatons are as follows (see the Aendx for the detals): t+ t where P s the one-ste transton robablty matrx. Let P P P () lm PK t,, k k (3) t Takng the lmt of E. () as t +,the steady state transformaton euatons can be obtaned

4 ,0 3,0 3,,0,0,0, m m m,0 m,0 m, m,0 m m m 3,0 m 3, m,0 (4) n n n m n,0 n,0 n, n,0 3 3, 4, 4,,0, 3, 3,,0,0 m m m m, m, m, m,0 m, m 3, m m m 3, m m,0 m, (5) np n m n m n, n, n, n,0 n, m m,, m, m m m m m m m m, m m, 0 m m m m, m m, m m, m m, m m, 3 3, m 3, m, m+ m m m m m m m m m m, m n n m m m n m n, m n, m n, m n, m n, m m m m m,,,, m m m m m m m m m, 0 m m, 3, m m m m m m 3, m m m, m m, m m, 0 n n n m m m n m n, m n, m n, m n, m n, m n, 0 (7) r r m r, r r, r r, r r, r r, 3, r 3, m, r m r r r r r r r r P r, r n n r r m n m n, r n, r n, r n, r n, r rm (8)

5 .3 Steady state degree dstrbuton euatons Let K be the steady state degree dstrbuton [5,3,4], and k be the robablty dstrbuton of K, that s, k PK k lm P, k t, k (9) t k k Combnng E.(9) and steady state transformaton euatons (4)-(8), we can obtan the steady state degree dstrbuton euatons as follows: m m 0,0 m,0 m m m m 0,0 m,0 m, m m m m m m m m, m m m m m m m m3 0 m m m m m m m m (0) r m r r r m r 3. Degree Dstrbuton of RBD wth etwork Sze Declne Let t be the number of nodes n RBD at tme t and 0 m. Let be the steady state network sze, and n be the robablty dstrbuton of, that s, Consderng n nk, n P n lm P t n = n t t, t 0 s an one dmensonal random walk wth a left bound,we have k 0 () n n n ()

6 As shown n E. (0),n the case of m=,, we only need to calculate,0 before the robablty generaton functon method s emloyed for the exact soluton of the degree dstrbuton [], n whch,0 can be obtaned drectly as follows: (3),0 However, n the case of m 3, k, s reured for the calculaton, rather than,0 whch s a secal cases. Obvously, k, s much more dffcult to obtan. Thus n the followng secton, we focus on the calculaton of k,. 3. Calculaton of k, To obtan k,, e n s ntroduced,.e. e n n k nk, (4) k 0 Snce k, E K k k k k k k k e k, k 0 (5) e n denotes the contrbuton of the network wth n nodes to the average degree EK. From E.(4) and the steady state transformaton euatons (4-8), we can obtan the euatons about,, as follows: e, nen n en n en n n n m nen n en n en m n n m (6) where To sum u the frst n n m tems n E.(6), we can get e 0 (7) ote n m n n n (8) e n e ne m m

7 then we have e 0 and e m (9) lm ne =0 n n (0) From E.(8), we have m + m e m = m m () m Then generaton functon for E. (6) can be rewrtten as satsfyng () 0 T x e x m x x x T x T x x x x x T e m m Solvng the dfferental euaton (3), (8),we have e can be obtaned. Combnng Es. (4) and (4)- e, 3, 3, e 3 (3),0, 3,0 3, 3, 3 (4) 3,0 3, =,0 = - - 3, 3,,,0,0 Then k, can be solved by E. (4). 3. Exact solutons of the degree dstrbutons for m=3, 0<</, Once k, s obtaned, robablty generatng functon aroach can be emloyed for E.(0) to obtan the steady state degree dstrbuton k. In ths secton, takng m=3 as an examle, we

8 exlan how ths aroach s used. From E. (0), we can obtan the degree dstrbuton euatons of RBD wth 0<</, m=3 as follows: 3 0,0, ,0,0, , (5) r 3 r r r 3 r From E. (5) we may fnd that,0 and, are needed before we solve the degree dstrbuton. In the case of m =3, E. (6) can be rewrtten as nen n en n en n n n 3 nen n en n en 6 n n 4 (6) Here we ntroduce the followng generatng functon Combnng Es.(6), we have: 3 (7) 0 T x e x x x x T x T x x x x x T Solvng the dfferental euaton (8), we have Thus T x t t t (8) x dt x (9) x t t t t e T 0 dt 0 (30) t Let then t y t (3)

9 y t, dt dy (3) y y So, we have y y y e dy 3 dy dy y y y y 3 dy 3 0 y (33) x 3 dx 3 0 x For, = e =,0, (34) Rewrtng (34), we have, = e = -,0, (35) To solve k,let the robablty generatng functon be Accordng to E.(5) and E.(35), we can get Solvng the E.(37), we can obtan G x x, G (36) 0 0 x 3x 3 Gx G x x x x x x 3,0 x x,,0,0 (37)

10 3,0 t t e x t 3 x 3 t G x e dt x,,0,0 6 6 e 3 k 3 e x 7 7 k 0 k! 3 e e x x 3 So,the degree dstrbutons of RBD for 0<</, m=3 are as follows (38) where k 3 3 c e a0, k 0! k 3 3 k! c e a, k 3 3 c e a k k!, 3 3 c e, k 3 k! 3 a0 e 9 9 a e 3 3 a e c 7 7 x e 3 dx 3 0 x Fgures llustrates the exact solutons and smulaton results of the degree dstrbutons for 0<</, m 3, where the horzontal and vertcal ordnates denote the degree of nodes and the robablty resectvely. Each smulaton number s the average value of 000 smulaton results for t As shown n Fg., the exact solutons match erfectly wth the numercal solutons, verfyng the correctness of our exact solutons. (39) (40)

11 es =0. ns =0. es =0. ns =0. es =0.3 ns =0.3 es =0.4 ns = k Fg. Exact solutons vs. numercal solutons (t=0000): degree dstrbutons of RBD for </, m=3 4. Posson Tal By E. (39), we can conclude that RBD wth 0<</,for the large k k m, the degree dstrbuton of RBD exhbts a backward accumulaton form of Posson dstrbuton, that s, where s a ostve constant and = r k k m (4) r! k m rk. For suffcent large r k k, we have ~ (4) r! k! r k Thus n the case of 0<</,the degree dstrbuton of RBD exhbts a Posson tal. Here we emloy the same aroach as n [] to verfy ths Posson tal for the degree dstrbuton. Let Then usng E.(4), we have r k k k- (43) r k ; ln r k ln ln k (44) k that s, rk s a lne wth sloe - for large k n the two-logarthm axs dagram.

12 r(k) m=4, =0. m=4, =0. m=4, =0.3 m=4, = k Fg.-a Fg.-b r(k) m=6, =0. m=6, =0. m=6, =0.3 m=6, = k Fg.-c Fg.-d Fg. The Posson tal of RBD wth 0<</ Fg. llustrates the Posson tals of the degree dstrbutons for varous (0<</). As we can see, n the case k m, the sloes of lnes tend to be -, showng the Posson tals for the degree dstrbutons of RBD. 5. Concluson In ths aer, we rovde a general aroach to obtan the exact solutons of the degree dstrbutons n the case of m 3, 0<</. Esecally, takng m=3 as an examle, we exlan the detaled solvng rocess, n whch comuter smulaton s used to verfy our degree dstrbuton solutons. In addton, the characterstcs of the degree dstrbuton are dscussed. Our fndngs suggest that the degree dstrbutons wll exhbt Posson tal roerty for the declnng RBD. Acknowledgments

13 Ths research was fnancally suorted by the atonal atural Scence Foundaton of Chna (o ) and the Chna Scholarsh Councl. Aendx : State transformaton euatons The state transformaton euatons of K t are as follows: P t P t P t P t m P,0 t m m P m,0 t P m, t np t np t P t n m P t,0,0,0, P t P,0 3,0 t P 3, t m P t m P t P t P t m,0 m3,0 m3, m,0,0,0, n n n n,0 P t P t P t P t P t 3P t P t P t P t np t n P, 3, 3,,0,0 3, 4, 4,,0 m P t mp t P t mp t m, m, m, m,0 m P t m P t P t mp t P t m, m3, m3, m,0 m,, n n t P t mp,,,0 t n m n n P n, t (45) (46) m mp t P t mp t m P t,,,, P, t m m m m m m m m m 0 m P, t P m m m, m t mp m, m t mp m, m t m P t 3P t mp t mp t m, m m3, m m3, m m, m m+ m P t m, m np n m P n, m t nm, t n m P t mp,, t n m n m mp n, m t (47)

14 m P t P t m P t mp t m, m m, m m, m m, m m P m, t 0 m P t P t m P t mp t m, m m3, m m3, m m, m m P, t P m m m, t 0 (48) np t n m P t m P t mp n n m P n, m t P n, t 0 n, m n, m n, m n, m t r P t P t r P t mp t r, r r, r r, r r, r r, r r3, r r3, r r, r r m P r, r t np t n r P t r P t mp t r P t P t r P t mp t n, r n, r n, r n, r n m P t n, r r m (49) References. A. L. Barábas and R. Albert, Emergence of scalng n random networks, Scence 86, 5439 (999).. R. Albert and A. L. Barabás, Statstcal Mechancs Of Comlex etworks, Rev. Mod. Phys. 74, 47 (00). 3. L. A. Adamc, B. A. Huberman, A. L. Barabas, R. Albert, H. Jeong, G. Bancon, Power-Law Dstrbuton of the World Wde Web, Scence 87, 546 (000). 4. S.. Dorogovtsev and J. F. F. Mendes, Evoluton of networks, Adv. Phys. 5, 079 (00). 5. G. Roger, A. Alex, D. G. Albert, G. Francesc, Dynamcal roertes of model communcaton networks, Phys. Rev. E 66, (00). 6.. Onuttom and S. Iraj, Scalng of load n communcatons networks, Phys. Rev. E 8, 0360 (00). 7. D. J. Watts and S. H. Strogatz, Collectve dynamcs of 'small-world' networks, ature (London) 393, 440 (998). 8. M. E. J. ewman, The structure and functon of comlex networks, SIAM Rev. 45, 67(003). 9. M. E. J. ewman, Scentfc collaboraton network: I. etwork constructon and fundamental results, Phys. Rev. E 64, 063 (00). 0. M. E. J. ewman, Scentfc collaboraton networks: II. Shortest aths, weghted networks, and centralty, Phys. Rev. E 64, 063 (00).. R. J. Wllams and. D. Martnez, Smle rules yeld comlex food webs, ature (London) 404, 80 (000).. L. A. Barbosa, A. C. Slva, and J. K. L. Slva, Scalng relatons n food webs, Phys. Rev. E 73, (006). 3. S. B. Otto, B. C. Rall and U. Brose, Allometrc degree dstrbutons facltate food-web stablty, ature (London) 450, 6 (007).

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2-Adic Complexity of a Sequence Obtained from a Periodic Binary Sequence by Either Inserting or Deleting k Symbols within One Period

2-Adic Complexity of a Sequence Obtained from a Periodic Binary Sequence by Either Inserting or Deleting k Symbols within One Period -Adc Comlexty of a Seuence Obtaned from a Perodc Bnary Seuence by Ether Insertng or Deletng Symbols wthn One Perod ZHAO Lu, WEN Qao-yan (State Key Laboratory of Networng and Swtchng echnology, Bejng Unversty