Random Birth-and-Death Networks

Transcription

1 Rando Birth-and-Death Networs Xiaojun Zhang *, Zheng He, Lez Rayan-Bacchus 3 School of Matheatical Sciences, School of Manageent and Econoics, Uniersity of Electronic Science and Technology of China, Chengdu, P. R. China, Uniersity of Winchester, Winchester, SO 5HT, UK Abstract In this aer, a baseline odel tered as rando birth-and-death networ odel (RBDN) is considered, in which at each tie ste, a new node is added into the networ with robability (0< <) connect it with old nodes uniforly, or an existing node is deleted fro the networ with robability =-. This odel allows for fluctuations in size, which ay reach any different discilines in hysics, ecology and econoics. The urose of this study is to deelo the RBDN odel and exlore its basic statistical roerties. For different, we first discuss the networ size of RBDN. And then cobining the stochastic rocess rules (SPR) based Maro chain ethod and the robability generating function ethod, we roide the exact solutions of the degree distributions. Finally, the characteristics of the tail of the degree distributions are exlored after siulation erification. Our results show that the tail of the degree distribution for RBDN exhibits a Poisson tail in the case of 0< / and an exonential tail as aroaches to. Key words: Rando birth-and-death networs, networ size, degree distribution, Poisson tail exonential tail *Corresonding author: Xiaojun Zhang Eail address: sczhxj@uestc.edu.cn

2 Rando Birth-and-Death Networs. Introduction In the real world, ost networs lie the World Wide Web [-3], friendshi networs [4-7], counications networs [8-0], and food-web [-3] are eoling with freuent node births (additions) and deaths (deletions), in which each agent is intelligent and has its own life cycle. Recently, these addition-deletion networs hae caught uch attention [4-4]. Various odels hae been deeloed to describe different eoling rocesses. Tyically, Moore et al. [8] discussed addition-deletion networs in which at each unit of tie one node is added, and with robability a randoly chosen node is deleted; Sarshar and Roychowdhury [6] considered an ac hoc networ where new nodes joining the networ ae lins referentially and existing nodes are uniforly deleted at a constant rate; Slater et al. [7] considered a growing tree odel allowing the ossibility of death; Saldaña [0] discussed growing rando networs with addition and deletion of nodes based on a differential ass balance euation; Ben-Nai and Kraisy [] discussed growing rando networs in which at each tie ste, with rate ( ), a node is added to the networ and with rate, a randoly selected node is deleted with its arent node inheriting the lins of its iediate descendants. Although all these odels deal with the deletion of nodes in the networ, they at the sae tie add nodes at each tie ste to ee the networ size growing or unchanged. In fact, the networ sizes of any eoling networs lie the Internet or social networs often fluctuate due to the aearance of new couters or ersons resectiely at one tie or the disaearance of reiously existing nodes at another tie. This dynaic rocess ay lead to the decreasing of the networ size, reflecting that shrining networs are also oular in reality. In this aer, a ore general rando birth-and-death networ (RBDN) is discussed, in which at each tie ste, a new node is added to the networ with robability (0 ) or an existing node is deleted fro the networ with robability. This RBDN odel is sile enough to constitute a baseline odel for any studies in dynaical networs, which reach any different discilines in hysics, ecology and econoics. Meanwhile, as its ain strength, it allows for fluctuations in size, granting it additional realis. Potentially, this odel ay allow scientists to decoule effects caused by dynais itself fro effects caused by the dynaic echanis used.

3 Thus the ai of this study is to deelo the RBDN odel and exlore its basic statistical roerties. This aer is organized as follows. Section introduces the RBDN odel. Section 3 discusses the networ sizes of RBDN for different and Section 4 calculates the degree distributions of RBDN for different. In Section 5, we first use couter siulations to erify our results in Section 4 and then exlore the tail characteristics of the degree distributions for RBDN. Section 6 concludes the aer and rooses further direction.. RBDN Model As noted aboe, reious odels of addition-deletion networs ee the size of the networ growing or unchanged, and all ignore that real-world networs ay either exand or contract oer tie. Here we introduce a ore general odel tered Rando Birth-and-Death Networ (RBDN) odel as follows. (i) The initial networ is an isolated node; (ii) At each unit of tie, add a new node to the networ with robability (0 ) and connect it with old nodes uniforly, or randoly delete a node fro the networ with robability. Note: (a) During the eoling rocess, there exists a networ size lower bound n 0, naely, if the nuber of nodes in the networ is n 0 at tie t, then at tie t, we add a new node to the networ with robability and randoly connect it to old nodes in the networ, or ee it unchanged with robability. To silicity, here we define n0.indeed, in our study, different n 0 will not affect the distribution tye and its roerties. (b) If at tie t, a new node is added to the networ and the networ size is less than, then the new node is connected to all old nodes. (c) If at tie t, a node is deleted, then all the edges incident to the reoed node are also reoed fro the networ, thus the degree of its neighbors decreases by one. Since corresonds to the ure addition networ (which has been studied in [8]), and 0 corresonds to the ure shrining networ which is a single node, here we assue 0. In the following sections, we will inestigate soe iortant roerties of RBDN.

4 3. Networ Size Networ size is fundaental to all networs since it ay affect any others roerties including the degree distribution, aerage ath length and clustering coefficient []. Thus in this aer, we first discuss the networ size of RBDN. For a growing networ, it is obious that the networ size will be infinite at the liit t. Howeer, for RBDN, the networ size is deterined by different. Let Nt be the nuber of nodes in RBDN at tie t and N 0. Let N n li P N t n n () t be the robability distribution of the networ size of RBDN in the liit Q i, j and Q satisfies be the one-ste transfor robability atrix of i, j N t, t 0, where t. Let P N t j N t i () i, j i, j j i, i j i, i (3) Since the one-ste transfor robability atrix Q is indeendent of tie, N t, t 0 is a Maro chain with stationary transition robabilities. Also because N t, t 0 is a one- diension rando wal with a left bound [5], we ay draw the following conclusions easily. (a) 0 N n n n (4) (b) N n n 0 else (5) (c) n N 0 n (6) As shown in Es. (4)-(6), the networ size of RBDN follows geoetric distributions in the

5 case of secial case 0. In the case of, the networ size tends to be infinite as, the size of the networ can uniforly be any ositie integer. t. For the 4. Degree Distribution Many ethods hae been used to calculate the degree distributions of eoling networs such as the ean-field ethod [6], the rate-euation ethod [7], the aster-euation aroach [8], the renoralization grou [9], and the Maro chain ethod [4,30,3]. In this section, the stochastic rocess rules (SPR) based Maro chain ethod [4] is eloyed since it is effectie for both node addition and deletion in the eoling networ. In addition, we enhance the SPR ethod by using a robability generating function aroach to sole the degree distribution euations for different. For the SPR ethod, there is an infinite nuber of isolated nodes at tie t 0. At each tie ste, the added nodes are roided by the other networs that hae the sae toologic structure, and the deleted nodes will construct new networs. Thus SPR ethod ees the nuber of nodes unchanged at any tie and aintains the toological structures and statistical characteristics [4]. Here for any node, we use n, to describe the state of node, where n is the nuber of nodes in the networ that contains, and is the degree of node. Let NK t denote the state of node at tie t and Pt be the robability atrix of NK t, i.e. Let P, n, P t P NK t n (7) be the one-ste transition robability atrix of NK t, t 0, P n,, n, (8) using SPR aroach, P can be obtained(see the Aendix), satisfying Pt P t P (9) Extending E.(9), we can deduce the state transforation euations of NK t as follows:

6 P t P t P t P t np t np t P t n P t,0,0,0, P t P,0 3,0 t P 3, t P t P t P t,0,0, P t P t P t P t,0 3,0 3,,0,0,0, n n n n,0 (0) P t P t P t P t P t 3P t P t P t P t, 3, 3,,0,0 3, 4, 4,,0 P t P t P t P t,,,,0 P t P t P t P t P t, 3, 3,,0, () np t n P, n n t P t P t,,,0 n n n P n, t P t P t P,,, t P, t P, i t i0 P t P t P t P t,,,, P, t 3P 3, t P 3, t P, t + P t, np n P n, t n, t n P t P,, t n n P n, t () P t P t P t P t,,,, P i, t i0 P t P t P t P t, 3, 3,, P, t P, i t i0 (3) np t n P t P t P n n P n, t P n, i t i0 n, n, n, n, t

7 r P t P t r P t P t r, r r, r r, r r, r r P, t P 3, t r r r r r P r 3, r t P r, r t r P t r, r np t n r P t r P t P t n, r n, r n, r n, r n P t n, r r (4) Let K be the steady state degree distribution [6-8], and be the robability distribution of K, that is, PK li P t (5) t i i, Cobining E.(5) with the state transforation Es.(0-4), we can obtain the degree distribution euations of RBDN for different. Case : 0 In this case, aing use of E.(4), we hae li P n, t N n t n n (6) So cobining the state transforation Es.(0-4), the degree distribution euations of RBDN can be written as 0,0 i i,0 i 0,0 i i,0 i i, i i, i i 3 i0 i N (7) r r r r r where

8 li PNK t, i, i (8) t For and,we can use the robability generating function ethod to directly calculate the degree distributions of RBDN. In the case, E.(7) can be silified to 0 0, (9) r r r r r Let the robability generating function be fro E. (9), we hae i G x i x, G i (0) i0 i0 x Gx Gx x x () Soling E. (), we obtain i / i / Gx e x e i jii! () Thus for, the degree distribution of RBDN is i / e i i!, 0 i / e, i i! (3) In the case,the degree distribution euations can be silified to 0, , (4) r r r r r

9 Using the sae ethod as for,the degree distribution of RBDN for is Different fro, i / e, 4 i i! i / e, 0 4 i i! i / e 4 i i! and, for 3 (5), fro E.(7), we can find that it is necessary to obtain i, before calculating the degree distribution of RBDN, in which other ethods are needed. Case : In this case, aing use of E.(5), the degree distribution euations of RBDN only deterined by the last ites of the state transforation Es.(0-4). Thus the degree distribution euations can be written as 0 0 r r r r r (6) i Constructing the robability generating function G x i x G hae,, fro E. (6), we i0 x Gx Gx x x (7) Soling E. (7), we get cx ce c ct G x t t e dt c (8) x x where c. Let

10 t y (9) so we hae cx ce x c cy G x y c y e dy 0 x ce e x c cx ce e x c i0 c cx c i x i ic cy 0 C y e dy j i c x i ic j C y dy j! 0 i0 j0 j c cx i i c i j i0 j0 j! i j c ce e C x j in n, i n c c cx j j i ce e x Cn C j0 n j i0 r c r j i ce x C C r0 j0 n j i0 c n ni i!!! in n, j i c c n n c n i r j n i r j (30) Case 3: Therefore the degree distribution is in n, ni j j c j i i c c n j0 n j n c i0 n i! j! (3) ce C C 0,,, In this case, aing use of E.(6) and the state transforation Es.(0-4). The degree distribution euations can be written as 0 0 r r r r r (3) We ay use the robability generating function aroach or the recursie ethod to sole the E.(3). The result is sae as that in Ref. [4].

11 j r! e 0 j j! r0 r! j r! e j0 j! r r! (33) 5. Siulation and Tail Characteristics Before discussing the tail characteristics of the degree distribution for RBDN, it is necessary to erify our theoretical results in Section 4. We do this by couter siulation. Figures -3 illustrate the exact solutions and siulation results of the degree distributions for different, where the horizontal and ertical ordinates denote the degree of nodes and the robability resectiely. Each siulation nuber is the aerage alue of 000 siulation results for t As shown in Fig. and, in the case of 0,,, the exact solutions atch erfectly with the siulation results and the correctness of our exact solutions can be erified es =0. cs = es =0. cs = es =0.3 cs = es = cs = Fig. Exact solutions (es) s. couter siulation (cs): the degree distributions of RBDN for 0, (All statistical errors are saller than 0-3 )

12 () es =0. cs =0. () es =0. cs = () es =0.3 cs = () es = cs = Fig. Exact solutions (es) s. couter siulation (cs): the degree distributions of RBDN for 0, (All statistical errors are saller than 0-3 ) Figures 3 roides the coarisons of exact solutions and couter siulation for the degree distributions of RBDN in the case of, 3. We ay find that the siulation results atch exact solutions ery well. () es =0.6 cs =0.6 () es =0.7 cs =0.7 () es =0.8 cs = () es =0.9 cs = Fig. 3 Exact solutions (es) s. couter siulation (cs):the degree distributions of RBDN for, 3 (All statistical errors are saller than 0-3 )

13 The tail characteristics of the degree distribution are iortant toics for colex networs. Currently, the literature on degree distribution discoers the ower-law tail for a scale-free networ [6], the Poisson tail for a sall-world networ [3], and the exonential tail for a growing exonential networ [8]. For the large, let r (34) If aroxiately follows a Poisson distribution,i.e. then we hae In other words, if ~ (35)! r, ln r ln (36) is subjected to a Poisson distribution,then the relationshi of ln r and of -. ln in the dual-logarith coordinate syste should be a straight line with sloe If is nearly subjected to an exonential distribution,i.e. ~ e (37) then r e (38) In other words, if aroxiately follows an exonential distribution, the relationshi of ln r and ln in the dual-logarith coordinate syste should be a straight line with sloe of 0. Thus we ay further exlore the tail characteristics of the degree distribution of RBDN. Fig. 4 illustrates the tails of the degree distribution for RBDN in the case of 0,. As shown in Fig.4, for different, when 0, the sloes of lines tend to be -. So exhibits Poisson tail.

14 0 - =0. =0. = =0.5 r() Fig. 4 The tails of the degree distribution of RBDN:, (Poisson tail) 0 0 =0.6 =0.7 =0.9 =0.95 r() Fig. 5 The tails of the degree distribution of RBDN:, 3 (Fro Poisson tail to exonential tail) In the case of changes with, as illustrated in Fig.5, the tail of the degree distribution also, exhibiting Poisson tail in the case close to (e.g. 0.7, 0.6 ). Howeer with the increasing of, Poisson tails disaear gradually. In the case 0.95, for

15 0 0, r tends to be a constant (for, r is a constant.),showing the tail of the degree distribution aroxiately exhibits an exonential tail as. 6. Conclusion In this aer, a rando birth-and-death networ (RBDN) odel is introduced, in which at each tie ste, a new node is added with robability or an old node is reoed fro the networ with robability. As a first ste in inestigating the roerty of RBDN, we hae exlored the networ size, the degree distribution and its tail characteristics. Exact solutions of degree distribution in the case of and 0 hae been calculated and coared with siulation results. We find that the tail of the degree distribution for RBDN exhibits a Poisson tail in the case of 0 and an exonential tail as aroaches to. For RBDN, this rototye inestigation could be deeloed further. First the relationshi between connectiity and should be considered. Second, other roerties lie aerage ath length and cluster coefficient need to be further exlored. Third, the dynaic behaior of RBDN needs further inestigation since it ay roide deeer understanding of eoling networs whose size aries oer tie. Acnowledgents This research is financially suorted by the National Natural Science Foundation of China (No ) and the China Scholarshi Council. Aendix: one-ste transition robability atrix P Using SPR, the one-ste transition robability atrix P has two ossibilities:. Add a node i. To be an isolated node, node connects to other networs at tie t, then the state of node turns fro n, to n, or n, n, and the one-ste transition robability is gien by P KV t,,, n, KV t n,, n,0 n n n n (39)

16 P KV t,,, n, n KV t n,, n,0 n n n n n (40) ii. Node does not connect to the new added node at tie t, then the state of node turns fro n, to n, and one-ste transition robability is gien by,, n P KV t n KV t n, n,0 n n,, n, n (4) iii. Node connects to the new added node at tie t, then the state of node turns fro n, to n, and one-ste transition robability is gien by,, P KV t n KV t n, n,0 n n,, n, n,, n P KV t n KV t n, n,0 n n,, n, n (4) (43). Delete a node Since any node in the networ with node ay be deleted with eual robability, we only need to coute the transition robability of nodes not being deleted. i. The degree of node is decreased by at tie t, Then the state of node turns fro n, to n,, and the one-ste transition robability is gien by,, P KV t n KV t n, n n,, n, n (44). The degree of node fro n, to n, P KV t,0,,0 n,0 KV t n,0, n n n (45) reains unchanged at tie t, Then the state of node, and the one-ste transition robability is gien by turns n P KV,,, t n, KV t n,, n n n n (46) References. A. L. Barábasi and R. Albert, Eergence of scaling in rando networs, Science 86, 509 (999).. R. Albert and A.L. Barabási, Statistical echanics of colex networs, Re. Mod. Phys. 74, 47 (00) 3. L.A. Adaic,B.A. Huberan,A.L. Barabasi, R. Albert,H. Jeong,G. Bianconi, Power-law distribution of the

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