Decay of Scale-Free Property in Birth-and-Death Network

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1 Decay of Scale-Free Proery n Brh-and-Deah Newor Xaojun Zhang *, Zheng He 2, Lez Rayan-Bacchus 3 School of Maheacal Scences, 2 School of Manageen and Econocs, Unversy of Elecronc Scence and Technology of Chna, Chengdu, P. R. Chna, Unversy of Wncheser, Wncheser, SO22 5HT, UK Absrac: Ths aer exlores he decay of he scale-free roery for a nd of brh-and-deah evolvng newors, n whch a each e se, a node s added wh robably and conneced wh old nodes by referenal aachen; or a node s randoly deleed fro he newor wh robably q=-. Eloyng he SPR-based Marov chan ehod, we calculae he degree dsrbuon of he roosed newor and dscuss s al feaures. Our resuls show ha he node deleon robably q has a srong ac on he ower-law al. As q 0, he ower-law roery s dsnc, whle wh he ncreasng of q o /2, he scale-free characer s gradually dsaeared. Ths ay ly for he real brh-and-deah newor, durng he eergen and growng sages, snce he robably of node deleon s very sall, he ower-law al of he degree dsrbuon s aaren. However, when he newor s n he aured sage, he deleon robably q wll be ncreenally ncreasng, ang he scale-free feaure decay gradually. Key words: Brh-and-deah newor, degree dsrbuon, scale-free roery, Marov chan, ower-law al

2 Decay of Scale-Free Feaure n Brh-and-Deah Newor. Inroducon In he real world, nodes n nearly all socal and anageral newors are nellgen agens le eole or enerrses whch have her lfe-cycles and decson caables. Faced wh he rad changes of exernal or nernal envronen, hese agens wll ae dfferen sraeges, ang frequen o ener or ex o a newor. Recenly hs nd of brh-and-deah newors has caugh uch aenon. Several odels have been develoed o descrbe varous evolvng newors and nvesgae her generc feaures [-7]. To be a sgnfcan roery, scale-free feaure [8,9] has been dscussed n hese brh-and-deah newors [0-6]. For exale, Moreno e al. [0] used a fber-bundle odel o sudy he nsably of a large scale-free newor; Sarshar [] suded an ad hoc newor, showng ha he referenal aachens for new nodes do no lead o a suffcenly heavy-aled degree dsrbuon; Moore e al. [4] furher oned ou ha he exonen of owerlaw dsrbuon ay dverge as he growh rae vanshes. Alhough all hese odels deal wh he deleon of nodes n he newor, hey a he sae e add nodes a each e se o ee he newor sze growng or unchanged. However, uch leraure has shown ha he sze of an nellgen newors le nnovaon dffuson newor and ndusral cluser newor wll exhbs S-shaed cuulave curve [7-22], ha s, he oal nuber of nodes wll frs exerence very slow and long ncreasng rocess and suddenly grow very fas n a shor erod of e, hen ener no a coaravely sable sage, followed by gradually declnng hase. Consderng he S-shae rules n any nellgen newors, n hs aer we consruc a brh-and-deah newor odel wh referenal aachen as well as rando node deleon o exane he scale-free roery n her evolvng rocesses. Dfferen fro revous newor odels [,4-6], our odel allows for flucuaon n sze, granng addonal reals. Usng he SPR ehod [23], we rovde he exac soluons of degree dsrbuon. Our heorecal and sulaon resuls sugges ha he scale-free feaure wll gradually dsaear wh he ncreasng robably of node deleon. Ths fndng exlans why he scale-free newor s no as coon as we execed before n he real socey even hough here exss 2

3 referenal aachen evolvng rules n hese newors. 2. Newor Model As shown n Fgure, for any nellgen newors, durng he forave and growng erods, he robably of node addon s uch hgher han ha of node deleng, whle n he aury and declne erod, he robably of node deleon s gradually equal o and fnally exceed ha of node addon. Thus he newor sze of any evolvng nellgen newors wll exhb S-shaed evenually. aury erod Newor Sze forave erod growh erod Fg. Sech of newor sze for evolvng newor: S-shaed curve In order o exlore he ower-law roery n he newors wh S-shaed evolvng ah, n hs secon, we frs roose a odel o descrbe hs nd of newor, whch s characerzed by referenal aachen as well as rando node deleon. Ths brh-and-deah newor odel s as follows: () The nal newor s an solaed node; () A each un of e, add a new node o he newor wh robably q and connec wh old nodes by referenal aachen, ha s, he robably ha he new node connec wh old node deends on he degree of node,.e. Or randoly delee a node fro he newor wh robably Noe: q0 q 2. () 3

4 (a) In order o connec he node whch degree s 0, n Eq.(), we use. (b) For he convenence of research, here we assue he low-bound of he newor sze n0, ha s, f he nuber of nodes n he newor s n 0 a e, hen a e, we only add a new node o he newor wh robably newor. and connec o he old node n he (c) If a e, a new node s added o he newor and he newor sze s less han, hen he new node s conneced wh all old nodes. (d) If a e, a node s deleed, hen all he edges ncden o he reoved node are also reoved fro he newor. Thus he degree of s neghbors decreases by one. 3. Exac Soluon of he Degree Dsrbuon Accordng o he sochasc rocess rules (SPR) ehod n [23],we use n, o descrbe he sae of node v, where n s he nuber of nodes n he newor ha conans v, and s he degree of node v. Le NK be he sae of node v a e, The sochasc rocess NK, 0 s an nhoogeneous Marov chan, and he sae sace,,,0. Le E n n n P be he one-se ranson robably arx of NK, 0 a e. n,, n2, 2 P (2) Usng SPR, he one-se ranson robably arx P roduces wo cases: A. Add a node and ln o he old nodes by referenal aachen. () The one-se ranson robably whch, urns o n, or n, n s gven by n P NK,,, n, NK n,, n,0 n n n n P NK,,, n, n NK n,, n,0 n n n n n () The one-se ranson robably whch, urns o n, s gven by n (3) (4) 4

5 P NK n NK n,, n,, n, PNK n, n n n * PNK n,, n,0 n (5) () The one-se ranson robably whch urns o s gven by n, n, P NK n NK n,, n,, n, PNK n, n * PNK n,, n,0 n n P NK,,, n, NK n,, n,0 n n n n B. Delee a node randoly (6) (7) (v) The one-se ranson robably whch urns o s gven by P NK,,, n, NK n, q, n n n n P NK n,0 NK n,0 q, n,0,,0 (9) n n (v) The one-se ranson robably whch, urns o n, s gven by n, n, n (8) Le n P NK,,, n, NK n, q, n n n n P 0 and P be he robably vecors of (0) NK 0 and NK resecvely, ha s, n, P P NK n () he nal robably vecor P 0 sasfes P NK 0,0 (2) We have P P P (3) Cobnng Eq.(3) and Eqs. (3-0),we ay derve he sae ransfer equaons for he roosed brh-and-deah newor(see Aendx). Le K be he average degree dsrbuon a e. Then we have 5

6 Thus he seady sae degree dsrbuon Noe n he case of,, (4) P K P NK P K [9,24,25] s gven by l P K l P NK, (5) 0 q 2, n l l P NK, (6) n So s deerned by he las ers of each equaon n he sae ransfer equaons (32-36). Snce l * P NK n, 2 2 q (7) n ang he ls of boh sdes n Eqs. (32-36), we ay oban he followng dsrbuon equaons 0 q 2 2 q 2q q q q q q 2q (8) r r rq r r q r r 2 2 Case : q=0 By Eq.(8), he degree dsrbuon can be obaned drecly as follows: (9) 6

7 Case 2: 0<q</2 In hs case, we consruc he robably generang funcon Fro Eq. (8), we have Gx x, G (20) 0 x q x G x x Gx x 2 2 (2) (a) If 2 q,fro Eq. (2), we have x c c 2 Gx G x 2 q x x 2 q x x (22) where c 2 2 q. The soluon of Eq. (22) under he condon G s as follows G x 2 x 2 q x c 2 q 2 c x c q d (23) c so we have 2 q 2 c 0 c 2 q 0 G0 d (24) 2 q Slar ehod can be used o oban he followng resuls. (b) If 2 q, he soluon of Eq. (2) under he condon G and G x (c) If 2 2 x 2 c c 2 q c x c c s as follows: d (25) q x 2 0 G0 c 0 c q, usng Eq.(2), we have 2 q d (26) 2 q c x Gx Gx x q x q x 2 2 (27) The soluon of Eq. (27) under he condon G s as follows: 7

8 q x q e G x e d x (28) x So G e e d (29) 0 q q 0 0 Cobnng (a)-(c) and Eq.(8), we can oban he degree dsrbuon as follows: q q 0 c 2 2 c q 2 2 q d, 0, 2 c q 0 c 2q, 0, 2 e e d q 2 q d, 0, 2 c q (30) 0 c 2 q a 0, where he sequence aj, j 0,,2, sasfes a0 q 2 a 2 j j jqa 2 j a 2 j a j j 0, j q qa a a q (3) 4. Decay of Scale-Free Proery Ths secon dscusses how he scale-free feaure changes wh he robably of node deleon. Fgures 2-4 exhb he changes of degree dsrbuon al wh he decreasng of node deleon robably q, where x-axs and y-axs reresens he degree of a node and s robably resecvely. As shown n Fg. 2, n he case of q 0, he degree dsrbuon al s a sragh lne, showng a srong scale-free roery. 8

9 Π () q=0, =2 q=0, =4 q=0, =6 q=0, = Fg.2 The degree dsrbuon for q=0 Fg.3 exhbs he change of degree dsrbuon wh q for dfferen. As shown n Fgs. 3.a- 3.d, wh he ncreasng q, he degree dsrbuon gradually bend downwards, showng he dsaear of scale-free. In addon, we also exlore he al shae of degree dsrbuon n he case q 0.5 he al of degree dsrbuons gradually exhb exonenal dsrbuon wh he ncreasng of q. Π () q=0.,=2 0-8 q=0.,=4 q=0.,=6 0-0 q=0.,= Fg. 3.a: The degree dsrbuons for q=0. Π () q=0.2,=2 0-8 q=0.2,=4 q=0.2,=6 0-0 q=0.2,= Fg. 3.b: The degree dsrbuons q=0.2 Π () q=0.3,=2 q=0.3,=4 q=0.3,=6 q=0.3,= Fg 3.c: The degree dsrbuons for q=0.3 Π () q=0.4,=2 0-8 q=0.4,=4 q=0.4,=6 q=0.4,= Fg. 3.d: The degree dsrbuons for q=0.4 9

10 Fg.3: The degree dsrbuons for q</2 Fg.4 llusraes he degree dsrbuon changes wh q n he case of =4. Fro Fg.4 we can fnd wh he ncreasng of q, he al of he degree dsrbuon gradually curve downward, showng he decay of he scale-free characerscs Π () =4,q=0 =4,q=0. =4,q=0.2 =4,q=0.3 =4,q= Fg.4 The degree dsrbuons for dfferen q (=4) Cobnng Fgs.-4, we ay fnd ha he scale-free feaure ay only aears n he forave and fas growng hases whle durng he aured or declnng sages, wll show an aaren decay of scale-free feaure wh he ncreasng robably of node deleon. 5. Concluson Ths aer dscusses he decay of scale-free feaure whch s oular n any nellgen newors. We frs roose a brh-and-deah newor odel wh referenal aachen and hen calculae s degree dsrbuons for varous node deleon robables. Sulaon resuls exhb ha coared o referenal aachen, he node deleon robably q also has srong ac on he scale-free roery. In he case q 0, he ower law roery s dsnc. However wh he ncreasng of q, he ower-law dsrbuon s gradually dsaeared. Ths sudy shows ha dese he referenal aachen for an evolvng newor ee unchanged, n real world, he scale-free feaure ay also gradually decay due o he deah of nodes n hs evolvng newor. Consderng he S-shaed newor sze for any evolvng newor, our fndngs ay ly ha durng he 0

11 eergen and growng sage, snce he robably of node deleon s very sall, he ower law of degree dsrbuon al s aaren. However, when he newor s n he aured sage, he deleon robably q wll be ncreenally ncreasng, ang he scale-free feaure decay gradually. Acnowledgens Ths research s fnancally suored by he Naonal Naural Scence Foundaon of Chna (No ) and he Chna Scholarsh Councl. Aendx: Sae ransfer equaons P qp qp qp 2 2,0,0 2,0 2, P 2,0 qp 3,0 qp 3, P qp qp,0 2,0 2, 2 P 2 2,0 qp 3,0 qp 3, PKV, P,0 * PKV, PKV n, np,0 n nqp qp n P n,0 n, n n * P KV n,,0 2P qp 2qP P P 2, 3, 3,2,0,0 (32) P qp 2qP P, 2, 2,2,0 PKV, 2 P 2, qp 3, 2qP 3,2 P,0 *, P KV 2PKV, P, * PKV, PKV n, np, n qp, 2qP,2 P n n n n,0 *, P KV n 2PKV n, n P n, n * PKV n,

12 (33) 2 P qp qp,,, P, 2 P, 0 P 2qP qp P, 2, 2,, 2 PKV, 2 3 * PKV, P qp qp P PKV, P, * PKV, 2, 3, 3,, 2 PKV n, n * PKV n, PKV n, * PKV n, np n qp qp P n, n, n, n, 2 n P n, (34) P qp qp P, 2, 2,, P, 0 P KV 2 P 2qP qp P, * PKV, 2, 3, 3,, PKV, P * PKV, P,, 0 PKV n, np n qp qp P,,,, n n n n * P KV n, PKV n, n2 n P n, P n, n * PKV n, 0 (35) 2

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