: [6,13,16].,,,,, 3 3, 2, 3 3 :, 3, 2, (BA), :,,,, [9 15]. Barabási Albert,,, ( ),, :,,,,,,, ( ), : [17 24],, [17 24]. 1 1 :,, p, [16 24,3

Size: px

Start display at page:

Download ": [6,13,16].,,,,, 3 3, 2, 3 3 :, 3, 2, (BA), :,,,, [9 15]. Barabási Albert,,, ( ),, :,,,,,,, ( ), : [17 24],, [17 24]. 1 1 :,, p, [16 24,3"

册乌
7 years ago
Views:

1 38 6 Vol. 38 No ADVANCES IN MECHANICS Nov. 25, 2008 *, ,, 3 :, 3,,, 3,,,, 1 E. O. Wilson [1] :,,,......,, 3,. 1,, 1736, [2], [1], 1 1,.,. 2 Edos( ) Renyi, [3],, ER,, ER 40 20, 1984,,, 20, 1998 ER (Cornell) Watts Strogatz, (small world, SW) [4 10]., 1999 (Notre Dame) Barabási Albert (scale-free, SF) [11 14],, 2006 Barabási Newmann Watts [15],,,, Barabási 2006 von Neumann( ) 3,,, : , : ( ), ( ) ( ) fjq96@126.com

2 : [6,13,16].,,,,, 3 3, 2, 3 3 :, 3, 2, (BA), :,,,, [9 15]. Barabási Albert,,, ( ),, :,,,,,,, ( ), : [17 24],, [17 24]. 1 1 :,, p, [16 24,36 39] : (1) Π j i = k i / m, j (w ij = w ji ): w ji = k i / i k i s j = i w ij = 1. i j (2) BA,, m i, j i (fitness) j k j / Π i = η i k i η j k j (3) (2), (4) : (i) m, i Π n i = j sw i e dni/rc j sw j e dnj/rc, R c, d ni n i. (ii) (n, i) w 0 ( w 0 = 1), : w ij w ij + δw ij /s w i. (5) (BBV ).. :,, n i Πn i BBV = s i = s ns i s j s n s j j s i, s j (i,j), m j, w 0, i l ji i l w il w il + w il, l w il, w il = δw il /s i : 3, δ w 0. j

3 6 : 665 γ w = 2 + 1/δ; δ,, δ, γ = 2. s(k) Ak β, A w, β = 1, γ = γ s = (4δ + 3)/ (2δ + 1). 1 DM : (1), 1,, δ; (2), 1. YJBT, j, m m 0 ( ),, j, γ w = 2 + 2/δ γ = γ s 2+1/(δ+1), γ = 3. γ s m, 3 C r c. [17] [18] BBPV, BA,, m i, j i (fitness),,. [19] YJBT p, 1 p p=1 YJBT ; p=0,, γ s p, p, p=0. [20] AK, j i,,, 1 +, [21] BB,. : w ij w ij + δw ij /s w i.. [22] BBV, BB,,. i l., 3, δ w 0.. δ = 0 BA [23] TDE :,,.,. m n BBV., C ω, -. w > 1. [24]

4 (6) (TDE ) [24]. :, ;, { wij + 1, w pij, w ij w pij = s is j w ij, 1 w pij, s a s b a<b w ij i j, w ij 0.. m n BBV. w 0., γ s = 2 + m/(m + 2ω); s(k) k β, β > 1; C ω (7) ( ),.,. (8) (9) ( ),,,,.,. :,, :, ( ),,,,,,.,, , :.,,, ( ),,,,,,.,,,.,,,, (average path length, APL), (Average clustering coefficient, ACC)., Watts Strogatz,,, :,,,,,

5 6 : 667,, 3, 1,,,, 1 3, (harmonious unifying hybrid preferential model, HUHPM); (large unifying hybrid network model, LUHNM); (large unifying hybrid variable growing model, LUHVGM unifying hybrid network model variable speed growth, UHNMVSG). 4, 3, 1 3 ( 1 ) HUHPM; ( 2 ) LUHNM; ( 3 ) LUHVGM, DP A: ; RP A: ; GRA: ; HP A: ; RA: ; DA: : [25 32] 1 HUHPM, BA [11], (CIAE),,,, HUHPM, dr dr = d r = DA RA (1) d (determinatic attachment, DA); r (random attachment, RA), d r [0, + ],. HUHPM 3 : (i) dr 1/1, ; (ii) dr = 1/1, (, ) ; (iii) dr 1/1, ; (i) (iii) 1 HUHPM, dr. HUHPM ( ),, BA(Barabási-albert) BBV(barrat-barthélemy-vespignani)

6 TDE(traffic-driven evolution) ( ), HUHPM-BA, HUHPM-BBV HUHPM-TDE dr,,, ( ). HUHPM : (1) :,. (2) : dr,,,. (i) :, : k 1 > k 2 > > k m > > k n, m (ii) :, HUHPM, : BA BBV TDE, HUHPM-BA, HUHPM-BBV HUHPM-TDE, :,,,, : HUHPM,,, HUHPM,, 1 HUHPM : (1) HUHPM ( HUHPM- BA HUHPM-BBV HUHPM-TDE), 3, γ dr, log(dr), dr = 1/1, ( ). dr 1/1, γ 3, ; dr > 1/1, BA BBV, δ, γ log(dr) ; HUHPM-TDE, w < 1, γ log(dr) γ dr, HUHPM-BA HUHPM-BBV [26 28] / [ ( γba HUHPM = 1 ) ] A3 dr β + 1 = A 1 exp + A 4 A 2 γba HUHPM = / [ ( ) ] ) / A3 dr (4δ + A 1 exp + A 4 (2δ + 1) A 2 (2) (3) HUHPM-TDE γ dr w [26 28]. : HUHPM-BA, HUHPM-BBV ( HUHPM- TDE ), γ dr (δ,w) m,, (d/r, δ, w, χ), ( ),,, (2) HUHPM, L C, [27,28]. C dr,, 3, (3) [29]. ( I) HUHPM, ( II) HUHPM HUHPM dr,., (4) HUHPM dr,

7 6 : 669 [31,32]., 1 HUHPM,,,, 3 (1) γ dr,,, (2),, dr,, (3) : [33 40] : 1 HUHPM,, :,,,,,,,,,,, HUHPM LUHNM [33 36] ( 1 ), : dr 2 : gr gr = g r = GRA RA fd fd = HP A DA (4) (5), : DA = HP A + DP A; RA = GRA + RP A, DA = f + d, RA = g + r.,,,, 2 dr 1/1, fd 1/1 gr 1/1, 2 1, 3, : (i) gr 1/1, ( ) ; (ii) gr = 1/1, ; (iii) gr 1/1, ; (i) (iii)., : (i) fd 1/1, ( ) ; (ii) fd = 1/1, ; (iii) fd 1/1, ;, (i) (iii)., 2 3 (dr, gr, fd), 2 1, 3 (dr, gr, fd)., i Π k i, dr i,, gr BA, (1 gr) ER ;,, fd, (1 fd), Π k i = r (1 gr)k i + gr d + r Σ j [(1 gr)k j + gr] + [ (1 fd) d d + r [[ ]] [ ki k max ] + fd [ kmin ]] (6) k i, t, N = m 0 + t, mt, (1) (4) (5) 3 (6),,, 8 : (i)fd=0/1 gr=0/1: (HUHPM); (ii) fd=0/1, gr :

8 (determinastic preferential attachment, DPA), (iii) fd = 1/1, gr : (helping preferential attachment, HPA); (iv) gr = 1/1, fd : (general random attachment, GRA), (v) gr=0/1: (randomness preferential attachment, RPA),. (vi) fd=0/0 gr = 0/1: BA, BBV TDE ; (vii) fd =0/0 gr = 1/0: ER (viii) fd 0 gr = 0/0:, LUHNM 3,,, LUHNM, 1,, - ( ) r c (assortative coefficient), dr 2 - r c (dr, m) (a) gr, r c log(dr) (b)(c) [41], fd = 1/1 2 : dr r c : BA r c 2 dr=0/1 gr=0/1 ; ER r c 2 dr=0/1 gr = 1/1 ; HUHPM 2 gr=0/0 Newman ER r c [42], Callaway (m=1) ER, [35], r c 0.35; LUHNM, ER r c 0.2, dr, gr, r c gr fd, r c,, 3 2 r c [35,41 51]. r c,, 2 (a) r c (dr, m) ; (b) (c) gr, - log(dr) [41]. fd = 1/1(), m = 3, N = 1 000

9 6 : 671, r c (0, 1) ( 1, 0) ; 2 3, r c, [41], r c, [33 37] r c,, 2 r c r c ( 1, 1) (LUHNM), 3 [35] [ 1, 0] [0, 1] 1), [41] 0 BA ER [42] [0, 0.5) Callaway, [43] [44] [45] ( 1, 0], [46] [47] ( 0.6, 0.4), [48] ( 0.21, 0.41) 22, N, [49], N= Penna, N = 500 [50] cul.arxiv.org [51] 1), 2 LUHNM : 3,,, 3 4, LUHNM-BBV r c dr gr fd, fd gr, δ=3. 3 : LUHNM-BBV r c,. dr > 1/1, fd 0.99/1 ( ), gr,., r c, fd gr, dr r c ;, gr =1/1 0.8/1, r c dr, dr 1/1( ) r c gr dr 1/10( ), r c, gr, r c 3 fd LUHNM- BBV r c (dr, gr)

10 dr r c fd gr, 2, (dr, gr, fd), 1. 3, 3, - ()r c ( 1, 1),, 3,,, r c.,, ,,,,,, LUHNM,, (fd 90% ),, r c, ( ) :, 2 LUHNM-BBV, -,,,,,,, 3,, :,,,,, ( ),,, [32 54]., 2, 3 : LUHVSG [55 57], 1,, dr 2 (fd, gr), vg vg = DV G RV G (7), [52 57] m(t) = p(n(t)) α (8) m(t) t, N(t) t, p, p(t), 0 <

11 6 : 673 p(t) < 1, ;α, α (α ) (0 < α < 1) (α 1), 3,, LUHVGS :, (8) p, p vg,, 3, 3, P (k) SF P (k) = k =k p(k ), k,,. p(k), k k 3 fd=0/1 gr=0/1 3 α P (k), (a) α=0( ); (b) α=0.3 (c) α=0.6, k, 3 (α=0), dr, P (k) : P (k) k γ,. (8) α=0.3 p(t), 5(b) dr :, (stretched), 3 [58] P (k) = e (k/k0)c (9) k 0, c, c 1 c 2. c = 1. c 0. 0 < c < 1 (, SF ) (SED) c,, SF. : α (α=0.3), dr 3 : (dr = 4/1, 49/1) (dr = 1/1) (dr = 1/49, 1/4), P (k), α P (k) α ( 0.6, 0.9), P (k), 5(b)(c). 5 3 γ c dr, 3 dr 5, γ dr, c dr ( ), α ; α, α. dr=1/1. fd=0.4/1( ) gr=1/1( ),, 5 5 fd=0/1 gr=0/1, P (k) k. γ c dr, 3

12 P (k) vg 3 vg 6 (dr=1/1, fd=0/1 gr=0/1) vg P (k) :, (Gaussian),, 4 3, dr vg 1/1,, (dr vg) 1/1 P (x) = y 0 + A w 2 π/2 e 2((x xc)/w) (10) y 0, x c, w A 6 (dr=1/1, fd=0/1 gr=0/1) P (k) vg, 1 w vg 3 6 3, (dr = 1/1, fd = 0/1 gr = 0/1),, 6 (dr=1/1, fd=0/1 gr=0/1) v g P (k) k 1 w v g 3 6. vg dr = 1/1, fd = 0/1, gr = 0/1, 1 2 c y 0 x c ω A y 0 x c ω A 1/ / / c = / c = / c = C α 3 C α. 7 C α. : fd=0/1 gr=0/1, C α : dr >> 1/1 (, 7(a)), α=0.3 C,, 1; dr << 1/1 (, 7(b)), α, C,, α 1; dr = 1/1(), α, α 0.3, C, α=0.5, ; α, C 1. : 3 C [0, 1], C,, 3 α, r c 3, r c 8 fd=1/1 gr=0/1 r c α

13 第 6 期方锦清等 : 网络科学中统一混合理论模型的若干研究进展 675 图 7 在对应图 5 相同情形 (固定 f d=0/1 和 gr=0/1) 下, 群聚系数 C 与混合比及变速指数 α 之间的复杂关系采用式 (8) 中确定性增长方式 (p(t) 为常数), 当工作在 dr >> 1/1 模式时, rc 出现一个最高波峰对应于 α = 之间, 随着 α 的增加, rc 变小, 此起彼伏, 变化不剧烈. 而随着 α 的增加 rc 出现非线性变化, 当工作在 dr < 1/1(随机性占主导) 模式时, 在 α=0.3 之前 rc 明显上升, 而后稍微下降; 当工作在 dr 1/1(确定性占主导) 模式情形, rc 一直在上升. 对于式 (8) 中随机性增长方式, 基本结果与图 8 的结果类似. 对于有权统一混合变速增长网络情形, rc 与混合比关系更加复杂, 即使在一些特殊情形下, 网络特性 rc 与 4 个混合比 (dr, f d, gr, vg) 之间存在着复杂的非线性关系 (见图 9), 主要有以下几点: (1) 当混合比 dr 为随机性占主导 (dr=1/49) 工作模式时, 网络特性随 vg 变化不明显, 有时趋于常数, 例如, 对于混合比 f d=0/1 和 gr=1/1 情形网络特性就变化不大. 图 8 第 3 曲中对于无权网络, 相称性系数 rc 与混合比及变速指数 α 之间的三维关系图

14 (2) dr (dr=49/1), ; gr vg. (3) vg, vg 1/1 r c C, ; (4) r c,, fd<0.9/1, (8) r c, dr, r c 1; fd 1/1, (8) r c, dr, r c, 4,, 9 r c (dr, fd, gr, vg), α = 0.5 4, 3 3,, 3,, [59 62],,,,,, 4,. 3, 3 ; [63], 2008 :,,,,,,

15 6 : 677., : ( ) [63].,,,, 3,,, 4,,,,,,,,,,, 1 Wilson E O. Consilience Knopf. New York, (Leonhard Euler, ) : www2.zzu.edu.cn/math/classes/2003/y1/oula.htm., :, Erdos P, Renyi A. On the evolution of random graphs. Publ Math Inst Hung Acad Aci, 1960, 5: Watts D J, Strogatz S H. Collective dynamics of smallworld networks. Nature, 1998, 393: Watts D J. Six Degrees: The Science of a Connected Age. New York: W. W. Norton & Company, Watts D J. The new science of networks. Annual Review of Sociology, 2004, 30: Watts D J. Small Worlds: The Dynamics of Networks between Order and Randomness. New Jersey: Princeton University Press, Newman M E J, Moore C, Watts D J. Mean-field solution of the small-world network model. Phys Rev Lett, 2000, 84: Newman M E J. Models of the small world. J Stat Phys, 2000, 101: Buchanan, Nexus M. Small Worlds and the Groundbreaking Science of Networks. New York: W W Norton and Company, Barabási A L, Albert R. Emergence of Scaling in Random Networks. Science, 1999, 286: Strogatz S H. Exploring complex networks. Nature, 2001, 410(8): Barabási A L. The New Science of Networks Cambridge: Prerseus, Albert R, Barabási A L. Statistical mechanics of complex networks. Rev Mod Phys, 2002, 74: Newman M, Barabási A L, Watts D J. The Structure and Dynamics of Networks. Princeton: Princeton University Press, Newman M E J. The structure and function of complex networks. SIAM Review, 2003, 45: Dorogovtsev S, Mendes J. Minimal models of weighted scale-free networks. Arxiv preprint, 2004, cond-mat/ Yook S H, Jeong H, Barabási A L, Tu Y. Weighted evolving networks. Phys Rev Lett, 2001, 86(25): Bianconi G, Barabási A L. Competition and multiscaling in evolving networks. Eur Phys Lett, 2001, 54: Zheng D, Trimper S, Zheng B, Hui P M. Stochastic weight assignments. Phys Rev E, 2003, 67: Antal P L. Krapivsky, Weight-driven growing networks. Phys Rev E, 2005, 71: Barrat A, Barthélemy M, Pastor-Satorras R, et al. The architecture of complex weighted networks. Proc Natl Acad Sci USA, 2004, 101(11): Barrat A, Barthélemy M, Vespignani A A. Weighted evolving networks: Coupling topology and weight dynamics. Phys Rev Lett, 2004, 92: Wang W X, Wang B H H, Hu B, et al. A weighted complex network model driven by traffic flow. Phys Rev Lett, 2005, 94: Fang J Q, Liang Y. Topological properties and transition features generated by a new hybrid preferential model. Chin Phys Lett, 2005, 22: Jin Q F, Qiao B, Yong L. Toward a harmonious unifying hybrid model for any evolving complex networks. Advances in Complex Systems, 2007, 10(2): ,, G, 2007, 3(2): Fang J Q, Bi Q, Li Y, et al. Sensitivity of exponents of three-power-laws to hybrid ratio in weighted HUHPM. Chi Phys Lett, 2007, 24(1): Lu X B, Wang X F, Li X, Fang J Q. Topological transition features and synchronizability of a weighted hybrid preferential network. Physica A, 2006, 370: Fang J Q, Bi Q, Li Y, et al. A harmonious unifying preferential network model and its universal properties for complex dynamical network. Science in China Series G, 2007, 50(3): Li Y, Fang J Q, Bi Q, Liu Q. Entropy characteristic on harmonious unifying hybrid preferential networks. Entropy, 2007, 9: Bi Q, Fang J Q. Entropy and HUHPM approach for complex networks. Physica A, 2007, 383: Fang J Q, Bi Q, Li Y. From a harmonious unifying hybrid preferential model toward a large unifying hybrid network model. International Journal of Modern Physics B, 2007, 21(30): Fang J Q, Bi Q, Li Y. Advances in theoretical models of network science. Front Phys China, 2007, 1: ,,, 2007, 25(11): , 2006, 24(12): , 2007, 17(7): ,, ( )., 2007, 27(3): ,, ( )., 2007, 27(4): Fang J Q. Theoretical research progress in complexity of complex dynamical networks. Progress in Nature Science, 2007, 17(7): Xul vi-brunet R, Sokolov I M. Reshuffling scale-free networks: From random to assortative. Phys Rev E, 2004, 70(6):

16 Newmanm E J. Assortative mixing in networks. Phys Rev Lett, 2002, 89(20): Newmanm M E J, Park J. Why social networks are different from other types of networks. Phys Rev E, 2003, 68(3): Soon-hyung Y, Filippo R, Hildegard M O. Self-similar scale-free networks and disassortativity. Phys Rev E, 2005, 72(4): Sumiyoshi A, Norikazu S. Complex earthquake networks: Hierarchical organization and assortative mixing. Phys Rev E, 2006, 74(2): Wang W X, Wang B H, Hu B, et al. General dynamics of topology and traffic on weighted technological networks. Phys Rev Lett, 2005, 94(18): Grabowski A, Kosinski R A. Evolution of a social network: The role of cultural diversity. Phys Rev E, 2006, 73(1): Wang W X, Hu B, Wang B H, et al. Mutual attraction model for both assortative and disassortative weighted networks. Phys Rev E, 2006, 73(1): Julian S, Janusz A H. Statistical analysis of 22 public transport networks in Poland. Phys Rev E, 2005, 72(4): Li C G, Philip K M. Complex networks generated by the Penna bit-string model: Emergence of small-world and assortative mixing. Phys Rev E, 2005, 72(4): Michele C, Guido C, Luciano P. Assortative model for social networks. Phys Rev E, 2004, 70(3): Mattick J S, Gagen M J. Accelerating networks. Science, 2005, 307: Gagen G M, Mattick J S. Accelerating, hyperaccelerating and decelerating probabilistic networks. Phys Rev E, 2005, 72: Sen P. Accelerated growth in outgoing links in evolving networks: Deterministic versus stochastic picture. Phys Rev E, 2004, 69: Fang J Q. Evolution features of large unifying hybrid network model with a variable growing speeds. In: The 4th International Workshop Hangzhou 2007 on Simulational Physics, Hangzhou, Zhejiang, Fang J Q. Some progresses in theoretical model of nonlinear dynamical complex networks. In: 2008 National Physics Conference, ,Nanjing, Fang J Q, Li Y. Large unified hybrid network model with variable growth. Chi Phys Lett, Laherère J, Sornette D. Stretched exponential distributions in nature and economy: fat tail with characteristic scales. Eur Phys J B, 1998, 2: ,,, 2008, 5(4) 60.., :,, : :,, , 2008, 1: ADVANCES IN UNIFIED HYBRID THEORETICAL MODEL OF NETWORK SCIENCE * FANG Jinqing LI Yong China Institute of Atomic Energy, Beijing , China Abstract Theoretical model has been one of the most significant issues in the network sciences. First of all, three milestones in this area are reviewed as breakthroughs in theoretical models, and progress in weighted evolution network model is presented. In order to describe the determinist and randomness of the harmony and unity of the real world and to mirror overall property of the real-world networks, the unified hybrid network theoretical frame, which formed the so-called hybrid network theoretical model trilogy, are reviewed and summarized in the main part of this article: the first of the trilogy is a harmonious unification hybrid preferential model (HUHPM), the second the large unification hybrid network model (LUHNM) and the third the large unified hybrid variable growth model (LUHVGM). Main features and topological properties of the hybrid network trilogy are then summarized, and diversity and complexity of the complex networks are revealed as well as their complicated transition relationship. Possible applications of these theories to actual networks are finally mentioned. Keywords network science, unified hybrid network theoretical model trilogy, complexity, universality, smallworld, scale-free The project supported by the National Nature Science Foundation of China ( , ) and Grand Theory Physics Special Project of NSFC ( ) fjq96@126.com

[6,13,16].,,,,, 3 3, 2, 3 :,, 2, (BA), :,,,, [9 15]. Baraási Albert,,, ( ),, :,,,,,,, ( ), : [17 24],, [17 24]. 1 1 :,, p, : (1) [18], k i Π

[6,13,16].,,,,, 3 3, 2, 3 :,, 2, (BA), :,,,, [9 15]. Baraási Albert,,, ( ),, :,,,,,,, ( ), : [17 24],, [17 24]. 1 1 :,, p, : (1) [18], k i Π 38 6 Vol. 38 No. 6 2008 11 25 ADVANCES IN MECHANICS Nov. 25, 2008 *, 102413 3,, :,,,,,,,, 1 E. O. Wilson [1] :,,,......,, 3,. 1,, 1736, [2], [1], 1 1,.,. 2 Edos( ) Renyi, 20 50 60 [3],, ER,, ER 40 20,