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, 1984,,, 20, 1998 ER (Cornell) Watts Strogatz, (small world, SW) [4 10]., 1999 (Notre Dame) Barabasi Albert (scale-free, SF) [11 14],, 2006 Barabasi M. Newmann D. J. Watts [15],,,, Barabasi 2006 von Neumann( ) 3,,, : : 2008-06-30, : 2008-07-10 (70431002) (10647001) E-mail: fjq96@126.com
2 2008 38 [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 j i =, k l m, j k i (w ij = w ji ): w ji = i k i s j = i w ij = 1. l (2) [19], BA,, m i, j i (fitness) : Π i = η ik i l η lk l (3) ; (2), (4) [22] : (a) : m, i : Π n i = sw i e dni/rc j sw j e dnj/rc, r c, d ni n i. (b) (n, i) w 0 ( w 0 = 1), : w ij w ij + δ w ij s w i (5) (BBV ) [23] : :,, n i n i = s i = s ns i s j s n s j Π BBV 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 = δ w il s i : 3, δ w 0. j
6 : 3 γ w = 2 + 1/δ; δ,, δ, γ = 2. 1 DM : (1), 3 [17] 1,,, δ; (2) γ w = 2 + 2/δ C, 1. γ = γ s 2+1/(δ+1) r c. YJBT, j, m m 0 ( ),, j., γ = 3. γ s m,. [18] BB, 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 +, 3 2.. [21] BB, : w ij w ij + δ w 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 2008 38 s(k) Ak β, A w, β = 1, γ = γ s = (4δ + 3)/(2δ + 1). (6) (TDE ) [24]. :, ;, { wij + 1, wp ij, w ij w pij = s is j. w ij, 1 wp ij, 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) ( ),,,,.,. :,, :, ( ),,,,,,.,,. 3, :.,,, ( ),,,,,,.,,,.,,,, (average path length, APL), (average clustering coefficient, ACC)., Watts Strogatz,,, :,,,,
6 : 5,,,, 1,,,, 1, (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,, 1 ( 1 ) HUHPM; ( 2 ) LUHNM; ( 3 ) LUHVGM 3.1 1 : [25 32] 1 HUHPM, BA [ [, (CIAE),,,, HUHPM, dr dr = d r = (DA) (RA) d (determinatic attachment, DA); r (random attachment, RA), d r [0, + ], HUHPM 3 : (a) dr 1/1, ; (b) dr = 1/1, (, ) ; (c) dr 1/1, ; (a) (b). 1 HUHPM, dr. HUHPM ( ),, BA(barabasi-albert) BBV(barrat-barthélemy-vespignani)
6 2008 38 TDE(traffic-driven evolution) ( ), HUHPM-BA, HUHPM-BBV HUHPM-TDE dr,,, ( ). HUHPM : (1) :,. (2) : dr,, (a) :, : k 1 > k 2 >... k m >... > k n, m. (b) :, 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(d/r), dr = 1/1, ( ). dr 1/1, γ 3, ; d/r > 1/1, BA BBV, δ, γ log(d/r) ; HUHPM-TDE, w < 1, γ log(dr) γ dr, HUHPM-BA HUHPM-BBV [26 28] BA = 1 β + 1 = A [ 1 (d/r ) ] A3 + A 4 (2) exp γ HUHPM γ HUHPM BA = A 2 A 1 4δ + [ (d/r ) ] A3 + A 4 exp A 2 2δ + 1 (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, [31,32].
6 : 7, 1 HUHPM,,,, 3 (1) γ dr,,, (2),, dr,, (3) 3.2 2 : [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, : (a) gr 1/1, ( ) ; (b) gr = 1/1, ; (c) gr 1/1, ; (a) (b)., : (a) fd 1/1, ( ) ; (b) fd = 1/1, ; (c) fd 1/1, ;, (a) (b), 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] + [ d [[ ki ]] [[ ]]] (6) kmin (1 fd) + fd d + r k max k i [[ ]], t, N = m 0 + t, mt, (4)(5)3 (6),,, 8 : (a)fd=0/1 gr=0/1: (HUHPM); (b) fd=0/1, gr :
8 2008 38 (determinastic preferential attachment, DPA), (c) fd = 1/1, gr : (helping preferential attachment, HPA); (d) gr = 1/1, fd : (general random attachment, GRA), (e) gr=0/1: (randomness preferential attachment, RPA),. (f) fd=0/0 gr = 0/1: BA, BBV TDE ; (g) fd =0/0 gr = 1/0: ER (h) fd 0 gr = 0/0:, LUHNM 3,,, LUHNM, 1,, - r c ) (assortative coefficient), dr 2 (a) - ( ) r c (dr, m) (b)(c) gr, r c log(dr), 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 [43 51] r c,, 2 (a) r c (dr, m) ; (b) (c) gr, - log(dr) [41]. fd = 1/1( ), m = 3, N = 100
6 : 9, r c (0, 1) ( 1, 0) ; 2 3, r c, [41], r c, r c,, 2 r c r c ( 1, 1) (LUHNM), 3 [35] [ 1, 0] [0, 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=500 0.495 2 Penna, N=500 [50] cul.arxiv.org [51] 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 2008 38 4 3 dr r c fd gr, 2, (dr, gr, fd), 1. 3, 3, - ( )r c ( 1, 1),, 3,,, r c,, 2008 5 12,,,,,, LUHNM,, (fd 90% ),, r c, ( ), :, 2 LUHNM-BBV, -,.,,,,,, 3,, 3.3 3 :,,,,, ( ),,, [32 54]., 2, 3 : LUHVSG [55 57], 1,, dr 2 (fd, gr), vg vg = DV G (7) RV G DV G(determinastic variable growing) ; RV G(random variable growing), [52 57] m(t) = p(n(t)) α (8) m(t) t, N(t) t, p(t)
6 : 11, 0 < p(t) < 1, ;α, α (α ) (0 < α < 1) (α > 0), 3,, LUHVGS :, (8) p, p vg,, 3, 3, 3.3.1 P (k) SF P (k) = p(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) k,, 3. [58] k P (k) = be ( ) k c 0 (9) b 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 k, 3 k 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. dr, 3
12 2008 38 3.3.2 P (k) vg 3 vg 6 (dr=1/1, fd=0/1 gr=0/1) vg P (k) :, (Gaussian), P (x) = y 0 + A π w 2 ( ) 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),,, 4 3, dr vg 1/1,, (dr vg) 1/1 6 (dr=1/1, fd=0/1 gr=0/1) P (k) vg. 1 w vg 3 6. vg dr = 1/1 fd = 0/1 gr = 0/1 1 1 c y 0 x c ω A y 0 x c ω A 1/49 0.98 36.67 11.43 13.72 0.04 2 198.40 1 945.43 290.42 1/4 0.94 38.27 16.12 19.03 0.05 1 993.96 2 238.00 267.4 1/1 0.93 45.69 30.66 35.49 c = 0.89 4/1 0.94 38.76 16.88 20.13 c = 2.13 49/1 0.98 37.03 11.98 14.55 c = 2.51 3.3.3 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,. 3.3.4 r c 3, r c 8 fd=1/1 gr=0/1 r c α
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 13 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 = 0.3 0.4 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 C 出现 非线性变化, 当工作 dr < 1/1(随机性占主导) 模式 时, 在 α=0.3 之前 C 明显上升, 而后稍微下降; 当 工作 dr 1/1(确定性占主导) 模式情形, C 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 2008 38 (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) (a) fd = 0/1, gr = 0/1; (b) fd = 0/1, gr = 1/1; (c) fd = 1/1, gr = 0/1; (d) fd = 1/1, gr =1/1. α=0.5 4, 3 3,, 3,, [59 62],,,,,, 4,. 3, 3 ;. [63], 2008 :,,,,
6 : 15,,, : ( ) [63].,,,,,,,,, 1 Wilson E O. Consilience Knopf. New York, P85, 1998 2 (Leonhard Euler,1707-1783) : http:// www2.zzu.edu.cn/math/classes/2003/y1/oula.htm, :, 2002 3 Erdos P, Renyi A. On the evolution of random graphs. Publ Math Inst Hung Acad Aci, 1960, 5: 17 61 4 Watts D J, Strogatz S H. Collective dynamics of smallworld networks. Nature, 1998, 393: 440 442 5 Watts D J. Six Degrees: The Science of a Connected Age. New York: W. W. Norton & Company, 2003 6 Watts D J. The new science of networks. Annual Review of Sociology, 2004, 30: 243 270 7 Watts D J. Small worlds: The Dynamics of Networks Between Order and Randomness. New Jersey: Princeton University Press, 1999 8 Newman M E J, Moore C, Watts D J. Mean-field solution of the small-world network model. Phys Rev Lett, 2000, 84: 3201 3204 9 Newman M E J. Models of the small world. J Stat Phys, 2000, 101: 819 841 10 Buchanan, Nexus M. Small Worlds and the Groundbreaking Science of Networks. New York : W. W. Norton and Company, 2002 11 Baraási A L, Albert R. Emergence of Scaling in Random Networks. Science, 1999, 286: 509 512 12 Strogatz S H. Exploring complex networks. Nature, 2001, 410(8): 268 276 13 Baraási A L. The New Science of Networks Cambridge: Prerseus, 2002 14 Albert R, Barabasi A L. Statistical mechanics of complex networks. Rev Mod Phys, 2002, 74: 47 98 15 Newman M, Barabási A L, Watts D J. The Structure and Dynamics of Networks. Princeton University Press, 2006 16 Newman M E J. The structure and function of complex networks. SIAM Review, 2003, 45: 167 256 17 Dorogovtsev S N, Mendes J F F. Cluster growth in two growing network models. Europhys Lett, 2000, 52: 33 18 Yook S H, Jeong H, Baraási A L, Tu Y. Weighted evolving networks. Phys Rev Lett, 2001, 86(25): 5835 5838 19 Bianconi G, Barabási A L. Competition and multiscaling in evolving networks. Eur Phys Lett, 2001,54:436 442 20 Zheng D, Trimper S, Zheng B, Hui P M. stochastic weight assignments. Phys Rev E, 2003, 67: 040102(R) 21 Antal P L. Krapivsky, Weight-driven growing networks. Phys Rev E, 2005, 71: 026103 22 Barrat A, Barthélemy M, Pastor-Satorras R, et al. The architecture of complex weighted networks. Proc Natl Acad Sci USA, 2004, 101(11): 3747 3752 23 Barrat A, Barthélemy M, Vespignani A A. Weighted evolving networks: Coupling topology and weight dynamics. Phys Rev Lett, 2004, 92: 228701 24 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: 188702 25 Fang J Q, Liang Y. Topological properties and transition features generated by a new hybrid preferential model. Chin Phys Lett, 2005, 22: 2719 2722 26 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): 117 141 27,, G, 2007, 3(2): 230 249 28 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): 279 282 29 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: 381 389 30 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): 379 396 31 Li Y, Fang J Q, Bi Q, Liu Q. Entropy characteristic on harmonious unifying hybrid preferential networks. Entropy, 2007, 9: 73 82 32 Bi Q, Fang J Q. Entropy and HUHPM approach for complex networks. Physica A, 2007, 383: 753 33 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): 5121 5142 34 Fang J Q, Bi Q, Li Y. Advances in theoretical models of network science. Front Phys China, 2007, 1: 109 124 35,,, 2007, 25(11): 23 29 36, 2006, 24(12): 67 72 37., 2007, 17(7): 841 857 38,, ( )., 2007, 27(3): 239 343 39,, ( )., 2007, 27(4): 361 448 40 Fang J Q. Theoretical research progress in complexity of complex dynamical networks. Progress in Nature Science, 2007, 17(7): 761 774 41 Xul vi-brunet R, Sokolov I M. Reshuffling scale-free networks: From random to assortative. Phys Rev E, 2004, 70(6): 066102 42 Newmanm E J. Assortative mixing in networks. Phys Rev Lett, 2002, 89(20): 208701 43 Newmanm M E J, Park J. Why social networks are different from other types of networks. Phys Rev E, 2003, 68(3): 036122
16 2008 38 44 Soon-hyung Y, Filippo R, Hildegard M O. Self-similar scale-free networks and disassortativity. Phys Rev E, 2005, 72(4): 045105 45 Sumiyoshi A, Norikazu S. Complex earthquake networks: Hierarchical organization and assortative mixing. Phys Rev E, 2006, 74(2): 026113 46 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): 188702 47 Grabowski A, Kosinski R A. Evolution of a social network: The role of cultural diversity. Phys Rev E, 2006, 73(1): 016135 48 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): 016133 49 Julian S, Janusz A H. Statistical analysis of 22 public transport networks in Poland. Phys Rev E, 2005, 72(4): 046127 50 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): 045102 51 Michele C, Guido C, Luciano P. Assortative model for social networks. Phys Rev E, 2004, 70(3): 037101 52 Mattick J S, Gagen M J. Accelerating networks. Science, 2005, 307: 856 858 53 Gagen G M, Mattick J S. Accelerating, hyperaccelerating and decelerating probabilistic networks. Phys Rev E, 2005, 72: 016123 54 Sen P. Accelerated growth in outgoing links in evolving networks: Deterministic versus stochastic picture. Phys Rev E, 2004, 69: 046107 55 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, invited talk, 2007-11-09 12. Hangzhou, Zhejiang, 2007 56 Fang J Q. Some progresses in theoretical model of nonlinear dynamical complex networks. Invited talk. In: 2008 National Physics Conference, 2008-09-18 10-02,Nanjing, 2008 57 Fang J Q, Li Y. Large unified hybrid network model with variable growth. Chi Phys Lett, 2008 58 Laherère J, Sornette D. Stretched exponential distributions in nature and economy: fat tail with characteristic scales. Eur Phys J B, 1998, 2: 525 539 59 Fang J Q. In: The 4th International Workshop Hangzhou 2007 on Simulational Physics, invited talk, 2007-11-09 12. Hangzhou, Zhejiang, 2007 60 Fang J Q. 2007 National Physics Conference, invited talk, 2007-09-18-10 02, Nanjing 61 Fang J Q. 2007 Chinese National Complex Conference, Plenary lecture, 2007, 2007-12-01 03. Shanghai 62 :,, 2008 63, 2008, 1: 13 17 ADVANCES IN UNIFIED HYBRID THEORETICAL MODEL OF NETWORK SCIENCE * FANG Jinqing LI Yong China Institute of Atomic Energy, Beijing 102413, 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 (70431002) and Grand Theory Physics Special Project of NSFC (10647001) E-mail: fjq96@126.com