38 6 Vol. 38 No. 6 2008 11 25 ADVANCES IN MECHANICS Nov. 25, 2008 *, 102413 3,, 3 :, 3,,, 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) Barabási Albert (scale-free, SF) [11 14],, 2006 Barabási Newmann Watts [15],,,, Barabási 2006 von Neumann( ) 3,,, : 2008-06-30, : 2008-07-10 (70431002), (60874087) (10647001) E-mail: fjq96@126.com
664 2008 38 : [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
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 +, 3 2.. [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]
666 2008 38 (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) ( ),,,,.,. :,, :, ( ),,,,,,.,,. 3 3 3, :.,,, ( ),,,,,,.,,,.,,,, (average path length, APL), (Average clustering coefficient, ACC)., Watts Strogatz,,, :,,,,,
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: 3.1 1 : [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)
668 2008 38 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,
6 : 669 [31,32]., 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, : (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 :
670 2008 38 (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
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=500 0.495 2 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)
672 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 RV G (7), [52 57] m(t) = p(n(t)) α (8) m(t) t, N(t) t, p, p(t), 0 <
6 : 673 p(t) < 1, ;α, α (α ) (0 < α < 1) (α 1), 3,, LUHVGS :, (8) p, p vg,, 3, 3, 3.3.1 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
674 2008 38 3.3.2 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 1/49 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 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 = 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 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
676 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), α = 0.5 4, 3 3,, 3,, [59 62],,,,,, 4,. 3, 3 ; [63], 2008 :,,,,,,
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