Dec. 18 20, 2006 in DEX-SMI 2006 DC http://www.smapip.is.tohou.ac.jp/ jun/ in collaboration with M. Yasuda and K. Tanaa 1/24
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scientific papers scientists Glucose 2 Lactate 2 ATP 2-Triose-P 2 P 2 NAD + i 2 NADH + 2 H+ 2 1,3-di-P-Glycerate 2 Pyruvate 2 ATP 2 ATP 3/24
2 1 3 4 5 : (e.g., 1 = 3, 3 = 2, 5 = 1) : P () = 1 N δ i, N i=1 2 1.0 0.5 0.8 0.4 10-1 P() 0.6 0.4 P() 0.3 0.2 P() 10-2 10-3 0.2 0.1 10-4 0.0 0 5 10 15 20 0.0 0 5 10 15 20 10-5 10 0 10 1 10 2 10 3 WWW etc. γ : P () γ 4/24
N: M: i : i a ij : r : r = 4M N i,j i j a ij [ N i,j ( i + j )a ij ] 2 2M N i,j (2 i + 2 j )a ij [ N i,j ( i + j )a ij ] 2 5/24
: or : bacbone : : : 6/24
BA (growing, dynamical) (nongrowing, dynamical) BA nongrowing, dynamical 7/24
Barabási-Albert (BA ) [Barabási and Albert, 1999] 1. m 0 2. m Π(i) i i i 3. m 1 m N = 200 = 2 4. 2 3 t N = t + m 0 t = 0 t = 1 t = 2 t = 3 t = 4 P() 10-1 10-2 10-3 10-4 10-5 10 0 10 1 10 2 10 3 N = 2000, = 4 averaged over 20 realizations 8/24
[Ohubo et al., 2005] 1. M N = 2M/N 2. φ(β) {β i } 3. l ij 4. m β = 0.10 β = 0.50 β = 0.15 β = 0.81 β = 0.72 β = 0.34 β = 0.45 m j β = 0.20 β = 0.62 i Π m = ( m + 1) β m j ( j + 1) β j β = 0.15 β = 0.81 β = 0.10 β = 0.72 β = 0.34 m m m j 5. l ij l im β = 0.62 6. 3,4,5 β = 0.50 β = 0.45 i β = 0.20 9/24
: φ(β) = δ(β 1) 10 0 10-1 P() 10-2 10-3 10-4 10-5 10 0 10 1 10 2 10 3 : φ(β) = 1, (0 β 1) P() 10 0 10-1 10-2 10-3 10-4 10-5 P() ~ -2.05 10 0 10 1 10 2 10 3 - (fat-tailed behavior) 10/24
f (β, t) : β [β, β + dβ] f (β, t) t Z(t) = ( + 1)β = f (β, t) + β Z(t) Z(t) f 1(β, t) dβ ( + 1) β f (β, t) N f (β, t) + ( + 1) N f +1(β, t) +1 Pref Rand Pref Rand -1 1 2 3 4 11/24
BA 12/24
1 3... 2 5 4 7 6 1 2 3 4 5 6 7... 1 2 3 4 5 6 7 13/24
...... N M ρ = M/N : E(n i ) = ln(n i!) High-energy Low-energy : p ni = e β ie(n i ) = (n i!) β i {β i } : φ(β) : W nl n l +1 (n l + 1) β l... 1 2 3 N-2 N-1 N 14/24
1 Z 1 = n 1 =0 n N =0 p n1 n 1! pn N n N! δ ( N ) n i, M i=1 = (n 1!) β1 1... (n N!) β N 1 1 2πi n 1 =0 n N =0 dz z P N i=1 n i M 1 (configuration) 1 f (β 1) = 1 Z 1 n 1 =0 n N =0 ( N ) δ(n 1, )(n 1!) β1 1... (n N!) β N 1 δ n i, M i=1 = (!) β 1 1 Z 2 Z 1 1 P (, β 1 ) = f (β 1) {β 2,...,β N } = (!)β 1 1 Z2 Z 1 {β 2,...,β N } P () = dβ φ(β)p (, β) = dβ φ(β)(!) β 1 Z2 Z 1 {β 2,...,β N } 15/24
2 ln Z 1 {β2,...,β N } ln Z i {β2,...,β N } = lim m 0 ( Z m i {β2,...,β N } 1 ) m (i = 1, 2) z s d ρ = G(z s ) G(z s ) dz s ( G(z) = ) dβ φ(β) (n!) β 1 z n n=0 P () = (!) β 1 zs φ(β) n=0 (n!)β 1 zs n dβ 16/24
N = 1000 (N = 4000) averaged over 20 realizations 0.5 0.4 10 0 10-1 ρ = 2 ρ = 5 P() 0.3 0.2 P() 10-2 10-3 0.1 10-4 P() 0.0 0 5 10 15 20 φ(β) = δ(β), ρ ρ P () = e! 10 0 ρ = 1.0 ρ = 4.0 10-1 10-2 10-3 10-4 10-5 10 0 10 1 10 2 10 3 10-5 φ(β) = 1, (0 β 1), P () = 1 0 10 0 10 1 10 2 10 3 φ(β) = δ(β 1), P () = 1 { ( 1 + ρ exp ln 1 + 1 )} ρ (!) β 1 z s n=0 (n!)β 1 z n s dβ z s = 0.660 (ρ = 1) z s = 0.967 (ρ = 4) P () 2 (ln ) 2 (ρ 1) 17/24
growing (dynamical) nongrowing (static) nongrowing & dynamical 1 2 BA 18/24
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Newman [Newman and Girvan, 2004] : modularity Q Q = [ M i=1 e i m ( ) ] 2 di 2m M: e i : i m: d i : i Q... 21/24
[Ohubo and Tanaa, 2006] : V nc 2 V = n e n: e: V 1 = 2, V 2 = 4.5, V 3 = 10 V 1 V 2 V 3 = V 1 + V 2 in contact σ 1 σ 2 σ 12 V 1 V 2 V 3 = V 1 + V 2 V... 22/24
arate club 27 19 30 24 26 28 32 25 10 27 19 30 24 26 28 32 25 10 23 16 15 21 34 33 31 29 9 3 14 8 4 13 23 16 15 21 34 33 31 29 9 3 14 8 4 13 20 2 1 5 11 20 2 1 5 11 18 22 12 7 6 18 22 12 7 6 17 17 arate club arate club 1 33 2 10 23/24
: : : etc. 24/24