103-8660 30 2003 7 30
e-mail: inaoka@eri.u-tokyo.ac.jp e-mail: takuto.ninomiya@boj.or.jp e-mail: ken.taniguchi@boj.or.jp e-mail: tokiko.shimizu@boj.or.jp e-mail: takayasu@csl.sony.co.jp *
2 1 2 1
2 Barabási 2
P(x) P x) -g ( µ x (1) 3
a g P( x) = a P( ax) (2) x x ax a g ax x a g x P ( ³ x) P ( ³ x) = ò P( x ) dx (3) x P(x) x P ( ³ x) x P ( ³ a) x a 4
x ³ a x x N(x) N ( ³ x) = ò N( x ) dx (4) x N ( ³ a) x a x P( ³ x) = N( ³ x) (5) N( ³ 0) Barabási Erdös Erdös N 0 p N 0-1 k N 0-1 - k p k N0-1-k ( 1 - p) N 0-1 k k P(k) 5
æ N 0-1ö k N -1-k 0 P( k) = ç p (1 - p) (6) è k ø N 0 ( N 0-1) p Erdös Barabási P k) -g ( µ k (7) Barabási m 0 m i k i p = å ki k j j (8) i k g Barabási 6
2001 2002 12 2001 7
A a B b A B 2001 546 15 733 14 710 7620 8
546 546 148,785 7,351 7,351 7,351 n N ( ³ n) n ³ 20 N( ³ n) µ n -1.3 (9) 21 21 9
-1.3 N( ³ n) µ n 10
546 21 354 1,727 62,481 1.1 11
N( ³ n) µ n -1.1 2001 6 2001 6 2002 12 3 2001 6 12
Barabási Barabási Erdös Barabási failure attack Erdös Erdös Barabási Erdös Barabási 13
Erdös failure Erdös attack Barabási failure Barabási Attack Erdös 1 8(a) 14
Barabási 8(b) 8(a) (a) (b) (a) (b) N ) -t ( ³ k µ k (10) 15
t t g t = g -1 t 1 Barabási t = 2 t 1.1 1.3 1.1 1.5 Barabási 2.0 t 9 9 Barabási log log 16
Barabási Barabási 1 1 1000 17
18
2001 6 10 11 7 2 2001 6 21 1 6,782 1 33 8 20 25 25 20 19
12 13 1 9 10 1 9 10 9 10 1 1 54 47 13 1 20
1 14 1 100 10 14 10 12 10 9 15 1 10 10 100 21
16 1 1 22
B N i j m m i j n m m ij i j n ij n = m m (11) ij i j n ij m i m j 11 n = m ij a i m a j (12) 23
a m 1 = m 2 = n a = 1/ 2 ij i j n ij n ij i j l ij l = n / n (13) ij ij ij l ij i j i j d i j l i j d ij N C a d (d ) n a ij N G N = C å i= 1 n ( d ) (14) a r G N G = 2N d r G a r G a a r G b r G 24
a b S ab = 2 N é æ d ö = ú ú ù ai + d bi expê- ç (15) 1 êë è N G ø û å i S ab a b S aa a M a = S aa - å b¹ a S ab (16) a R. N. Mantegna and H. E. Stanley, An introduction to econophysics: correlation and complexity in finance (Cambridge Univ. Press, 1999). Empirical science of financial fluctuations: The advent of econophysics, edited by H. Takayasu (Springer Verlag, 2002). H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, New York, 1987). M. Faloutsos, P. Faloutsos, and C. Faloutsos, Comput. Commun. Rev. 29, 251 (1999). R. Albert, H. Jeong, and A.-L. Barabási, Nature (London) 401, 130 (1999). A.-L. Barabási and R. Albert, Sience 286, 509 (1999). B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982). R. Albert, H. Jeong, and A.-L. Barabási, Nature 406, 378 (2000). 25