2012.08.10 START
3 B(3, 0.5) N =8 3 2 1 0 1 3 3 1 w, =1, 2,...,N w 3 2 1 1 N 0 1 N 1( )
w, =1, 2,...,N w > 0 w = c log N, =1, 2,...,N, c>0 P[ [x, ) ]= N 1 { w x} = 1 N { Ne x/c } e x/c X w x 1 N 2( )
Zpf Pareto ( ) N 1/b w = a, =1, 2,...,N, a, b > 0 P[ [x, ) ]= N 1 { w x} = ( a ) b x 1 N 3( )
( ) N 1/b w = a logw = 1 b logn + log a a: b: w 1 = N 1/b b w N 1 b>1 b=1 b<1 N b>1 1 b=1 N b<1 w vs log w vs log 4( )
(Pareto) (2.0) 5( )
2ch.net ameblo You Tube Tunes DL AKB48 Amazon DL 6( )
web w 2011 3 AKB48 1 1000 0 1 5 30 100 1 100 b =2.3 ( 29) b =1.1 (29 < 100) 7( )
107 0 1 30 1209 1 1209 b =2.8 ( 28) b =2.2 (28 < 1209) 8( )
1 100 N 1 1209 N N =7 10 6, w =4.8 10 4, b =0.44 (w 1 = 3697, w 100 = 147, >100 1.2 104 ) N =7 10 6, w =2.8 10 10, b =1.3 (w 1 = 1 10 7, w 1209 = 8 10 5, >1209 2.8 1010 ) 9( )
2011 AKB48 N = 150, 120 b =1.7 ( 11), b =0.59 (11 < 40), b =0.35 (40 < 150) 10 5 1 40 N 0 1 11 40 AKB b (b >1) b <1 10( )
11( )
b 1 1 w w w 1 8 w 8 b <1 b >1 b>1 12( )
1 w (Web2.0) c 1 c 1 c 2 n 1 n 2 c 2 n 1 n 2 b>1 (c 2 c 1 ) 13( )
b<1 b>1 b 1 b >1 w (log ) 2 b 1 b <1 14( )
N N 3 2 4 1 5 1 3 2 4 5 2 1 3 4 5 1 2 3 4 5 3 1 2 4 5 1 1, 2, 1, 3 M.L.Tsetln (1963) 15( )
t X (N) (t) t 0, =1, 2,...,N, X (N) : Ω R + {1, 2,...,N} X (N) =(X (N) 1,,X (N) N ) X (N) (0) = x (N), =1, 2,,N Posson X (N) (t) (0) (X (N) (t) =1) 1 (X (N) j (t) =X (N) j (t )+1) 16( )
(1) (s, t] ν (N) ((s, t]) (t s) P[ ν (N) ((s, t]) = k ]=e λλk k! λ = w (N) (2) 1 1 X (N) (t) (3) ν (N) ((1, 2]) ν (N) ((3, 4]) 1 1 17( )
N =1, 2,...,N ν (N) : (Ω, R 2 + ) Z + ds dξ 1 X (N) (t) = x (N) N + j=1 s (0,t] 1 (N) ξ R X + (s )<X (N) j (s ) 1 ξ w (N) j + (1 X (N) (s )) 1 (N) s (0,t] ξ R ξ w + 1 A A A (X (N) (X (N) j (s ),s) ν(n) j (dξds) (s ),s) ν(n) (dξds) w (N) [ ] [ ] 18( )
J (N) (0,t)={ν (N) ((0,t]) > 0} t N Y (N) C (t) = 1 N Y (N) C λ (N) = 1 N N =1 δ w (N) Y (N) C (t) y C(t) =1 1 (N) J =1 (0,t) λ (N ) R + e wt λ(dw) (N ) Y (N) (N) C (t) E[ Y C (t) ] 0 P[ J (N) (0,t)]=1 P[ ν (N) ((0,t]) = 0 ] = 1 e wt 19( )
w (N) N µ (N) t = 1 N =1 Y (N) δ (w (N),Y (N) (t)) = 1 N (X(N) 1) µ (N) 0 µ 0 (N ) t >0 µ (N) t µ t (N µ t µ 0 skp 20( )
y C (y 0,t 0 ; t) Γ = {(y, 0) [0, 1) R + y 0} Γ b = {(0,t) [0, 1) R + t 0} Γ =Γ Γ b (y 0,t 0 ) Γ Y (N) (N) C (t) =Y C (y 0,t 0 ; t) y C (t) =y C (y 0,t 0 ; t) Y (N) C (y 0,t 0 ; t) := y 0 + 1 (N) N (t 0,t) ; Y (N) (t 0 ) y 0 1 J y C (y 0,t 0 ; t) := 1 e w (t t 0) µt0 (dw [y 0, 1)) R + y 0 y Y (N) C (y 0,t 0 ; t) y C (y 0,t 0 ; t) t 0 t 21( )
U (N) (dw, y, t) =µ (N) t (dw [y, 1)) = 1 N Y (N) C (y 0,t 0 ; t) =y 0 + 1 N ; Y (N) (t 0 ) y 0 1 J ; Y (N) (t) y (N) δ (N)(dw) w (t 0,t), (y 0,t 0 )=γ Γ ϕ (N) (dw, γ, t) =U (N) (dw, y (N) C (γ,t),t) N y (N) C y (N) C y = y C (γ,t) =y C (y 0,t 0,t) γ ˆγ U(dw, y, t) = U(dw, y C (ˆγ(y, t),t)=ϕ(dw, ˆγ(y, t),t) 22( )
µ t t>0fx y = y C (γ,t), γ Γ t =Γ {(0,t 0 ) Γ b t 0 t} ˆγ(y, t) =(y 0 (y, t),t 0 (y, t)) = γ µ t (dw [y, 1)) = e w (t t0(y,t)) µ 0 (dw [y 0 (y, t), 1)) { e wt µ = 0 (dw [y 0, (y, t), 1)) y>y C (γ,t) e w (t t 0(y,t)) µ0 (dw [0, 1)) y<y C (γ,t) (λ(dw) =µ 0 (dw [0, 1))) 23( )
λ = β r β δ wβ, β r β =1,w β,r β > 0 u α = µ 0 ({w α } ): [0, 1) R + : U α (y, t) =U({w α },y,t) U α t (y, t)+ w β U β (y, t) U α y (y, t) = w αu α (y, t) β U α (0,t)=r α U α (, 0) = u α ( ), 24( )
Amazon Amazon.co.jp Amazon.co.jp Amazon Internet retalers are extremely hestant about releasng specfc sales data 25( )
26( )
200,000 1 22fr Dec 06 29fr 27( )
rankng rankng 100 1,000 Jan 11 May 11 date Jan 11 May 11 date rankng rankng 10,000 100,000 Jan 11 May 11 date Jan 11 May 11 date vs. 28( )
x (N) X (N) (t) =X (N) = X (N) (0) = 1 (t)+1 NY(N) (t) N N R e wt λ(dw) + C w λ λ Zpf (Pareto ) w (N) = a C ( ) N 1/b ; a: b: X (N) (t) N Nb(at) b Γ( b, at)); Γ(z, p) = p e x x z 1 dx N, a, b N Amazon (Pareto ) 29( )
2 (t l,x l ), l =1, 2,...,n d E = E(N, a, b) = n d (x l Ny C (t l )) 2 /x l l=1 n d l=1 N[0, 1] 2 b <1 30( )
Amazon.co.jp rankng 500,000 Jun 07 Sep 07 Dec 07 Mar 08 date 1 O(1 ) 1 O(100 ) 31( )
rankng 500,000 Jun 07 Sep 07 Dec 07 Mar 08 date 3 98 (N, a, b )=(8 10 5, 6 10 4, 0.81) b<1 o(n) 32( )
2ch.net (b <1 ) skp 33( )
X (N) (t) N N e t t w(s) ds 0 Λ(dw) L 1 loc (R +) w (N) (t) = w (N) A(t), w (N) 0 34( )
A(t) λ A(t) =t+ X (N) (24n + t 0 ) N N e wn+a 0λ(dw)) R + 35( )
Amazon.co.jp 250000 0 1 web1 1 36( )
2ch.net web 1 move-to-front sage N 10 3 1 37( )
400 400 12:00 18:00 00:00 06:00 12:00 0 6 12 18 24 24 0 1 y C (t) 2008 M2 2009 M2 38( )
Pareto X (N) (t) N(1 0 e ws(n) (t) λ(dw)); λ([w, )) = ( aw ) b, w a 400 400 1000 2000 3000 4000 1000 2000 3000 4000 X (N) (A 1 (t)) A(t) S (N) (t) b =0.872 < 1 Amazon.co.jp 2ch.net b <1 39( )
4000 12:00 18:00 00:00 06:00 12:00 12:00 18:00 00:00 06:00 12:00 20 01 03 09 40( )
(t ) µ (dw [y, 1)) = e wt 0(y) λ(dw); 1 y = e wt 0(y) λ(dw) (**) t 0 (y) > 0, y>0 (cf. t 0 (0) = 0) tal [y, 1) R wµ (dw [y, 1)) + t R wλ(dw) + (**) λ (b <1) λ (b >1) 0 41( )
w (N) w (N) λ (b <1) λ (b >1) 42( )
rankng 500,000 N =80 (2007 ) Jun 07 Sep 07 Dec 07 Mar 08 date rankng 500,000 N =90 (2009 ) rankng Jan 09 May 09 Sep 09 date 500,000 N =95 (2010 ) Jan 10 May 10 Sep 10 date w(t) =0 43( )
rankng 500,000 Jan 10 May 10 Sep 10 date 1 7 1 skp 44( )
w (N) 1/w (N) 1 45( )
End of sldes. Clck [END] to fnsh the presentaton. Amazon 2011.5 K. Hattor, T. Hattor, Stochastc Processes and ther Applcatons 119 (2009) 966 979. K. Hattor, T. Hattor, Funkcalaj Ekvacoj 52 (2009) 301 319. K. Hattor, T. Hattor, RIMS Kokyuroku Bessatsu B21 (2010) 149 162. Y. Harya, K. Hattor, T. Hattor, Y. Nagahata, Y. Takeshma, T. Kobayash, Tohoku Mathematcal Journal 63 1 (2011) 77 111. 2011 Google END Bye