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1 START
2 3 B(3, 0.5) N = w, =1, 2,...,N w N 0 1 N 1( )
3 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( )
4 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( )
5 ( ) 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( )
6 (Pareto) (2.0) 5( )
7 2ch.net ameblo You Tube Tunes DL AKB48 Amazon DL 6( )
8 web w AKB b =2.3 ( 29) b =1.1 (29 < 100) 7( )
9 b =2.8 ( 28) b =2.2 (28 < 1209) 8( )
10 1 100 N N N =7 10 6, w = , b =0.44 (w 1 = 3697, w 100 = 147, > ) N =7 10 6, w = , b =1.3 (w 1 = , w 1209 = , > ) 9( )
11 2011 AKB48 N = 150, 120 b =1.7 ( 11), b =0.59 (11 < 40), b =0.35 (40 < 150) N AKB b (b >1) b <1 10( )
12 11( )
13 b 1 1 w w w 1 8 w 8 b <1 b >1 b>1 12( )
14 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( )
15 b<1 b>1 b 1 b >1 w (log ) 2 b 1 b <1 14( )
16 N N , 2, 1, 3 M.L.Tsetln (1963) 15( )
17 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( )
18 (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]) ( )
19 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( )
20 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( )
21 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( )
22 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 (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( )
23 U (N) (dw, y, t) =µ (N) t (dw [y, 1)) = 1 N Y (N) C (y 0,t 0 ; t) =y 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( )
24 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( )
25 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( )
26 Amazon Amazon.co.jp Amazon.co.jp Amazon Internet retalers are extremely hestant about releasng specfc sales data 25( )
27 26( )
28 200, fr Dec 06 29fr 27( )
29 rankng rankng 100 1,000 Jan 11 May 11 date Jan 11 May 11 date rankng rankng 10, ,000 Jan 11 May 11 date Jan 11 May 11 date vs. 28( )
30 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( )
31 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( )
32 Amazon.co.jp rankng 500,000 Jun 07 Sep 07 Dec 07 Mar 08 date 1 O(1 ) 1 O(100 ) 31( )
33 rankng 500,000 Jun 07 Sep 07 Dec 07 Mar 08 date 3 98 (N, a, b )=(8 10 5, , 0.81) b<1 o(n) 32( )
34 2ch.net (b <1 ) skp 33( )
35 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( )
36 A(t) λ A(t) =t+ X (N) (24n + t 0 ) N N e wn+a 0λ(dw)) R + 35( )
37 Amazon.co.jp web1 1 36( )
38 2ch.net web 1 move-to-front sage N ( )
39 :00 18:00 00:00 06:00 12: y C (t) 2008 M M2 38( )
40 Pareto X (N) (t) N(1 0 e ws(n) (t) λ(dw)); λ([w, )) = ( aw ) b, w a X (N) (A 1 (t)) A(t) S (N) (t) b =0.872 < 1 Amazon.co.jp 2ch.net b <1 39( )
41 :00 18:00 00:00 06:00 12:00 12:00 18:00 00:00 06:00 12: ( )
42 (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( )
43 w (N) w (N) λ (b <1) λ (b >1) 42( )
44 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( )
45 rankng 500,000 Jan 10 May 10 Sep 10 date skp 44( )
46 w (N) 1/w (N) 1 45( )
47 End of sldes. Clck [END] to fnsh the presentaton. Amazon K. Hattor, T. Hattor, Stochastc Processes and ther Applcatons 119 (2009) K. Hattor, T. Hattor, Funkcalaj Ekvacoj 52 (2009) K. Hattor, T. Hattor, RIMS Kokyuroku Bessatsu B21 (2010) Y. Harya, K. Hattor, T. Hattor, Y. Nagahata, Y. Takeshma, T. Kobayash, Tohoku Mathematcal Journal 63 1 (2011) Google END Bye
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A hydrodynamc lmt of move-to-front rules and ts applcaton to web rankngs 2010.07.15 START Move-to-front cf. Move-to-front N N 1( ) (move-to-front ) N =1,,N t 0 t X (N) (t) X (N) (0) = x (N) X (N) (t) =
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Amazon.co.jp 2008.09.02 START Amazon.co.jp Amazon.co.jp Amazon.co.jp Amazon Internet retailers are extremely hesitant about releasing specific sales data 1( ) ranking 500,000 100,000 Jan.1 Mar.1 Jun.1
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