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Effects of Connections and Initial Conditions

on Spreading Phenomena

24

31-116903

物性研究・電子版　Vol, 2, No. 3, 023602 （2013年8月号）《修士論文》


i

2013 6 1


ii

31-116903

1927 McKendrick

[1]. S( ),I( ),R( ) 3 ,

S, I,R .

I S ,

,

.

. . ,

1.1 ( ) ( ) .

1 [2]

, ( )

. ,SIR

SIR

.

SIR ,

λ . N

, 1 I N − 1 S

,λ ,

. ,

. 2002 Moreno [3] , (

) ( )

λ ∼ 0 ,

.

,

.

, SIR

, .

.

, ,

( ) R

. , ,


iii

.

, SIR ,

, 0 1

[4, 5]. ,

, 1

, .

, .

.

-1

-0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3

U(Y

)

Y

Y*

❍❍

λ=0.51.2

2 U(Y )

.N → ∞, t � O(N)

, SIR I

Y

Y = −∂Y U(Y ) +√2Dξ (1)

. , D = (λ+ 1)/8 , U(Y )

U(Y ) = −1

4(λ− 1)Y 2 +

1

8(λ+ 1) log(Y ) (2)

. 3.3 , U(Y ) λ ≤ 1 , Y (t)

0 . 1 . λ > 1( ) U(Y ) Y∗, ,Y → ∞( )

. U(Y )

. [6].

, ,

SIR ,

.

,2012 10 ( )

. .

1. Y. Kermack. Bulletin of mathematical biology, Vol. 53, No. 1, pp. 33–55, 1991.

2. V. Krebs. Mapping the spread of contagions via contact tracing, 2003.

3. Y.Moreno, et al. Eur. Phys. J. B, Vol. 26, No. 4, pp. 521–529, 2002.

4. L.K. Gallos, et al. Physica A, Vol. 330, No. 1, pp. 117–123, 2003.

5. A. Lancic, et al. Physica A, Vol. 390, No. 1, pp. 65–76, 2011.

6. J. Iwai and S. Sasa. To be submitted soon.



v

1 1

1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 SIR . . . . . . . . . . . . . . . . . . . . . 4

1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 SIR 9

2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 11

. . . . . . . . . . . . . . . . . . . . 11

. . . . . . . . . . . . . . . . . . . . 12

2.3 . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

λc . . . . . . . . . . . 13

λ→ ∞ 〈tf 〉, σ(tf )2 . . . . . . . . . . . . 15

2.4 m=1 , . . . . . . . . . . . . . 20

2.5 m ∝ N , . . . . . . . . . . . 20

2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 . . . . . . . . . . . . . . . . . . 24


vi

2.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 27

3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5.1 SIR . . . . . . . . . . . . . 36

Newman . . . . . . . . . . . . 37

. . . . . . . . . . . . . . . . . . 44

SIR . . . . . . . . . . . . . . . . . . . . . . 45

3.5.2 . . . 47

3.5.3 . . . . . . 47

3.5.4 . . . . . . . . . . . . . . . . . . . 51

3.5.5 . . . . . . . . . . . . . . . . 52

3.5.6 Q(Y, t) (3.10) . . . . . . . 54

3.5.7 SIS . . . . . . . . . . 54

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 59

4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.1 . . . . . . . . . . . . . . . . . . . . . . 62

4.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 67

5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68


vii

A 69

A.1 . . . . . . . . . . . . . . . . . . . . . . 69

A.1.1 . . . . . . . . . . . . . . . . . . . . . 69

A.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

A.2 SIR . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.2.2 §4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

73



1

1

,

, . ,

, . , ,

, , .

1.1

2 , 2

, ,

,

, , ,

. ,

, ,

. ,

, , ,

.

. [1] .

. ,

, .

, , 2

.


2 1

1.2

,

.

. ,

[2]. ,

. ,

.

, . ,

,

,

. , .

, . ,

,

, ,

.

,

[3]. .

, .

, . 2

. . G(V,E)

V = v1, v2, · · · , vN E = e1, e2, · · · em. ei 2 ,ei = {j, k} j k

. . N × N A

Aij =

{1 for ∃k, ek = {ij}0 for ∀k, ek = {ij} (1.1)

. 2 i, j Aij = 1,

Aij = 0 . , i, j Aij

. Aij = 0 . ,

, .

. ,

,

[4] . .


1.2 3

( ) Liu

Controllability of complex networks[4]

. .

,

. G(V,E) . i xi

, x .

:

dx(t)

dt= Ax+Bu(t) (1.2)

A G ,N ×M B N ≤ M ,

M u . (controllablity) ,N ×NM

C = (B,AB,A2B, · · ·AN−1B) rank(C) = N

, x(0) τ x(τ)

u . G

.

m 2N − 1

, N . , ,

(structural controllability) . A ,0 A

, rank(C) = N N ×M B .

B M ”maximum matching” ,

ND . ND N nD

. M u(t) ,

M , .

,

, .

, ,

,

. ,

,B , .

,ND = 1 , , .

,

, , ,

ND .


4 1

, .

, ,

, .

§1.3 , .

1.3

.

( 1.1),

. , ,

, ,

. SIR .

1.1

[5]

1.3.1 SIR

SIR .

.

, 3 S,I,R

. S ,I S

, R I

.

Susceptible,Infected,Recoverd .

. S I

, S-I , λ S I ,I-I .

. S-I . ,I

μ R . .

. , λ , t

t + Δt Δt S-I I-I dPI

dPI = λΔt+O(Δt2) . , I R dPR

dPR = μΔt+O(Δt2) . Δt→ 0 O(Δt2) .

,N , 1

I , N − 1 S .

I ,S R . .


1.4 5

R nR , .

N [6] , .

,

, , N N → ∞ ,

. , εN ,N → ∞ε → 0 , N m

. 1 ,

. Boguna [7] m = 1

N → ∞ ρ0 ρ0 = 1/N

, ρ0 N ρ0 = ε .

§3 .

1.4

.

SIR , SIR 1927

McKendrick [8]. S,I,R

S, I, R , .⎧⎨⎩S = −λSII = λSI − μI

R = μI

(1.3)

S + I +R = N . SIR

.

SIR

Mollison[9] Grassberger[10] . 1998 ,

[11]

[12] , .

, k P (k) P (k) ∼ k−γ

.

,

. SIR

, λ/μ ,

nR = 0 , nR 0

. 2002 , Moreno [13]


6 1

. ,

λ/μ ∼ 0 .

λ ,

. , SIS

[14]

, 0 . ,

,

[15].

1.4.1

SIR .

SIS SIR I R ,

SIS S , . SIR

, SIS

.

SIR

,

[16]. §3 .

λ λΔt .

,

.

, S

I SEIR [17] ,

[18] .

1.5

.

§2§2 , ,

SIR . nR tf ,


7

, .

§3§3 ,SIR

. ,2013 6 1

. ,

, SIR

.

§4§4 ,SIR ,

. , . S

I , I

, SIR

.

.

§5§5 , .

, .

[1] Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari,

S. Tadaki, and S. Yukawa. Traffic jams without bottlenecks–experimental evi-

dence for the physical mechanism of the formation of a jam. New Journal of

Physics, Vol. 10, No. 3, p. 033001, 2008.

[2] H. Haken. Information and self-organization: A macroscopic approach to complex

systems, Vol. 40. Springer, 2006.

[3] . .

, 3 2012.


8 1

[4] Y.Y. Liu, J.J. Slotine, and A.L. Barabasi. Controllability of complex networks.

Nature, Vol. 473, No. 7346, pp. 167–173, 2011.

[5] V. Krebs. Mapping the spread of contagions via contact tracing, 2003.

[6] J.C. Miller. Epidemics on networks with large initial conditions or changing

structure. arXiv preprint arXiv:1208.3438, 2012.

[7] M. Boguna, C. Castellano, and R. Pastor-Satorras. Langevin approach for the

dynamics of the contact process on annealed scale-free networks. Physical Review

E, Vol. 79, No. 3, p. 036110, 2009.

[8] WO Kermack and AG McKendrick. Contributions to the mathematical theory

of epidemics-i. Bulletin of mathematical biology, Vol. 53, No. 1, pp. 33–55, 1991.

[9] D. Mollison. Spatial contact models for ecological and epidemic spread. Journal

of the Royal Statistical Society. Series B (Methodological), pp. 283–326, 1977.

[10] P. Grassberger. On the critical behavior of the general epidemic process and

dynamical percolation. Mathematical Biosciences, Vol. 63, No. 2, pp. 157–172,

1983.

[11] D. Watts and S. Strogatz. The small world problem. Collective Dynamics of

Small-World Networks, Vol. 393, pp. 440–442, 1998.

[12] A.L. Barabasi and R. Albert. Emergence of scaling in random networks. science,

Vol. 286, No. 5439, pp. 509–512, 1999.

[13] Y. Moreno, R. Pastor-Satorras, and A. Vespignani. Epidemic outbreaks in com-

plex heterogeneous networks. The European Physical Journal B-Condensed Mat-

ter and Complex Systems, Vol. 26, No. 4, pp. 521–529, 2002.

[14] C. Castellano and R. Pastor-Satorras. Routes to thermodynamic limit on scale-

free networks. Physical review letters, Vol. 100, No. 14, p. 148701, 2008.

[15] A.L. Barabasi and E. Bonabeau. Scale-free. Scientific American, 2003.

[16] N. Boccara and K. Cheong. Automata network sir models for the spread of

infectious diseases in populations of moving individuals. Journal of Physics A:

Mathematical and General, Vol. 25, No. 9, p. 2447, 1999.

[17] J. Stehle, N. Voirin, A. Barrat, C. Cattuto, V. Colizza, L. Isella, C. Regis, J.F.

Pinton, N. Khanafer, W. Van den Broeck, et al. Simulation of an seir infectious

disease model on the dynamic contact network of conference attendees. BMC

medicine, Vol. 9, No. 1, p. 87, 2011.

[18] M.J. Keeling and K.T.D. Eames. Networks and epidemic models. Journal of the

Royal Society Interface, Vol. 2, No. 4, pp. 295–307, 2005.


9

2

SIR

2.1

,

[1]. Barabasi[2] , WWW

. , [3] ,

, [4], [5], SNS

[6] . ,

. ,E

, ,

,

.

,

. ,

.

, ,

[7].

, .

, ,

. [8] .

,

, . , ,

,


10 2 SIR

.

.

, .

SIR , m m = 1 . ,

SARS , 1

. , 1

, ,

, . ,

.

, ,

. , m = 1

.

. ( 2.1) , 10

,2 .

, 10

,

[9]. ,

. ,

,

. 10 ,

. , m = 1

m > 1 .

. , ,

. SIR

[10, 11] , ,

m = 1 . ,

. ,

. ,

. ,

.

, ,

, .

, nR

tf .


2.2 11

. nR 〈nR〉 λ ,

λc 0,λc ,

. . ,λ→ ∞ 〈tf 〉, σ(tf )2

. m = 1 nR, tf

, .

N ,

. , N , nR tf

, ,

.

2.2

,

.

2.2.1

,

m = 1 . ,

,m = εN . ε� 1 .

2.2.2

N .

. 2

. ,

.

. ,

,

. ,

, .

, . Hi, i = 1, 2, · · ·n i

, Yj , j = 1, 2, · · ·n j .

. c . ρ1(l), ρ2(l)


12 2 SIR

2.1

2.2 HY

[1, 2, 3, · · ·N ] → [1, 2, 3, · · ·N ] .σ(a, b) a b

. l, l = 1, 2, · · ·N Hσ(ρ1(l),c), Yσ(ρ2(l),c) .

l 2 Hσ(ρ1(l),c)Yσ(ρ2(l),c) ,

H,Y c ,

. HY .

λ . 2.2 c = 4

.

, ,

.

,

. λ/N

. .

2.3

. k ,

.


2.3 13

2.3.1

, .

m NR tf .〈·〉,σ(A)2 A 〈A2〉 − 〈A〉2 , Σ(A)2 = Nσ(A)2 = σ(NA)2/N .

nRNR

N 〈nR〉, σ(nR),Σ(nR), 〈tf 〉, σ(tf ) , λ .

2.3 k = 3,m = 1 N = 1024, 2048, 4096, 8192 N

. 〈nR〉, σ(nR)2 N . 〈tf 〉 λ < λc

, λ→ ∞ N . σ(tf )2

λc , . 2.4 N = 8192,m = 1

k = 3, 4, 5 k . 2.3 λc ≈ 1/(k − 2)

. 2.5 k = 3 m = N/256, N = 1024, 2048, 4096, 8192

N . 2.4 Σ(nR)2 N → ∞

. m = 1 2.3.1

(〈N2R〉 − 〈NR〉2)/N2 m = N/256 2.4 (〈N2

R〉 − 〈NR〉2)/NN → ∞ . 1

( ) ,m = N/256

. m = 1 .§3.

2.3.2

λc 〈tf 〉, σ(tf )2 .

λc

k m = 1 λc

. . k

, .

. S-I I-I λ ,I R

1 . t S,I,R S, I,R

.I ∼ O(N) dI/dt

. I ∼ O(N) I I

, I ∼ O(N) . I ∼ O(N) dI/dt = 0 λ


14 2 SIR

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

n R

λ

m=1,N=1024N=2048N=4096N=8192

(a) 〈nR〉

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 1 2 3 4 5

σ(n R

)2

λ

N=1024N=2048N=4096N=8192

(b) σ(nR)2

0

2

4

6

8

10

12

14

16

18

0 1 2 3 4 5

tf

λ

N=1024N=2048N=4096N=8192

(c) 〈tf 〉

0

50

100

150

200

250

300

350

400

450

500

0 1 2 3 4 5

σ(tf)

2

λ

N=1024N=2048N=4096N=8192

(d) σ(tf )2

2.3 k = 3,m = 1, N = 1024, 2048, 4096, 8192

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5

n R

λ

k=3k=4k=5

(a) 〈nR〉

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 1 2 3 4 5

σ(n R

)2

λ

k=3k=4k=5

(b) Σ(nR)2

0

2

4

6

8

10

12

14

16

18

20

0 1 2 3 4 5

<tf>

λ

k=3k=4k=5

(c) 〈tf 〉

0

50

100

150

200

250

300

350

400

450

500

0 1 2 3 4 5

σ(tf)

2

λ

k=3k=4k=5

(d) σ(tf )2

2.4 k = 3, 4, 5, N = 8192

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5

n R

λ

N=1024N=2048N=4096N=8192

(a) 〈nR〉

0

5

10

15

20

25

30

35

40

45

0 1 2 3 4 5

Σ(n R

)2

λ

N=1024N=2048N=4096N=8192

(b) Σ(nR)2

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5

<tf>

λ

N=1024N=2048N=4096N=8192

(c) 〈tf 〉

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5

σ(tf)

2

λ

N=1024N=2048N=4096N=8192

(d) σ(tf )2

2.5 m = N/256, N = 1024, 2048, 4096, 8192

.

dI

dt= [ S I ]− [ I R ] (2.1)

. . S I

,

[S − I ]× λ (2.2)

. I R ,

I (2.3)


2.3 15

. S-I . R

t � O(1) , S I , I

m = 1 1 .

S . I kI .

I − 1 , 1 2

, S

kI − 2(I − 1)

. (2.2),(2.3) 2.1

dI

dt= λ [kI − 2(I − 1)]− I

. dI/dt = 0

λc [kI − 2(I − 1)] = I

⇒ λc =1

k − 2(1− 1/I)

.N → ∞ 1/I → 0

λc =1

k − 2(2.4)

. (2.4) 2.4 .

λ→ ∞ 〈tf 〉, σ(tf )2N 〈tf 〉N 〈tf 〉, σ(tf )2N = σ(tf )

2 N

. λ , λ � λc

〈tf 〉 . , m ≥ 1

I .

. m = N , R tf

PN (tf ) . 1 I R t p(t)

. . , μ δt

μδt . δt 1 − μδt

. t , t δt tδt

. (1− μδt)tδt

. t

p(t)

p(t) = limδt→0

(1− μδt)t/δt

= μe−μt


16 2 SIR

.

PN (tf ) . i R ti ,PN (tf )

PN (tf ) =

∫0≤ti≤tf ,Max[ti]=tf

dt1dt2 · · · dtNp(t1)p(t2) · · · p(tN )

= N !

∫ tN

0

dtN−1

∫ tN−1

0

dtN−2 · · ·∫ t2

0

dt1p(t1)p(t2) · · · p(tN )

.i = j ti = tj R ,

0 < t1 < t2 < · · · < tN = tf

,PN (tf ) N ! .

2.5 ,pi(t) .

p1(t) = p(t)

pi(t) =

∫ t

0

pi−1(t′)p(t′)dt′

PN (t) pi(t)

PN (t) = N !pN (t)

. pi(t) .

p1(t) = μe−μt

p2(t) =

∫ t

0

μe−μt′dt′μe−μt

= 1− e−μt

p3(t) =

∫ t

0

(1− e−μt′

)dt′μe−μt

=1

2

(1− e−μt

)2μe−μt

...

pN (t) =1

(N − 1)!

(1− e−μt

)N−1μe−μt

PN (t) = N(1− e−μt

)N−1μe−μt (2.5)

.


2.3 17

〈tf 〉N , σN (tf )2 .

〈tf 〉N =

∫ ∞

0

tPN (t)dt

=N

μ

∫ ∞

0

x(1− e−x

)N−1e−xdx

X = e−x .

〈tf 〉N =N

μ

∫ x

0

x(1− e−x

)N−1e−xdx

=N

μ

∫ 0

1

(1−X)N−1

logXdX

∫ 0

1(1−X)

N−1logXdX

∫ 0

1

(1−X)N−1

logXdX =[(1−X)

N−1X (logX − 1)

]01

−∫ 0

1

(N − 1)(−1) (1−X)N−2

X (logX − 1)

=

∫ 0

1

(N − 1) (1−X)N−2

X (logX − 1)

= (N − 1)

∫ 0

1

[− (1−X)

N−1logXdX

]dX

+(N − 1)

∫ 0

1

[− (1−X)

N−2X + (1−X)

N−2logX

]dX

⇔ N

∫ 0

1

(1−X)N−1

logXdX = (N − 1)[− (1−X)

N−2X + (1−X)

N−2logX

]dX

= (N − 1)

[1

N (N − 1)+

∫ 0

1

(1−X)N−2

logXdX

]

=1

N+ (N − 1)

∫ 0

1

(1−X)N−2

logXdX

⇔ 〈tf 〉N =1

μN+ 〈tf 〉N−1

〈tf 〉N =1

μ

N∑j=1

1

j(2.6)

.


18 2 SIR

. .

I N 1 R N − 1 I

∫ ∞

0

tNμe−Nμtdt =1

μN

.I N −1 〈tf 〉N−1

, 2.6 .

〈t2f 〉N .

〈t2〉N = μN

∫ ∞

0

(1− e−μt

)N−1e−μtdt

=N

μ2

∫ ∞

0

x2(1− e−x

)N−1e−xdx

∫ ∞

0

x2(1− e−x

)N−1e−xdx =

∫ 1

0

(logX)2(1−X)

N−1dX

X = e−x .∫(logX)

2dX = X

[(logX)

2 − 2logX + 2]

= −(1 − X)[(logX)

2 − 2logX + 2]+[

(logX)2 − 2logX + 2

].

∫ 1

0

(logX)2(1−X)

N−1dX =

[∫(logX)

2dX (1−X)

N−1

]10

+(N − 1)

∫ 1

0

∫(logX)

2dX (1−X)

N−2dX

= (N − 1)

[∫ 1

0

(logX)2{− (1−X)

N−1+ (1−X)

N−2}dX

+2

∫ 1

0

{logX (1−X)

N−1 − logX (1−X)N−2

}dX

+ 2

∫ 1

0

{− (1−X)

N−1+ (1−X)

N−2}dX

]

⇔ N

∫ 1

0

(logX)2(1−X)

N−1dX = (N − 1)

∫ 1

0

(logX)2(1−X)

N−2dX

+2 (N − 1)

NN

∫ 1

0

logX (1−X)N−1

dX

−2 (N − 1)

∫ 1

0

logX (1−X)N−2

dX

+2

N


2.3 19

〈t2〉N = 〈t2〉N−1 − 2 (N − 1)

μN〈t〉N +

2

μ〈t〉N−1 +

2

μ2N

= 〈t2〉N−1 +2

μN〈t〉N

⇔ 〈t2〉N =2

μ

N∑i=1

1

i〈t〉i

=2

μ2

N∑i=1

1

i

i∑k=1

1

k

=1

μ2

⎡⎢⎣⎛⎝ N∑

j=1

1

j

⎞⎠

2

+

N∑j=1

1

j2

⎤⎥⎦

. σN (tf )2

σN (tf )2 = 〈t2〉N − 〈t〉2N

=1

μ2

⎡⎣( N∑

k=1

1

k

)2

+N∑

k=1

1

k2

⎤⎦−

(1

μ

N∑i=1

1

i

)2

σN (tf )2 =

1

μ2

N∑j=1

1

j2(2.7)

.

(2.6),(2.7) . μ = 1, N = 1024, 2048, 4096, 8192

(2.6),(2.7) 〈tf 〉N , σN (tf )2 . HN,n =

∑Nk=1 1/k

n (harmonic

number) . MathWorld[12] n = 1

HN,1 ∼ lnN + γ (2.8)

. γ � 0.577 . (2.8)

H1024,1 � 7.51, H2048,1 � 8.20, H4096,1 � 8.90, H8192,1 � 9.59

H1024,2 � 1.64, H2048,2 � 1.64, H4096,2 � 1.64, H8192,2 � 1.64

, 2.3,2.4,2.5 (c),(b) .

(2.6),(2.7) .

mathematica ,∫∞0dteatPN (t)

a a = 0 .


20 2 SIR

2.4 m=1 ,

m = 1 . 2.6, 2.7,

2.9 , ,HY .

, 〈k〉. *1 .

〈nR〉 2.4,2.6,2.7,2.8 , 〈nR〉 λc 0

, 〈nR〉 = 1 .

λc λc � 0.25,HY

λc � 0.35 , ( )

.

σ(nR)2 2.4,2.6,2.7,2.8 , σ(nR)

2 λc 0

λ � 0.6 ,

.

〈tf 〉 2.4,2.6,2.7,2.8 , 〈tf 〉 λc

, N .

σ(tf )2 2.4,2.6,2.7,2.8 , λ < λc 0 ,λc

0 . N

, .

2.5 m ∝ N ,

m = εN . 2.10,

2.11, 2.12, 2.13 , ,HY

. ε = N/128 .

〈nR〉 2.5,2.10,2.11,2.12 , 〈nR〉 λc 0

, 〈nR〉 = 1

. λc

λc � 0.25,HY λc � 0.35 ,

*1 , p ,N → ∞〈k〉N/2 = NC2p [13].


2.5 m ∝ N , 21

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

n R

λ

m=1,N=8192N=16384N=32768N=65536

(a) 〈nR〉

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

σ(n R

)2

λ

N=8192N=16384N=32768N=65536

(b) σ(nR)2

0

2

4

6

8

10

12

14

16

18

20

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

tf

λ

N=8192N=16384N=32768N=65536

(c) 〈tf 〉

0

200

400

600

800

1000

1200

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

σ(tf)

2

λ

N=8192N=16384N=32768N=65536

(d) σ(tf )2

2.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

n R

λ

m=1,N=8192N=16384N=32768N=65536

(a) 〈nR〉

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

σ(n R

)2

λ

N=8192N=16384N=32768N=65536

(b) σ(nR)2

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

tf

λ

N=8192N=16384N=32768N=65536

(c) 〈tf 〉

0

100

200

300

400

500

600

700

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2σ(

tf)2

λ

N=8192N=16384N=32768N=65536

(d) σ(tf )2

2.7

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

n R

λ

m=1,N=8192N=16384N=32768N=65536

(a) 〈nR〉

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

σ(n R

)2

λ

N=8192N=16384N=32768N=65536

(b) σ(nR)2

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

tf

λ

N=8192N=16384N=32768N=65536

(c) 〈tf 〉

0

100

200

300

400

500

600

700

800

900

1000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

σ(tf)

2

λ

N=8192N=16384N=32768N=65536

(d) σ(tf )2

2.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

n R

λ

m=1,N=8192N=16384N=32768N=65536

(a) 〈nR〉

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

σ(n R

)2

λ

N=8192N=16384N=32768N=65536

(b) σ(nR)2

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

tf

λ

N=8192N=16384N=32768N=65536

(c) 〈tf 〉

0

200

400

600

800

1000

1200

1400

1600

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

σ(tf)

2

λ

N=8192N=16384N=32768N=65536

(d) σ(tf )2

2.9 HY


22 2 SIR

.

Σ(nR)2 2.5,2.10,2.11,2.12 , Σ(nR)

2 λc 0

,

0 . N .

〈tf 〉 2.5,2.10,2.11,2.12 , 〈tf 〉 λ < λc N

λc , N

. N

.

σ(tf )2 2.5,2.10,2.11,2.12 , λ < λc 0

,λc

. N .

2.6

λc k λc

λc = 1/(k − 2) . 2.6,2.10 . HY

,k2, k ,

. ,

.

.HY ,

.

.

m = 1 σ(NR)2 Σ(nR)

2 σ(nR)2 , 2.4,2.6,2.8

, m = 1 σ(NR)2 ≈ N2 .

2.5,2.10,2.12 , m = εN σ(NR)2 ≈ N . NR

, ,σ(NR)2 ≈ N ,m = 1 NR

. §3 .

λ→ ∞ 〈tf 〉 2.4,2.6,2.8, 2.5,2.10,2.12 , λ→ ∞〈tf 〉 N . 2.3.2

.

m = εN Σ(nR)2, σ(tf )

2 2.5,2.10,2.12 , Σ(nR)2

N


2.6 23

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

n R

λ

m=1,N=8192N=32768N=65536

(a) 〈nR〉

0

2

4

6

8

10

12

14

16

18

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Σ(n R

)2

λ

N=8192N=32768N=65536

(b) Σ(nR)2

5

10

15

20

25

30

35

40

45

50

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

tf

λ

N=8192N=32768N=65536

(c) 〈tf 〉

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

σ(tf)

2

λ

N=8192N=32768N=65536

(d) σ(tf )2

2.10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

n R

λ

m=1,N=8192N=32768N=65536

(a) 〈nR〉

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Σ(n R

)2

λ

N=8192N=32768N=65536

(b) Σ(nR)2

5

10

15

20

25

30

35

40

45

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

tf

λ

N=8192N=32768N=65536

(c) 〈tf 〉

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2σ(

tf)2

λ

N=8192N=32768N=65536

(d) σ(tf )2

2.11

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

n R

λ

m=1,N=8192N=16384N=32768N=65536

(a) 〈nR〉

0

2

4

6

8

10

12

14

16

18

20

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Σ(n R

)2

λ

N=8192N=16384N=32768N=65536

(b) Σ(nR)2

0

10

20

30

40

50

60

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

tf

λ

N=8192N=16384N=32768N=65536

(c) 〈tf 〉

0

20

40

60

80

100

120

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

σ(tf)

2

λ

N=8192N=16384N=32768N=65536

(d) σ(tf )2

2.12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

n R

λ

m=1,N=8192N=32768N=65536

(a) 〈nR〉

0

2

4

6

8

10

12

14

16

18

20

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Σ(n R

)2

λ

N=8192N=32768N=65536

(b) Σ(nR)2

5

10

15

20

25

30

35

40

45

50

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

tf

λ

N=8192N=32768N=65536

(c) 〈tf 〉

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

σ(tf)

2

λ

N=8192N=32768N=65536

(d) σ(tf )2

2.13 HY


24 2 SIR

. , 2.5,2.10,2.12 σ(tf )2

.

. §2.7 .

2.7

, , ,HY

. .

2.7.1

2.142.15 , , ,HY

,m = εN Σ(nR)2, σ(tf )

2 λ λc

.〈k〉 = 6, N = 16384, 32768, 65536. N < 70000

. N ,λ ≈ λc*2 .

2.8

, SIR

. m m = 1 m = εN nR, tf

. , , ,

,HY 4 ,

, ,

. HY 3 , λc

. λc,λ → ∞〈tf 〉, σ(tf )2 . m = εN ,σ(nR)

2, σ(tf )2

, .

, N

. ,

, .

*2 ,

, .


25

0

2

4

6

8

10

12

14

16

18

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Σ(n R

)2

λ

N=8192N=16384N=32768N=65536

(a)

0

2

4

6

8

10

12

14

16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Σ(n R

)2

λ

N=8192N=16384N=32768N=65536

(b)

0

2

4

6

8

10

12

14

16

18

20

0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34

Σ(n R

)2

λ

N=8192N=16384N=32768N=65536

(c)

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Σ(n R

)2

λ

N=8192N=16384N=32768N=65536

(d) HY

2.14 Σ(nR)2

0

10

20

30

40

50

60

70

80

90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

σ(tf)

2

λ

N=8192N=16384N=32768N=65536

(a)

0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

σ(tf)

2

λ

N=8192N=16384N=32768N=65536

(b)

0

20

40

60

80

100

120

0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34

σ(tf)

2

λ

N=8192N=16384N=32768N=65536

(c)

0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

σ(tf)

2

λ

N=8192N=16384N=32768N=65536

(d) HY

2.15 σ(tf )2

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complex networks. Reviews of Modern Physics, Vol. 80, No. 4, p. 1275, 2008.

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[7] , , , , , , , ,

, . . , Vol. 46,


26 2 SIR

No. 6, pp. 235–245, 2005.

[8] . ( )–(

). , Vol. 98, No. 8, pp. 1287–1291, 2010.

[9] (2012 ) ,

[10] Y. Chen, G. Paul, S. Havlin, F. Liljeros, and H.E. Stanley. Finding a better

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Mat. Kutato Int. Kozl, Vol. 5, pp. 17–61, 1960.


27

3

[1]

. ,

§3.5 . , ,

1950 70 .

, [2].

3.1

,

. , ,

. ,

,

. , ,1

, , ,

. ,

.

SIR .

,SIR λ

. , ,

[3]. , ,

λ λc .

, .


28 3

, [4, 5], [6, 7]

.

SIR , ,

.SIR nR ,

λ > λc 0 ,λ = λc

. , ,

1 . , SIR

. ,

SIR , 0

[8], [9]. ,SIR

[10, 11]. ,Ising

, . ,

,

. , ,

0 1 .

, ,

. , , 0

.

,SIR ,

. ,

. SIR ,

. ,

. ,

Langevin . Langevin

, 0 .

3.2

N k G . x ∈, k B(x) . σ

,σ = S, I, and R , ,Susceptible, Infectible, ,

Recovered . ,[σ(x)]x∈G , , σ

. SIR ,σ W (σ → σ′)

. , W ,I


3.3 29

λ I μ ,

W (σ → σ′) =∑x∈G

w(x),

w(x) = λ[δ(σx, S)δ(σ′x, I)

∑y∈B(x)

δ(σy, I)] + μδ(σx, I)δ(σ′x, R) (3.1)

. μ = 1 .

, , G I

, . . R

N nR ,

. , I

, S .

,k = 3, N = 8192 , λ , ,

. nR , p(nR;λ)

. 3.2 log p ,λ

. , ,nR

. 3.2 λ = 1.5

log p , .

,[10] . , 3.2

,nR < 1/16 λ . ,

, λc , 1 nR = 0 ,

q(λ) , nR ,1 − q(λ)

nR = 0 . , .

3.3

, s ≡∑x δ(σx, S)/N i ≡∑

x δ(σx, I)/N

. ,(s, i) → (s, i−1/N)

Ni , (s, i) → (s − 1/N, i + 1/N) λkNsψ

, ψ . ,ψ , σx = S x

,σy = I y x . , I

, loop

O(log(N) , ( )

.

Ni(k − 2) [12], ,ψ Nk


30 3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

nR

0

1

2

3

4

5

λ

-12

-10

-8

-6

-4

-2

0

0 0.2 0.4 0.6 0.8 1

nR

0

1

2

3

4

5

λ

-12

-10

-8

-6

-4

-2

0

-12-10-8-6-4-2 0

0 0.2 0.4 0.6 0.8 1nR

log(p)

λ=1.5

(a)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

p(n R

<1/1

6)

λ

N=8192N=16384N=32768N=65536

(b) λ− p(nR < 1/16)

3.1 SIR . , nR,

λ,log p . λ log p

. , λ, log p.λ = 1.5.

.N = 8192,m = 1. , λ, p(nR < 1/16). N → ∞ ,

nR < 1/16 .

. ,k = 3 .

, t (s, i) P (s, i, t) .

,P (s, i, t)

∂P (s, i, t)

∂t= N

(i+

1

N

)P

(s, i+

1

N, t

)−NiP (s, i, t)

+Nλ

(s+

1

N

)(i− 1

N

)P

(s+

1

N, i− 1

N, t

)−NλsiP (s, i, t)(3.2)

. ,N , 1/N

,

∂P (s, i, t)

∂t= −∂i

{[(λs− 1) iP (s, i, t)]− ∂i

[(λs+ 1) i

2NP (s, i, t)

]+ ∂s

[λsi

2NP (s, i, t)

]}

−∂s{− [λsiP (s, i, t)]− ∂s

[λsi

2NP (s, i, t)

]+ ∂i

[λsi

2NP (s, i, t)

]}(3.3)

+O

(1

N2

)(3.4)

.O(1/N2) ,

[13]. ,


3.3 31

. ,

3.5.2 .

(3.4) ,

. 3.5.3

. ,⎧⎨⎩ s(t) = −λsi−

√λsiN · ξ1

i(t) = λsi− i+√

λsiN · ξ1 +

√iN · ξ2

(3.5)

. ξi , 〈ξi (t)〉 = 0, 〈ξi (t) ξj (t′)〉 =

δijδ (t− t′) . (3.5) ξ1, ξ2 · .

(3.5) . (3.5)

⎧⎨⎩ s(t+Δt) = s(t)− λs(t)i(t)Δt−

√λs(t)i(t)

N ΔtN1(0, 1)

i(t+Δt) = i(t) + (λs(t)i(t)− i(t))Δt+√

λs(t)i(t)N ΔtN1(0, 1) +

√i(t)N ΔtN2(0, 1)

(3.6)

. Ni(0, 1) 0, 1 . 3.2 (s0, i0) =

(1 − 1/N, 1/N) . SIR

. ,(3.5) .

, Y =√iN ,(3.5) ,

{s = 1

N

[−λsY 2 −

√λsY 2 · ξ1

]Y = 1

2

{(λs− 1)Y − 1

4 (λs+ 1) 1Y

}+ 1

2

√λs · ξ1 + 1

2

√1 · ξ2

(3.7)

.(3.5) ξ1, ξ2 i ,(3.7)

. 3.5.4 . ,

nR ,

. , nR > 0 q(λ) .nR = 0 ,

Y � O(N1/2) . , (3.7)

.

, (3.7) s Y N

. ,N , Y s

. , O(N) ,s = 1 .

, (3.7) Y

Y = −∂Y U(Y ) +√2Dξ (3.8)


32 3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5

n R

λ

N=512N=1024N=2048N=4096N=8192

(a) < nR >

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5

sig2 (n

R)

λ

N=512N=1024N=2048N=4096N=8192

(b) σ(nR)2

0

2

4

6

8

10

12

14

0 1 2 3 4 5

<tf>

λ

N=512N=1024N=2048N=4096N=8192

(c) < tf >

0

50

100

150

200

250

300

350

400

0 1 2 3 4 5

sig2 (tf

)

λ

N=512N=1024N=2048N=4096N=8192

(d) σ(tf)2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

nR

0

1

2

3

4

5

λ

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

0 0.2 0.4 0.6 0.8 1

nR

0

1

2

3

4

5

λ

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

-10-9-8-7-6-5-4-3-2-1 0

0 0.2 0.4 0.6 0.8 1

log(p)

λ=1.5

(e) log p .N = 1024,

N = 8192.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

p(n R

<1/1

6)

λ

N=1024N=2048N=4096N=8192

(f) p(nR < 1/16)

3.2 (3.5) .m = 1, N =

1024, 2048, 4096, 8192. SIR (

2.3,3.2,3.2) .

. , D = (λ+ 1)/8 , U(Y )

U(Y ) = −1

4(λ− 1)Y 2 +

1

8(λ+ 1) log(Y ) (3.9)

.ξ 1 ,√λ/2 · ξ1 + 1/2 · ξ2 =√

λ+ 1/2 · ξ . Y (0) = 1 .(3.8) N

. , Y → ∞ ,(3.7) Y � O(N1/2)

. ,(3.8) Y → ∞q(λ) , .

3.5.5 (3.4) s = 1, I = Ni

.

t Y Q(Y, t) ,Q(Y, t)

∂tQ(Y, t) = −∂Y [{∂Y U(Y )}Q(Y, t)] +D∂Y2Q(Y, t) (3.10)


3.3 33

. Q(Y, 0) = δ(Y, 1) . (3.10) (3.4) s = 1, Y =√Ni .3.5.6 .

, U (Y ) . λ limY→+0

U (Y ) = −∞.λ < 1 U (Y ) lim

Y→∞U (Y ) → ∞ .1 < λ

limY→∞

U (Y ) = −∞ ,Y = Y∗ .

Y∗ =1

2

√λ+ 1

λ− 1(3.11)

. ,λ = 0.5, 1.2 3.3 .

,λ < 1 . ,U(Y ) Y ,

(3.8) Y → ∞ . ,

1 Y = 0 . ,q(λ) = 0 .

,λ > 1 . (3.8) Y Y∗, Y → ∞ . 3.3 , Y Y∗ Y → ∞

(3.8) , Y = Y∗ Y → ∞. , λ , Y Y∗

. Y = 1 Y = Y∗ W ,

t = 0 Y = 1 ,t = τ Y = Y∗ p(Y (τ) = Y∗|Y (0) = 1)

p(Y (τ) = Y∗|Y (0) = 1) = Wτ τ τ .

3.4 (3.8) τ .λ ≈ λc

, p(Y (τ) = Y∗|Y (0) = 1) = Wτ τ .

τ , SIR Y = 1 Y = 0 1

, W Y∗ ,q = W/(1 +W ) . ,

q(λ) 0 1 .

. λc ,

λc = 1 (3.12)

. 3.3 (3.8) ,Y = 1

Y∗ λ .

,Y∗ < 1 λ > 5/3 1 .

, q(λ) . , , λ = λc + ε

ε . Y∗ � O(ε−1/2), U(Y∗) � O(log ε)

ε .(Y∗, U(Y∗)) (1, U(1)) ε → 0

U(Y∗)− U(1))/(Y∗ − 1) � √ε→ 0 , Y = 1 Y = Y∗ ,

D = (λ+1)/8 . , τtr


34 3

-1

-0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3

U(Y

)

Y

Y*

❍❍

λ=0.51.2

3.3 U(Y ) .λ = 0.5

red line , Y ,

1 Y = 0 .

λ = 1.2 green line , Y∗ =√11/2

, Y Y∗ Y →∞ .

0

0.002

0.004

0.006

0.008

0.01

0.012

0 5 10 15 20 25 30 35 40

p(Y

(τ)=

Y*|Y

(0)=

1)

τ

λ=1.004λ=1.008λ=1.016

λ=1.02

3.4 (3.8)

, Y = 1 Y = Y∗τ pp(Y (τ) =

Y∗|Y (0) = 1) .

pp(Y (τ) = Y∗|Y (0) = 1) ∝ Wτ

τ .

0

0.2

0.4

0.6

0.8

1

1 1.2 1.4 1.6 1.8 2

p(Y

->∞

|Y(t=

0)=Y

*)

λ

p(Y-> ∞|Y=Y*)

(a)

0

0.2

0.4

0.6

0.8

1

1 1.2 1.4 1.6 1.8 2

p(Y

-> Y

*|Y(t=

0)=1

)

λ

p(Y=Y*|Y=1)

(b)

3.5 (3.8) p(Y∗ → 0).λ >

λc λ p(Y∗ → 0) = 0.4 .

(3.8) . ,Y = 1 Y∗p.


3.4 35

Y 2∗ /(2D) = 1/ε+O(ε0) . W ,W = ε+O(ε2)

. ,

q(λ) = ε+O(ε2). (3.13)

. 3.4 (3.7) nR > 0.003 p(nR > 0.003)(�q(λ)) ε = λ − 1 . ,

p(nR > 0.003) = ε + O(ε2) . 3.4 (3.7)

nR > 0.003 p(nR > 0.003)(� q(λ)) ε = λ− 1 .

(3.13) p(nR > 0.003) � ε1 .

3.4

, ,

. ,

, U(Y ) ,

. ,

.

, ,

. , m 1 ,

. Y∗ , N

. ,m = cN ,

,Y (0) , .

, . , 3.4 SIR

,c = 1/2561 nR < 1/16 p λ

. λ q(λ) = 1 , .

,

. , ,

. , ,

. , .


36 3

0.001

0.01

0.1

1

0.001 0.01 0.1 1

n R

λ-1

N=16384N=32768N=65536fit (λ-1)1

(a)

0.001

0.01

0.1

1

0.001 0.01 0.1

n R

λ-1

N=131072N=262144

N=2097152N=16777216

fit (λ -1)1

(b)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

p(n R

<1/1

6)

λ

N=8192N=16384N=32768

(c)

3.6 , SIR , m = 1

, p(nR > 0.003) ε = λ−1 . p(nR > 0.003) < ε

, . , (3.7)

, p(nR > 0.003) ∼ q(λ) ε = λ − 1 . N → ∞q(λ) � ε . , SIR

, m N/128 nR < 1/16 p λ . λ

1 q(λ) = 1 .

3.5

3.5.1 SIR

Newman SIR

[6] , SIR

,Newman ,

. ,SIR

nR

.

, 1 .

, 1 .

. Newman .

, λc

. ,

. λ < λc 0,λ > λc 1

.


3.5 37

1 SIR ,SIR

. ,SIR

, , τ

,

, .

Newman

Newman[6] Durrett[14] , SIR

. .

i, j ,i I ,j S . i s

γij ,i τi

. i j i 1− Tij

1− Tij = limδt→0

(1− γijδt)τi/δt = e−γijτi (3.14)

. i j Tij

Tij = 1− e−γijτi (3.15)

.

i, j γij τi i, j , P (γ), P (τ)

. T (γ, τ) = 1 − e−γτ . SIR λ,

1 ,{P (γ) = δ(λ− γ)P (τ) = 1− e−τ) (3.16)

, 〈T 〉 =∫dγ∫dτT (γ, τ) .

〈T 〉 =∫ ∞

0

dγ

∫ ∞

0

dτP (γ)P (τ)(1− e−γτ )

= 1−∫ ∞

0

dτP (τ)e−(λ+1)τ

= 1− 1

λ+ 1

=λ

λ+ 1(3.17)

. T = T (γ, τ) .


38 3

p(k)

.

N → ∞ ,

.

, ,

.

G0(x) .

G0(x) =∑k

p(k)xk (3.18)

< k >,< k2 > .

∂xG0(1) = < k >

∂2xG0(1) = < k2 > − < k >

G0(x) .

. 0 , n

n . 0

k 1 p1(k) , 1

k p(k) ,

p1(k) ∝ kp(k)

. .

p1(k) =kp(k)∑k k

′p(k′)=

kp(k)

< k >(3.19)

. p1(k) ∑k kp(k)x

k

< k >=x∑

k kp(k)xk−1

G0(1)= x

G0(x)

G0(1)(3.20)

. 1 0 2

k p′1(k) , 1 k + 1

(k + 1)p(k + 1) p′1(k)

p′1(k) =(k + 1)p(k + 1)∑k(k + 1)p(k + 1)

=(k + 1)p(k + 1)

G′0(1)

(3.21)


3.5 39

.p′1(k) G1(x)

G1(x) =

∑k(k + 1)p(k + 1)xk

G0(1)=G′

0(x)

G′0(1)

(3.22)

.

2 k p′2(k) G2(x) .

p′2(k) =∑n

p(n)∑

k1+k2+···+kn=k

p′1(k1)p′1(k2) · · · p′1(kn) (3.23)

G2(x) =∑k

p′2(k)xk =

∑k

∑n

p(n)∑

k1+k2+···+kn=k

p′1(k1)p′1(k2) · · · p′1(kn)xk(3.24)

, (3.24)∑

k

∑k1+k2+···+kn=k

∑k1

∑k2

· · ·∑kn

G2(x) =∑n

p(n)∑k1

∑k2

· · ·∑kn

p′1(k1)p′1(k2) · · · p′1(kn)xk1+k2+···+kn

=∑n

p(n)∑k1

p′1(k1)xk1

∑k2

p′1(k2)xk2 · · ·

∑kn

p′1(kn)xkn

=∑n

p(n)n∏

i=1

∑ki

p′1(ki)xki

=∑n

p(n)

(∑k

p′1(k)xk

)n

=∑n

p(n)G1(x)n

= G0 (G1(x))

. G0(x) x 1 , 1

0 G1(x) x

.

T (γ, τ) , T (γ, τ)

. T (γ, τ)

, 1− T (γ, τ) .

G0(x;T ) k m

kCmTm(1− T )k−m (3.25)


40 3

.G0(x;T )

G0(x;T ) =∞∑k

k∑m=0

p(k)kCmTm(1− T )k−mxm

=

∞∑k=0

p(k)

k∑m=0

kCm(xT )m(1− T )k−m

G0(x;T ) =∞∑

m=0

p(k) (1− T + xT )k

(3.26)

.G1(x;T )

G1(x;T ) =

∑∞k=0(k + 1)P (k + 1)

∑km=0 T

m(1− T )k−mxm∑∞k=0(k + 1)P (k + 1)

=

∑∞k=0(k + 1)P (k + 1) (1 + (x− 1)T )

k

G′0(1)

.< a >=∫∞0ap(T )dT G0(x) =< G0(x;T ) >, G1(x) =< G1(x;T ) >{G0(x) = <

∑∞k=0 p(k) (1 + (x− 1)T )

k>

G1(x) = <∑

k p(k)k(1+(x−1)T )k−1

G′0(1)

>(3.27)

.

0

s P0(s) , H0(x)

H0(x) =∑s

P0(s)xs (3.28)

. 1 s

P1(s), H1(x)

H1(x) =∑s

P1(s)xs (3.29)

. O(N) .

, 1 s

, 2 . 1 0

(3.22) G1(x) . 2

s P1(s) . H1(x)

H1(x) = xG1 (H1(x)) (3.30)


3.5 41

. 1 1 x . H0(x)

H1(x), G0(x)

H0(x) = xG0 (H1(x)) (3.31)

. 0 x .

< s > . (3.31)

< s > = H ′0(1)

=[G0 (H1(x)) + G′

0 ([H1(x)]x=1)H′1(x)

]x=1

=[G0 (1) + G′

0 (1)H′1(x)

]x=1

< s > = 1 + G′0 (1)H

′1(1) (3.32)

. (3.30)

[H ′1(x)]x=1 =

[G1 ([H1(x)]x=1) + G′

1 (H1(x))H′1(x)

]x=1

=[1 + G′

1(x)H′1(x)

]x=1

H ′1(1) =

1

1− G′1(1)

(3.33)

(3.32),(3.33)

< s > = 1 +G′

0(1)

1− G′1(1)

(3.34)

.G′0(1), G

′1(1) .<> ∂x .

G′0(1) =

[G′

0(x)]x=1

=∞∑k=0

p(k)[< ∂x (1 + (x− 1)T )

k>]x=1

=

∞∑k=0

p(k)k[< (1 + (x− 1)T )

k−1T >

]x=1

=∞∑k=0

kp(k)k−1∑m=0

k−1Cm [(x− 1)m]x=1 < Tm+1 >

= 〈T 〉∞∑k=0

kp(k)


42 3

G′0(1) = 〈T 〉G′

0(1) (3.35)

.

G′1(1) =

[G′

1(x)]x=1

=

[∂x <

∑k p(k)k (1 + (x− 1)T )

k−1

G′0(1)

>

]x=1

=

[<

∑k p(k)k∂x (1 + (x− 1)T )

k−1

G′0(1)

>

]x=1

=

[<

∑k p(k)k(k − 1)T (1 + (x− 1)T )

k−2

G′0(1)

>

]x=1

=

[<

∑k p(k)k(k − 1)T

∑k−2m=0(x− 1)mTm

G′0(1)

>

]x=1

=

∑k p(k)k(k − 1)

∑k−2m=0 [(x− 1)m]x=1 < Tm+1 >

G′0(1)

=

∑k p(k)k(k − 1)

∑k−2m=0 [(x− 1)m]x=1 < Tm+1 >

G′0(1)

=〈T 〉∑k p(k)k(k − 1)

G′0(1)

G′1(1) =

〈T 〉G′′0(1)

G′0(1)

(3.36)

. (3.35),(3.36) (3.34) . G′′0(1) =< k2 > − < k >,G′

0(1) =< k >

< s > = 1 +〈T 〉G′

0(1)

1− 〈T 〉G′′0 (1)

G′0(1)

= 1 +〈T 〉 < k >

1− 〈T 〉(<k2>−<k>)<k>

.< s > 〈T 〉 = Tc

1− Tc(< k2 > − < k >)

< k >= 0

⇔ Tc =< k >

< k2 > − < k >(3.37)


3.5 43

. (3.37),(3.17) < s > λ < k >

< k >

< k2 > − < k >=

λcλc + 1

⇔ λc =< k >

< k2 > −2 < k >(3.38)

.

< k >= k,< k2 >= k2 ,

λc =1

k − 2(3.39)

. (3.12) 2.3.2 . (

) §2.

,p(k′) = δ(k − k′) (3.27) G0(x), G1(x) ,

{G0(x) = 〈(1 + (x− 1)T )

k〉G1(x) = 〈k(1+(x−1)T )k−1

G′0(1)

〉 (3.40)

(3.35)

G′0(1) = 〈T 〉k (3.41)

. (3.30),(3.31)

H1(x) = x〈[1 + (H1(x)− 1)T ]

k−1〉〈T 〉 (3.42)

H0(x) = x〈[1 + (H1(x)− 1)T ]k〉 (3.43)

(3.42) k . k = 3 ,

H1(x)k=3 =1− 2(1− T )x−√1− 4(1− T )x

2Tx(3.44)

,

H0(x)k=3 =

[1−√1− 4(1− T )x

]38x2

(3.45)


44 3

. x = 1 H0(1) =∑

s P0(s) = 1 ,

H1(1)k=3 =1− 2T +

√4T − 3

2T(3.46)

H0(1)k=3 =1− 2T +

√4T − 3

8(3.47)

,T ≥ 3/4 λ ≥ 2 .

λ > 0 , . ,

[15] .

,

, 〈T 〉 > Tc , 1 . ,

1

. , ,SIR 1

R 1 , SIR

. k ≥ 3

,

[15] , .

3.5.1 ,

. k .

〈T 〉 . q

. q , 1 − 〈T 〉 ,

k − 1 ,

〈T 〉qk−1 ,

q = 1− 〈T 〉+ 〈T 〉qk−1

⇔ (1− q)− 〈T 〉(1− qk−1) = 0

⇔ (1− q)(1− 〈T 〉k−2∑j=0

qj) = 0 (3.48)

. q k ≥ 3 1 1 . 1 q∗.

1 − p . p , k


3.5 45

p = qk

⇒ p =

{1 for q = 1qk∗ for q = q∗

(3.49)

.

:k = 3 k = 3 . (3.48)

q = 1,1− 〈T 〉〈T 〉 (3.50)

. (3.49) p

p =

{1 for q = 1(

1−〈T 〉〈T 〉

)3for q = 1−〈T 〉

〈T 〉(3.51)

.q = 1, p = 1 〈T 〉 < Tc λ < λc , q = q∗, p = q3∗〈T 〉 > Tc λ > λc .

SIR

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

p

λ

percSIR

3.7 (3.53)

p.

.k = 3.

SIR nR <

1/16 p(nR < 1/16).

k = 3,m = 1, N = 65536.

(3.17)

〈T 〉 = λ

λ+ 1(3.52)

. p =(

1−〈T 〉〈T 〉

)3

p =

{1 for λ < λc(1λ

)3for λ > λc

(3.53)

.

SIR ( 3.2) ( 3.7).

λ > λc

p .

. .

. .

. , S I

. ,λ > λc

. ,S I a


46 3

0.0001

0.001

0.01

0.1

1

1 10 100 1000 10000

P(N

R)

NR

N=8192,lam=0.4N=4096,lam=0.4N=2048,lam=0.4N=1024,lam=0.4N=512,lam=0.4

(a) λ = 0.4

0.0001

0.001

0.01

0.1

1

1 10 100 1000 10000

P(N

R)

NR


(b) λ = 1

0.0001

0.001

0.01

0.1

1

1 10 100 1000 10000

P(N

R)

NR


(c) λ = 2

0.0001

0.001

0.01

0.1

1

1 10 100 1000 10000

P(N

R)

NR

N=8192,lam=3.0N=4096,lam=3.0N=2048,lam=3.0N=1024,lam=3.0

N=512,lam=3.0

(d) λ = 3

3.8 SIR NR . m = 1, N =

1024, 2048, 4096, 8192.

I , I b , a S

. 〈T 〉a, 〈T 〉b,

. λ > λc

.

. nR

, . ,

I S

.

O(logN) , N → ∞ .

3.2) N , . 3.8

SIR NR p(NR;λ,N) . λ > λc

N . .

. .

, SIR

.SIR ,τ I

, τ P (τ) τ , I

.

SIR .

,SIR 1

, . ,

,

, .


3.5 47

3.5.2

(3.4) i = 0 , ,

. .

(3.4) P (s, i)

0 = −∂i{[(λs− 1) iP (s, i)]− ∂i

[(λs+ 1) i

2NP (s, i)

]+ ∂s

[λsi

2NP (s, i)

]}

−∂s{− [λsiP (s, i)]− ∂s

[λsi

2NP (s, i)

]+ ∂i

[λsi

2NP (s, i)

]}(3.54)

. SIR , i = 0 .

i > 0 P (s, i) = 0 .

Pε (s, i) = P (s) δ(i− ε) (3.55)

. i = 0 , i = ε Pε (s, i)

, ε→ 0 . (3.54)

i . i [0, 1] .

f(s) = −{[(λs− 1) iP (s) δ(i− ε)]− ∂i

[(λs+ 1) i

2NP (s) δ(i− ε)

]+ ∂s

[λsi

2NP (s, i)

]}i=1

i=0

−∫ 1

0

di∂s

{− [λsiP (s) δ(i− ε)]− ∂s

[λsi

2NP (s) δ(i− ε)

]+ ∂i

[λsi

2NP (s) δ(i− ε)

]}

= −∂s{− [λsεP (s)]− ∂s

[λsε

2NP (s)

]}(3.56)

.f(s) , P (s) . f(s) = 0

c

P (s) = ce−2Ns

s(3.57)

, . i = 0 δ(i)

.

3.5.3

2 .

2 x(t), y(t){x(t) = ax + bxx · ξx + bxy · ξyy(t) = ay + byx · ξx + byy · ξy (3.58)


48 3

. ξi , .

〈ξi〉 = 0 (3.59a)

ξi(t)ξj(t′) = δijδ(t− t′) (3.59b)

ξj · . ,2 x, y

∂tP (x, y, t) = −∂x [PAx + ∂xPBx + ∂yPCx]− ∂y [PAy + ∂yPBy + ∂yPCy](3.60)

. x, y ax, ay, bxx, bxy, byx, byy Ax, Bx, Cx, Ay, By, Cy

.

P (x, y, t) x, y .

P (x, y, t) = 〈δ(x− x(t))δ(y − y(t))〉 (3.61)

< · > . P (x, y, t +Δt) Δt . ,xi(t+

Δt) = xi(t) + Δxi Δxi .

Δxi = xi(t+Δt)− xi(t)

= [ai + byj · ξj ] Δt+O(Δt3/2) (3.62)

Δt→ 0 O(Δt3/2) .

P (x, y, t + Δt) = 〈δ(x− x(t+Δt))δ(y − y(t+Δt))〉 (3.62)

.

δ(xi − xi(t+Δt)) = δ(xi − xi(t)−Δxi)

= δ(xi − xi(t))− δ′(xi − xi(t))Δxi +1

2δ′′(xi − xi(t))Δx

2i

= δ(xi − xi(t))− δ′(xi − xi(t)) [ai + byj · ξj ] Δt+1

2δ′′(xi − xi(t)) [ai + byj · ξj ]2 Δt2 (3.63)


3.5 49

(3.59) δ(x− x(t+Δt))δ(y − y(t+Δt))

δ(x− x(t+Δt))δ(y − y(t+Δt))

=

[δ(x− x(t))− δ′(x− x(t)) [ax + bxj · ξj ] Δt+ 1

2δ′′(x− x(t)) [ax + bxj · ξj ]2 Δt2

]

×[δ(y − y(t))− δ′(y − y(t)) [ay + byj · ξj ] Δt+ 1

2δ′′(y − y(t)) [ay + byj · ξj ]2 Δt2

]= δ(x− x(t))δ(y − y(t))

−δ(x− x(t))δ′(y − y(t)) [ay + byj · ξj ] Δt+ δ(x− x(t))1

2δ′′(y − y(t)) [ay + byj · ξj ]2 Δt2

−δ(y − y(t))δ′(x− x(t)) [ax + bxj · ξj ] Δt+ δ(y − y(t))1

2δ′′(x− x(t)) [ax + bxj · ξj ]2 Δt2

+δ′(x− x(t))δ′(y − y(t)) [ax + bxj · ξj ] [ay + byj · ξj ] Δt2 (3.64)

. j j = x, y . (3.64) < · >P (x,y,t+Δt)−P (x,y,t)

Δt . (3.64) Δt

,

〈δ′(xi − xi(t))δ′(xj − xj(t))〉 = ∂xi∂xjP (xi, xj , t) (3.65a)

ξjΔt = 0 (3.65b)

ξjξjΔt = 1 (3.65c)

.Δt→ 0

∂P (x, y, t)

∂t= −∂Pax

∂x− ∂Pay

∂y

+1

2

∂2P(byx

2 + byy2)

∂y2+

1

2

∂2P(bxx

2 + bxy2)

∂x2

+∂2P (bxxbyx + bxybyy)

∂x∂y+O(Δt1/2) (3.66)

.O(Δt1/2 , . ∂∂y ,

∂∂x

∂P (x, y, t)

∂t= −∂x

[Pax − 1

2∂x[P(bxx

2 + bxy2)]− 1

2∂y [P (bxxbyx + bxybyy)]

]

−∂y[Pay − 1

2∂y[P(byy

2 + byx2)]− 1

2∂y [P (bxxbyx + bxybyy)]

](3.67)


50 3

.

Ax = ax (3.68a)

Bx = −1

2

(bxx

2 + bxy2)

(3.68b)

Cx = −1

2(bxxbyx + bxybyy) (3.68c)

Ay = ay (3.68d)

By = −1

2

(byy

2 + byx2)

(3.68e)

Cy = −1

2(bxxbyx + bxybyy) (3.68f)

.

x→ s, y → i (3.4) (3.68)

−λsi = as (3.69a)

− 1

2N(λs+ 1)i = −1

2

(bss

2 + bsi2)

(3.69b)

1

2Nλsi = −1

2(bssbis + bsibii) (3.69c)

(λs− 1)i = ai (3.69d)

1

2Nλsi = −1

2

(bii

2 + bis2)

(3.69e)

1

2Nλsi = −1

2(bssbis + bsibii) (3.69f)

. as, ai, bss, bsi, bis, bii . as, ai

as = −λsi (3.70a)

ai = (λs− 1)i (3.70b)

.bss, bsi, bis, bii ,

bss = −√λsi

N(3.71a)

bsi = 0 (3.71b)

bis =

√λsi

N(3.71c)

bii =

√i

N(3.71d)

.


3.5 51

3.5.4

x, y ,(x, y) (u(x, y), v(x, y)

. x, y {x(t) = ax + bxx · ξx + bxy · ξyy(t) = ay + byx · ξx + byy · ξy (3.72)

.

,dx(t), dy(t) .{dx(t) = axdt+ bxx · ξxdt+ bxy · ξydt+O(dt3/2)dy(t) = aydt+ byx · ξxdt+ byy · ξydt+O(dt3/2)

(3.73)

,dx2, dy2, dxdy dt1 .⎧⎨⎩dx2 = (b2xx + b2xy)dtO(dt2)dy2 = (b2yx + b2yy)dt+O(dt2)dxdy = (bxxbyx + bxybyy)dt+O(dt2)

(3.74)

, u, v .du, dv x, y

. {du = uxdx+ 1

2uxxdx2 + uydy +

12uyydy

2 + uxydxdydv = vxdx+ 1

2vxxdx2 + vydy +

12vyydy

2 + vxydxdy(3.75)

. uxy = ∂2u∂x∂y . (3.74)

⎧⎪⎪⎨⎪⎪⎩du =

[uxax + 1

2uxx(b2xx + b2xy) + uyay +

12uyy(b

2yx + b2yy) + uxy(bxxbyx + bxybyy)

]dt

+(uxbxx + uybyx) · ξxdt+ (uxbxy + uybyy) · ξydt+O(dt3/2)dv =

[vxax + 1

2vxx(b2xx + b2xy) + vyay +

12vyy(b

2yx + b2yy) + vxy(bxxbyx + bxybyy)

]dt

+(vxbxx + vybyx) · ξxdt+ (vxbxy + vybyy) · ξydt+O(dt3/2)

(3.76)

. dt u, v .⎧⎪⎪⎨⎪⎪⎩u = uxax + 1

2uxx(b2xx + b2xy) + uyay +

12uyy(b

2yx + b2yy) + uxy(bxxbyx + bxybyy)

+(uxbxx + uybyx) · ξx + (uxbxy + uybyy) · ξy +O(dt1/2)v = vxax + 1

2vxx(b2xx + b2xy) + vyay +

12vyy(b

2yx + b2yy) + vxy(bxxbyx + bxybyy)

+(vxbxx + vybyx) · ξx + (vxbxy + vybyy) · ξy +O(dt1/2)

(3.77)

uxbxy + uybyy, vxbxy + vybyy u, v ,

.


52 3

x→ s, y → i u = s, v =√Ni .

us = 1 (3.78a)

uss = ui = uii = usi = 0 (3.78b)

vs = vss = vsi = 0 (3.78c)

vi =1

2

√N

i=N

2

1

v(3.78d)

vii = −1

4

√N

i3= −N

2

4

1

v3(3.78e)

(3.70),(3.71)

{u = 1

N

[−λuv2 −

√λuv2 · ξ1

]+O(dt1/2)

v = λu−12 v − λu+1

81v +

√λu2 · ξ1 + 1

2 · ξ2 +O(dt1/2)(3.79)

. (3.7) (3.79)(u, v) (s, Y ) .

3.5.5

(3.4) s = 1, I = Ni

∂tP (I, t) = −∂I[(λ− 1)IP (I, t)− ∂I

[(λ+ 1)I

2P (I, t)

]](3.80)

. . (3.80) ,{∂tP (I, t) = −∂IJ(I, t)J(I, t) = (λ− 1)IP (I, t)− ∂I

[(λ+1)I

2 P (I, t)]

(3.81)

.

. ∂+f(x) = (f(x + Δx) − f(x))/Δx, ∂−f(x) =

(f(x)− f(x−Δx))/Δx , (3.81){∂tP (I, t) = −∂+J(I, t)J(I, t) = (λ− 1)IP (I, t)− ∂−

[(λ+1)I

2 P (I, t)]

(3.82)

. .

P (I, t+Δt) = P (I, t)− ∂+J(I, t)Δt (3.83)


3.5 53

1e-005

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10 12 14 16 18 20

P(I)

I

lam=0.5,t=0t= 1.2t= 2.4t= 3.6t= 4.8t= 6.0t= 7.2t= 8.4t= 9.6

t=10.8

(a) λ = 0.5

1e-005

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10 12 14 16 18 20

P(I)

I

lam=1,t=0t=0.6t=1.2t=1.8t=2.4t=3.0t=3.6t=4.2t=4.8t=5.4

(b) λ = 1

1e-005

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10 12 14 16 18 20

P(I)

I

lam=2,t=0t=0.3t=0.6t= 0.9t= 1.2t= 1.5t= 1.8t= 2.1t= 2.4t=2.7

(c) λ = 2

3.9 λ (3.80) .

. λ = 0.5 . I = 0

. λ = 1, λ = 2 .λ

I = L .

I [0,∞] , L [0, L] .

I = 0, I = L .{J(0, t) = 0J(L, t) = 0

(3.84)

. (3.82)⎧⎨⎩J(0, t) = [(λ− 1)IP (I, t)]I=0 −

[(λ+1)I2ΔI P (I, t)− (λ+1)(I−ΔI)

2ΔI P (I −ΔI, t)]I=0

J(L, t) = [(λ− 1)IP (I, t)]I=L −[(λ+1)I2ΔI P (I, t)− (λ+1)(I−ΔI)

2ΔI P (I −ΔI, t)]I=L

⇔{P (−ΔI, t) = 0

P (L, t) = (λ+1)/2(λ−1)Δt−(λ+1)/2

L−ΔIL P (L−ΔI, t)

(3.85)

. .

, I [0, 2] μ = 1, σ2 = 0.1

, 0 .ΔI = L/128, L = 20,Δt = 1/4096, N = 128

.

λ = 0.5, 1, 2 3.9 .


54 3

3.5.6 Q(Y, t) (3.10)

(3.4) s = 1, Y =√Ni (3.10) . s .s = 1 ,s

0 . P (i, t) = P (1, i, t) .

∂P (i, t)

∂t= −∂i

{[(λ− 1) iP (i, t)]− 1

2N∂i [(λ+ 1) iP (i, t)]

}(3.86)

Y =√Ni . ∂i = N

2Y ∂Y , P (i, t)di = Q(Y, t)dY Q(Y, t) =2YN P (Y

2

N , t) .Q(Y, t) .

∂P(

Y 2

N , t)

∂t= − N

2Y∂Y

{[(λ− 1)

Y 2

NP

(Y 2

N, t

)]− 1

2N

N

2Y∂Y

[(λ+ 1)

Y 2

NP

(Y 2

N, t

)]}(3.87)

2YN . ∂tQ(Y, t) .

∂ 2YN P

(Y 2

N , t)

∂t= −∂Y

{[λ− 1

2Y2Y

NP

(Y 2

N, t

)]− λ+ 1

8Y∂Y

[Y2Y

NP

(Y 2

N, t

)]}∂Q (Y, t)

∂t= −∂Y

{[λ− 1

2Y Q (Y, t)

]− λ+ 1

8Y∂Y [Y Q (Y, t)]

}∂Q (Y, t)

∂t= −∂Y

{[λ− 1

2Y − λ+ 1

8Y

]Q (Y, t)−

[λ+ 1

8

]∂YQ (Y, t)

}(3.88)

(3.10) .

3.5.7 SIS

SIS

P (I, t) =λ

N(S + 1) (I − 1)P (I − 1, t)− λ

NSIP (I, t) (3.89)

+ (I + 1)P (I + 1, t)− IP (I, t) (3.90)

. S = N − I, i = IN

1N .

P (i, t) = −∂i[(λ(1− i)− 1) iP (i, t)− ∂i

(λ(1− i) + 1) i

2NP (i, t)

](3.91)

i = a+ b · ξ ⇔ P (i, t) = −∂[aP (i, t)− ∂i

b2

2P (i, t)

](3.92)


3.5 55

(3.91)

a(i) = (λ(1− i)− 1) i (3.93)

b(i) =

√(λ(1− i) + 1) i

N(3.94)

.

3.5.4 .

df(i) = f ′di+1

2f ′′di2 (3.95)

df(i)

dt= f ′a+

1

2f ′′b2 + f ′b · ξ (3.96)

f ′(i)b(i) = const. y = f(i)

y(i) = ArcTan

(√i

λ+1λ − i

)(3.97)

i(y) =λ+ 1

λsin2(y) (3.98)

.

a(i(y)) =(λ+ 1) (λ− 3 + (λ+ 1)cos(2y)) sin2(y)

2λ(3.99)

b(i(y)) =(λ+ 1) cos(y)sin(y)√

λN(3.100)

f ′(i(y)) =λ

2 (λ+ 1) cos(y)(3.101)

f ′′(i(y)) = − λ2(1− 2sin2(y)

)4 (λ+ 1)

2cos3(y)sin3(y)

(3.102)

(3.103)

. f ′(i(y))a(i(y)), f ′′(i(y))b(i(y))2, f ′(i(y))b(i(y))

f ′(i(y))a(i(y)) =sin(y)

4cos(y)(λ− 3 + (λ+ 1) cos(2y)) (3.104)

f ′′(i(y))b(i(y))2 = − λcos(2y)

4Nsin(y)cos(y)(3.105)

f ′(i(y))b(i(y)) =1

2

λ

N(3.106)

. y

y = f ′(i(y))a(i(y)) +1

2f ′′(i(y))b(i(y))2 + f ′(i(y))b(i(y)) · xi (3.107)

=sin(y)

4cos(y)(λ− 3 + (λ+ 1) cos(2y))− 1

2

λcos(2y)

4Nsin(y)cos(y)+

1

2

λ

N· xi (3.108)


56 3

.u(y)

u(y) =((1 + λ)Ncos(2y) + (λ− 8N)log(cos(y)) + λlog(sin(y)))tan(y)

8Ntan(y)(3.109)

. Q(y) ∝ e−u(y)

λ8N , i

P (i) P (i) = Q(y)dydi . ,

.

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59

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, SIR

, ,

. ,

.

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4.1.1

,

, ,

. ,

, , ,

,

. ,

,

, .

,

, .

, , ,

.

,

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,


60 4

, .

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SIS .

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, .

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4.1.2

,

. ,

SIR , .


4.1 61

(a) twitter . [7] .

(b) google [8]. , .

(c)

[2]

(d)

[3]

.

. ,

.

.

. ,

.

,

. , ,


62 4

[9].

,

.

.

, , .

4.2

,

, 3

. , .

4.2.1

, SIR . ,

S,I,R , S ,I

,R . S I

I .I R

( 4.1). .

4.2.2

, .

.

, λ, μ , k I

kI kIkI

k λ(kI), μ(kI) . λ(kI), μ(kI)

4.2.1 kI .

λ(kI) =

⎧⎨⎩

λk2

(kI

cλ

)αλ

for kI ≤ cλ

λk − λk2

(1−kI

1−cλ

)αλ

for kI > cλ(4.1)

μ(kI) =

⎧⎨⎩μ− μ

2

(kI

cμ

)αμ

for kI ≤ cμ

μ2

(1−kI

1−cμ

)αμ

for kI > cμ(4.2)

. 0 ≤ cλ ≤ 1, 0 ≤ cμ ≤ 1, αλ ≥ 0, αμ ≥ 0 .


4.2 63

4.1 .

,

. (a) λ(kI), μ(kI)

(b) ( ) ( ) .

4.2 (a) λ(kI), μ(kI) . α > 1. (b)

. .

. .

.

SIR ,S j I kI ,

j I I

λkI (4.3)

, I . ,I

, μ . (4.2) cλ = 1/2, αλ =

1, cμ = 1, αμ = 0 .


64 4

4.2.3

2.2.2 HY .

.

, , HY

, [10]

.

. 4.2.1 ,

. I , S

I ,R , I

R .

.

I ,

. 4.2.1 . I

, ,μ(kI) ,R

.

.

4.3

,

,

.

[1] . http://www.google.co.jp.

[2] tryworks. . http://ameblo.jp/tryworks, 2012.

[3] . . http:

//www.pref.nara.jp/secure/62295/riyou.pdf, 2011.

[4] , ITmedia. .


65

http://www.itmedia.co.jp/news/articles/0803/07/news044.html, 3 2008.

[5] . . http://www.pref.nara.jp/

secure/69837/top.html, 2011.

[6] J-CAST . 15 . http://www.

j-cast.com/2008/04/21019250.html, 2008.

[7] Alastair Dant and Jonathan Richards. Behind the rumours: how we built our

twitter riots interactive. http://www.guardian.co.uk/news/datablog/2011/

dec/08/twitter-riots-interactive, 2011.

[8] . google . http://www.google.com/trends,

2012.

[9] E. Noelle-Neumann, , . .

, , 1993.

[10] Duncan J Watts and Steven H Strogatz. Collective dynamics of ‘small-world’

networks. Nature, Vol. 393, No. 6684, pp. 440–442, June 1998.



67

5

5.1

§2 , ,

SIR . nR tf ,

, .

nR , nR tf

, . ,

λc λ → ∞ tf, σ(tf)2 .

. , ,

εN nR ,

,

.

§3 ,SIR

. SIR

, , ,

. 1 ,

.

U(Y ) λ > 1 Y = 12

√λ+1λ−1 ,

Y = 1 , nR = 0

nR > 0 . , (λ = λc + ε)

q(ε) q(ε) = ε ,

. , ,

, .


68 5

, λ .

,

. ,SIS

, SIR .

§4 ,SIR ,

. ,S ,I

I . ,

, , ,

.

5.2

.

§2 , nR, tf

,

. , ,

, , .

,§3 , ,

. ,SIS

. ,

.

§4 , ,

. ,

,

, ,

.

,§1 .

,

. §3 ,

,

. ,

, ,

,

. , .


69

A

, .

A.1

. N 〈k〉, 〈k〉N/2 .

, 〈k〉N . N = 4 , 1

2,3 4, 1 3 , {1, 2, 3, 4, 1, 3} .

A.1.1

N k .

. ,k � N ,k � N

i k .i ei[j], j = 0, 1, · · · k − 1

. f [i] i , .

k , f = k, k, · · · , k .

g .

0, 1, · · · , N − 1 k kN . rand(j) 0 ∼ j

. kN/2 .

1. j = 0 .

2. j .

3. 0 ≤ i ≤ N − 1 f [i] f i , imax


70 A

0

50

100

150

200

250

300

350

400

450

0 10 20 30 40 50 60 70 80 90 100

sec.

k

N=1024N=2048N=4096

(a)

0

500

1000

1500

2000

2500

0 10000 20000 30000 40000 50000 60000 70000

sec.

N

k=3k=4k=5

(b)

A.1 N, k .

.

4. r = rand(kN − 2j − 1) .

5. g[r] eimax , r = imax 4

. r j′ .

6. eimax j′ ,ej′ imax .f [imax]

−1 . g imax, j′ . (imax, j

′) .

.

7. f [imax] = 0 4 . f [imax] = 0 j+ = 1 2

.j = kN/2 .

c++ g [multiset] ,ei [set] .f [i] = k − ei.size() .

,f .

k � N ,

,2∼7 N

, . A.1 , N, k

. ,

N = 65536, k = 3 15 ,N = 32768, k = 40 8 , N = 16384, k = 80 7

, .

A.1.2

.


A.2 SIR 71

1. j = 0 .

2. j .

3. 2 r0 = random(N − 1), r1 = random(N − 1) .

4. r0 = r1 er0 r1 3 .

5. er0 r1 ,er1 r0 . (r0, r1)

j . .

6. j+ = 1 2 .j = kN/2 .

A.2 SIR

A.2.1

I 1, S-I

λ . I I , I .

S − I , I P [I − 1]

λP [I − 1] . P [i] P [i] =∑i

0 k[Ilist[i]] . ,I

0 ∼ I − 1 , j

j = Ilist[i] .

1. I = m,S = N − I .

.P, Ilist .

2. I = 0 . rate . rate = λP [I−1]+ I

.

3. . rate

, τ (0, 1) r τ = −log(1 − r)/rate

. τ .

4. (0, rate) rrate ,rrate < I 5 . rrate ≤ I

6 .

5. Ilist[(int)(rrate) R . Ilist, P, I .2

.

6. P [ir] ≥ (rrate− I)/λ < P [ir + 1] ir .

7. Ilist[ir] .

8. S 2 . S I

.Ilist, P, I .2 .


72 A

A.2.2 §4,

.

, . . kI

I ,λpsj [kI , k] S I

. pij [kI , k] I R .st[j]

j , rate =∑

j [λpsj [kI , k]δ(S, st[j]) + pij [kI , k]δ(I, st[j])]

. δ(S, st[j]), δ(I, st[j]) , Ilist Slist ,

P .


73

, , ,

.

,

, A.3 ,

, , .

A.2 2012

.

,

15 16

,

.

2 ,

,

.

,

. ,

,

.

,

,

. ,

,

.

, ,

.

.

. ( A.2:sasa ...)

.


74

A.3

. ,

( )

,

.

, ,

.

, ,M2

5 ,

. ,

,

. 3

, .

,

. , , C C++

. ,

, ,

. .

, , ,

,

. 3 , ,

.

A.4

.

. ,

, ,

,

, , ,

. ( )

( )

, .

,twitter

,

.

,

, ,

.

,

. ,

. .

,


75

( ) , *1

. , .

.

2013 6 1

*1 2012 9 13 ( ) 14 ( )CMRU ” ”


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