¶ æ m s U q s 8 Ú E U t b v æ q Å w è ¹mercury.yukawa.kyoto-u.ac.jp/~bussei.kenkyu/pdf/02/3/...iii U å ï ¼ Ü t ¤ ¿ ´ Á l h É ¿ Ä ë « ò U ¿ U t X M q M O A L

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

[1] S.N. Dorogovtsev, A.V. Goltsev, and J.F.F. Mendes. Critical phenomena in

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

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

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

[3] F. Liljeros, C.R. Edling, L.A.N. Amaral, H.E. Stanley, and Y. Aberg. The web

of human sexual contacts. arXiv preprint cond-mat/0106507, 2001.

[4] R. Albert and A.L. Barabasi. Statistical mechanics of complex networks. Reviews

of modern physics, Vol. 74, No. 1, p. 47, 2002.

[5] M.E.J. Newman, S. Forrest, and J. Balthrop. Email networks and the spread of

computer viruses. Physical Review E, Vol. 66, No. 3, p. 035101, 2002.

[6] K. Yuta, N. Ono, and Y. Fujiwara. A gap in the community-size distribution of

a large-scale social networking site. arXiv preprint physics/0701168, 2007.

[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

immunization strategy. Physical Review Letters, Vol. 101, No. 5, p. 58701, 2008.

[11] R. Cohen, S. Havlin, and D. Ben-Avraham. Efficient immunization strategies for

computer networks and populations. Physical review letters, Vol. 91, No. 24, p.

247901, 2003.

[12] Jonathan Sondow and Eric W Weisstein. ”harmonic number” from mathworld–

a wolfram web resource. http://mathworld.wolfram.com/HarmonicNumber.

html, 2012.

[13] P. Erdos and A. Renyi. On the evolution of random graphs. Magyar Tud. Akad.

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 . ,

.

[1] 2012 11

(2012 12 )

[2] Ping Yan. Distribution theory, stochastic processes and infectious disease mod-

elling. Mathematical Epidemiology, pp. 229–293, 2008.

[3] S.N. Dorogovtsev, A.V. Goltsev, and J.F.F. Mendes. Critical phenomena in

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

[4] V.M. Eguiluz and K. Klemm. Epidemic threshold in structured scale-free net-

works. Physical Review Letters, Vol. 89, No. 10, p. 108701, 2002.

[5] DA Rand. Correlation equations and pair approximations for spatial ecologies.

Advanced ecological theory: principles and applications, pp. 100–142, 1999.

[6] M.E.J. Newman. Spread of epidemic disease on networks. Physical Review E,

Vol. 66, No. 1, p. 016128, 2002.

[7] E. Kenah and J.M. Robins. Network-based analysis of stochastic sir epidemic

models with random and proportionate mixing. Journal of theoretical biology,

Vol. 249, No. 4, p. 706, 2007.

[8] L.K. Gallos and P. Argyrakis. Distribution of infected mass in disease spreading

in scale-free networks. Physica A: Statistical Mechanics and its Applications, Vol.

330, No. 1, pp. 117–123, 2003.

[9] A. Lancic, N. Antulov-Fantulin, M. Sikic, and H. Stefancic. Phase diagram of

epidemic spreading–unimodal vs. bimodal probability distributions. Physica A:

Statistical Mechanics and its Applications, Vol. 390, No. 1, pp. 65–76, 2011.

[10] H.C. Tuckwell, L. Toubiana, and J.F. Vibert. Epidemic spread and bifurcation

effects in two-dimensional network models with viral dynamics. Physical Review

E, Vol. 64, No. 4, p. 041918, 2001.


57

[11] D.H. Zanette. Critical behavior of propagation on small-world networks. Physical

Review E, Vol. 64, No. 5, p. 050901, 2001.

[12] 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.

[13] C. Gardiner. Handbook of stochastic methods: for physics, chemistry & the

natural sciences,(series in synergetics, vol. 13). 2004.

[14] R. . : , pp.

95–100. , 2011.

[15] D. Stauffer, A. Aharony, . , pp. 48–53.

, 2001.



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 ” ”


Documents

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