Simulações quase estacionárias do processo de contato em redes complexas

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RONAN SILVA FERREIRA

SIMULAÇÕES QUASE ESTACIONÁRIAS DO PROCESSO DE CONTATO EM REDES COMPLEXAS

Dissertação apresentada à Universidade Federal de Viçosa, como parte das exigências do Programa de Pós-Graduação em Física Aplicada, para obtenção do título de Magister Scientiae.

APROVADA: 13 de fevereiro de 2009.

Prof. José Arnaldo Redinz (Coorientador)

Prof. Marcelo Lobato Martins (Corientador)

Prof. Everaldo Arashiro Hallan Souza e Silva

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δ

η

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{A} ⇔ {X} ⇔ {F} {A}

{X} {F}

{A} {X} {X} {A}

{X} {F}

A B C D

D

A B C D

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i N

σi = 0,1

σi = 0,

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N t i nλ/z n λ z i

σ = (σ1, σ2, σ3, ..., σi, ..., σN)

wi(σ) =

λ

z (1−σi)

j

σj +σi,

j i i

wi(σ) =

λ z

j

σj,

i

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d

dtP(ξ, t) =

N

i=1

[wi(ξi)P(ξi, t)−wi(ξ)P(ξ, t)],

ξ = (ξ1, ξ2, ξ3, ..., ξi, ..., ξN)

t ξi

i ξi _{= (ξ1, ξ2, ξ3, ...,}_(ξ

i)⋆, ..., ξN) i

t ξ P(ξ, t)

f(ξ)

f(ξ) =

{ξ}

f(ξ)P(ξ, t),

d

dtf(ξ) = d dt

{ξ}

f(ξ)P(ξ, t)

=

{ξ}

f(ξ)d

dtP(ξ, t)

d

dtf(ξ) =

{ξ} f(ξ) N i=1

wi(ξi)P(ξi, t)−wi(ξ)P(ξ, t)

= {ξ} N i=1

[wi(ξ)P(ξ, t)−wi(ξ)P(ξ, t)]f(ξ)

= N i=1 {ξ}

f(ξi)wi(ξ)P(ξ, t)−f(ξ)wi(ξ)P(ξ, t)

= N i=1 {ξ} {

f(ξi)−f(ξ)

wi(ξ)P(ξ, t)}

=

N

i=1

f(ξi)−f(ξ)

wi(ξ)

.

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σ = (σ1, σ2, σ3, ..., σi, ..., σN) σi = (σ1, σ2, σ3, ...,(σi)⋆, ..., σN)

d

dtσi =

σi−σi

wi(σ)

= [1−σi−σi]wi(σ)

= [1−2σi]wi(σ).

d

dtσi =

[1−2σi]

λ

z (1−σi)

j

σj+σi

= λ z

j

(1−σi)σj − σi

= λ(ρ−φ)−ρ

= (λ−1)ρ−λφ,

ρ≡ σi φ≡ σiσj

φ

σ= (σ1, σ2, σ3, ...)

P(σ) φ

i j

φ=

σσiσjP (σ)

σP(σ)

.

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P (σ) =P (σi₎_P _(σj₎

φ =

σσiσjP (σi)P (σj)

σP (σi)P (σj)

= σi σj.

φ ≡ σi2 ≡ρ2

d

dtρ= (λ−1)ρ−λρ 2

.

λ =λ = 1

λc = 1

⎧ ⎨

⎩

¯

ρ = 0, λ≤1

¯

ρ = 1−λ−1

, λ >1

¯

ρ t→ ∞

¯ ρ= 0

¯ ρ

λ > λc

∆ =λ−λc >0

¯

ρ= ∆β+O(∆2).

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λ ρ λc λ ρ λ_c

λ λc

λ= 1

ρ(t)∼ ∆

λ−e−∆t,

τ

λ → λc |∆|−1

ν

τ ∝ |∆|−ν_.

ξ

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λ→λc

ξ ∝ |∆|−ν⊥_,

ν⊥ n(t) = _N =1 σ (t) , σ R2

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λ= 3.0 λ= 3.29785 λ= 3.5

λc = 3.297848(20)

∆ → 0 ξ → ∞

τ → ∞

t= 0

ρ(r, t; 0,0)≃tξ−dz/2

f(r/tz/2

,∆t1/ν_),

d f

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P(t)

P(t) → P∞ > 0

t → ∞ P∞ → ρ¯

P →aβP,

∆→a∆,

t→a−ν_t.

P(∆, t) =aβP(a∆, a−ν_t).

a >0 ta−ν = 1

P(∆, t) =t−β/ν_P_(∆t1/ν_,_1).

P(t)≃t−δΦ(∆t1/ν_),

Φ ∆t1/ν _δ_≡_β/ν

tξ−dz/2

∆ = 0

n(t)∝tξ

n(t) = =1 σ (t) =

ρ(r, t)ddr

=

tξ−dz/2

f(r/tz/2

,∆t1/ν_)dd_r

= tξ−dz/2

f(r/tz/2

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u=r/tz/2

n(t)∼tξf′(∆t1/ν_).

P(t) ∝ tδ,

n(t) ∝ tξ,

R2(t) ∝ tz.

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δ

10

2

10

4

10

6

t

10

1

10

2

n(t)

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δ 0.15947(3)a _0.4505(10)b _0.730(4)c ₁d

η 0.31368(4)a _0.2295(10)b _0.114(4)c ₀d

z 1.26523(3)a _1.1325(10)b _1.052(3)c ₁d

d dc = 4

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N ∆

ξ

N/ξ

∆N1/ν⊥

λc

N → ∞ ρ¯ ∝ ∆β _N

∆

ρq(∆, N)∝N−β/ν⊥f(∆N1/ν⊥),

f(x) f(x) → xβ

x→ ∞ x→0 ρq

∆ = 0

¯

ρq(0, N)∝N−β/ν⊥,

β/ν⊥

ν⊥

∂

∂tρ(x, t) = (λ−1)ρ(x, t)−λρ 2

(x, t) +D∇2ρ(x, t).

ρ(t) −→ ζ−β/ν⊥_{ρ(ζx, ζ}2/z_t)

ζ2/z ∂

∂tρ(x, t) = (λ−1)ρ(x, t)−ζ

−β/ν⊥_λρ2_{(x, t) +}_Dζ2_∇2_{ρ(x, t).}

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β ν

ν⊥ 12

pc = λ/(1 +λ)

pa = 1−pc

pc

pa

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λ = λ_c N

δ =−0.16

N → ∞

λ = λc

λc = 3.297848(20)

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2 2,5

3 3,5

4 4,5

5

λ

0 0,2

0,4

0,6

0,8

ρ

q

_{L = 20}

L = 100

L = 500

L = 1000

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R R = (υ, ε)

υ

ε

i j

(i, j) i j

M = {aij} N ×N

aij =

⎧ ⎨

⎩

1, (i, j)∈ε;

0, (i, j)∈/ε.

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N(N −1)/2 p

k

ki i

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p

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N

n n

n/2

n= 2

p

n > 4

k

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P(k) k k k k k k′ k

P(k) =

k′

P(k′|k) = 1.

p= 0.01 p= 1.0 p= 0.45

p= 1.0

p = 0.01 0.045 0.1

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p n= 4

C

ki ki(ki − 1)/2

ci

Ci i i

Ci =

2ci

ki(ki−1)

.

p = 0

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p = 0

L

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L

C

L L

p2/Nk N

N → ∞ p

L C

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P(k)∼k−γ.

m0

m

Π

Π(ki) =

ki

jkj

,

t m0 m

N =t+m0 mt

m0 = 3

m= 3

k2

=

∞

m0

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m0 = 3 m= 3

m0 1 k

k2

∼

∞

m0

k2−γ_dk.

γ >3

k

k=

∞

m0

kP(k)dk.

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P(k) k

k ≫ k k ≪ k

∂

∂tρ(t) =−ρ(t) +λkρ(t)[1−ρ(t)].

λ

∂tρ(t) = 0

λc =k

⎧ ⎨

⎩

ρ = 0, λ < λc

ρ ∼ ∆≡λ−λc, λ > λc

ρ

λ

λc = 1/K K

λc = 1/K = 0.1666

K = 3 λc =

0.1643±0.01

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∂

∂tρk(t) =−ρ(t) +λk[1−ρk(t)]Θ(ρ(t)),

ρk(t) k

k [1−ρk(t)]

λ k Θ(ρ(t))

ρ

λ Θ

∂tρ(t) = 0

ρk =

kλΘ(λ) 1 +kλΘ(λ).

Θ

s

sP(s)

Θ(λ) =

k

kP(k)ρk

ssP(s)

.

ρ

ρ=

k

P(k)ρk.

k

P(k) = 2m2 /k−3

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k=

∞

m

kP(k)dk = 2m,

Θ(λ) =mλΘ(λ)

∞

m

1 k3

k2

1 +kλΘ(λ)dk,

Θ(λ) = e

−1/mλ

mλ (1−e

−1/mλ₎−1 .

Θ

ρ∼e−1/mλ.

λc = 0

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k

ρ(t) =

k

ρk(t)P(k),

ρ(t)

[1−ρk(t)] k

k

k′

P(k′_|_k)_ρ k′(t)

ρk(t)

∂ρk(t)

∂t =−ρk(t) +λk[1−ρk(t)]

k′

P(k′_|_k)ρ k′(t) k′ .

1/k′

k′

λ

k

P(k′_|_k)

k knn

k

knn(k) ≡

k′

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k = K/2 K

k

k ∈[k −σ,k+σ]

σ k

ρk(t)≃ρ(t).

k ∈[k −σ,k+σ]

P(k′_|_k)

P(k′|k) = k

′_P_(k′₎

k ,

knn(k)= k2

k,

k

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∂ρk(t)

∂t = −ρk(t) +λ k

k[1−ρk(t)]ρ(t)

= −ρ(t) + λρ(t)

k

(kP(k)−kP(k)ρk(t))

= −ρ(t) + λρ(t)

k [k −

k

kρk(t)P(k)]

≃ −ρ(t) + λρ(t)

k [k −

kρk(t)P(k)dk]

≃ −ρ(t) + λρ(t)

k [k −

k+σ

k−σ

kρ(t)P(k)dk]

≃ −ρ(t) + λρ(t)

k [k −ρ(t)

k+σ

k−σ

kP(k)dk]

≃ −ρ(t) + λρ(t)

k [k − kρ(t)].

k k

∂ρ(t)

∂t =−λρ 2

(t) + (λ−1)ρ(t).

β ν ν⊥

ρ

ρq(∆ = 0, N)∼N−β/ν⊥F(∆N1/ν⊥),

F(x)

ν⊥

k k2 _ν

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n

pa = 1/(1 +λ) a

pc = 1−pa

c

t → t+ 1/No No

λ

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ρq(0, N)∝N−β/ν⊥.

β/ν⊥

λ=λc

−β/ν⊥

λ < λc λ > λc

λ=λc

τρq

N ∆→0

τrhoq ∝N

ν/ν⊥_g(∆N1/ν⊥_).

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x→ ∞ g(x)→ x→0 ∆ = 0

τρq(0, N)∝N

ν/ν⊥_.

λc

k k2

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p= 0.045

λ N

p= 0.45

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N λ= 1.8202

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λc 1.821±0.001 1.409±0.001

ν/ν⊥ 0.50±0.01 0.49±0.03

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N = 200

λ = 1.72 λc ≃ 1.8202 λ = 1.92

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k k2

→ ∞

knn k

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knn k

0 50 100 150 200

k 18 20 22 24 26 28 <knn>

knn k

∆ ≡ λ−λc = 0

β/ν⊥

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λ N

λc β/ν⊥

1

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