Punctuated Equilibrium and Criticality on Network Structures

12004 년 봄 물리학회성균관대학교 ( 수원 )

Punctuated Equilibrium and

Criticality on Network Structures

I. Introduction

II. Random Neighbor Model

III. BS Evolution Model

on Network Structures

IV. Results

V. Summary

Sungmin Lee, Yup KimKyung Hee Univ.


I. Introduction

Self-organized critical steady state

S.J.Gould (1972)

Instead of a slow, continuous movement, evolution

tends to be characterized by long periods of virtual

standstill ("equilibrium"), "punctuated" by episodes

Of very fast development of new forms

The "punctuated equilibrium" theory


The Bak-Sneppen evolution model

0.2 0.30.15

0.40.45

0.7 0.90.35

0.10.55

0.75

0.5 0.80.65

0.60.25

Fitness - An entire species is represented by a single fitness - The ability of species to survive - The fitness of each species is affected by other species to which it is coupled in the ecosystem.

At each time step, the ecology is updated by (i) locating the site with thelowest fitness and mutating it by assigning a new

randomnumber to that site, and

10 if ),,1( Ni PBC

P.Bak and K.sneppenPRL 71,4083 (1993)

0.2 0.30.15

0.40.45

0.7 0.90.95

0.47

0.22

0.75

0.5 0.80.65

0.60.25

Lowest fitness

(ii) changing the landscapes of the two neighbors by assigning new random numbers

to those sites

New lowest fitness

Snapshot of the stationary

state66702.0cf

M.Paczuski, S.Maslov, P.BakPRE 53,414 (1996)


Gap and Critical fitness

)(min sf : The lowest fitness at step s

)](,),0(max[)( minmin sffsG

cfsG )(


Avalanche - subsequent sequences of mutations through fitness below a certain threshold

Distribution of avalanche

sizes in the critical state

Punctuated equilibria - long periods of passivity interrupted by sudden bursts of activity

The activity versus time in a local segment

of ten consecutive sites.

SSP ~)(

1d 2d

1.07(1) 1.245(10)


(1) biospecies 의 연관구조가 Scale-free Network 나 Random Network 일 때 evolution 의 특성 연구

Motivation of this study

(2) Evolution and Punctuated

Equilibrium 의 Network

structures 에서의 Self-Organized

Criticality (3) What is the best structure

for the adaptation of species-correlation? (Is there evolution-free

network?)


Exactly solvable model

0.2 0.30.15

0.40.45

0.7 0.90.35

0.10.55

0.75

0.5 0.80.65

0.60.25

- At each time step, the ecology is updated by (i) locating the node with the lowest fitness and mutating it by assigning a new random number to that site (ii) changing the landscapes of randomly

selected K-1 sites by assigning new random

numbers to those sites.

II. Random Neighbor Model

0.2 0.30.15

0.77

0.45

0.34

0.90.35

0.52

0.55

0.75

0.50.22

0.65

0.60.89

Lowest fitness

New lowest fitness

ix: the i-th smallest fitness value

)(xpi : the distribution for ix)(xp : the distribution of all fitness values in the ecology

)()()()( 1)!()!1(

! xQxpxPxp iNiiNi

Ni

x

xpxdxP0

)()( 1

)()(x

xpxdxQwhere


),(),()1,( 11 txptxptxp N

NK

NNK txptxp

),(),( 11

11

The evolution equation for ),( txp

NKxp )(

1)( KKxp

)( 11NK x

)( 11NKx

for

for

SSP ~)( 5.1

(1) identifying each burst with a node(2) and each of K new fitness values resulting from a burst - with either a branch rooted in that node (if the fitness value is less than the threshold value) - with a leaf rooted in the same node (if the fitness value is above threshold)

Avalanches

Kcf10 1

The limit is necessary to obtain the tree structure.N

t


- generate network structures with N nodes- A random fitness equally distributed between 0 and

1, is assigned to each node.- At each time step, the ecology is updated by (i) locating the node with the lowest fitness and

mutating it by assigning a new random number to that site (ii) changing the landscapes of the linked neighbors

by assigning new random numbers to those nodes.

III. BS Evolution Model on Network Structures

0.2

0.75

0.6

0.7

0.8

0.9

0.4

0.3

0.5

0.1

0.25

0.45

0.2

0.21

0.6

0.7

0.62

0.9

0.4

0.3

0.5

0.31

0.25

0.98


dkkPNk

max

)(1

dkkNk

max

1

11

~max Nk

kkP ~)(Scale-free network

3

dkkPkkk

max

0

22 )(

13

~~ 3max

2

Nkk

13

21

1~

1~

Nkfc

Prediction for our model

)(kP: degree distribution

Mean degree 4 kin this work

- We predict the critical value is proportional to the fluctuation of degree

13

21

1~

1~

Nkfc

3

- We predict the critical behavior of random network is similar to random neighbor model.

Random network

Scale-free network

( logarithmic)3

similar to random network

kkckk f 11

11

kcf 1~ )1( kk


IV. ResultsRandom Network

10-6 10-5 10-4 10-30.3

0.4

0.5

0.6 N f

c(N)

-------------------------1000 0.425410000 0.3603100000 0.33221000000 0.3325

f c(N

)

1/N

3/1cf

1 10 100 1000

10-6

10-5

10-4

10-3

10-2

10-1

100

-1.650(8)

S

P(S

)

)8(650.1

4 kMean degree

510N


Scale-free Network

3 4 kMean degree

10-6 10-5 10-4 10-3

0.01

0.1

=2.34 =2.75

0.26(1)

0.379(4)

f c(N

)

1/N

10-6 10-5 10-4 10-3

0.05

0.10

0.15

0.20

0.25

0.30 =3.01

f c(N

)

1/N

13

21

1~

Nfc

2462.0

1~N

0714.0

1~N

)34.2(

)75.2(

xc Nf

1~

Logarithmic(?)


0 500 1000 1500 2000

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

1 10 100 1000

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

P(S

)

S

-1.88(1)

P(S

)

S

(?)0cf

34.2

510N

10-6 10-5 10-4 10-3

0.0

0.1

0.2

0.3

f c(N

)

1/N

------------------------------- 3.01 0.035(6) 2.75 0.021(5) 2.34 0.004(2)

)( Nfc


0 2000 4000 6000 8000 1000010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

1 10 100 1000 1000010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

S

P(S

)-1.74(1)

S

P(S

)

75.2

01.3

510N

0 10000 2000010-6

10-5

10-4

10-3

10-2

10-1

100

1 10 100 1000 1000010-6

10-5

10-4

10-3

10-2

10-1

100

S

P(S

)

-1.64(1)

P(S

)

S

510N


Scale-free Network

10-6 10-5 10-4 10-30.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

f c(N

)

1/N

3

1 10 100 1000 1000010-6

10-5

10-4

10-3

10-2

10-1

100

-1.65(1)

S

P(S

)

))1(15.0(finitefc

30.44 kMean degree

510N


10-6 10-5 10-4 10-30.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

f c(

N)

1/N

))1(28.0(finitefc

68.5

1 10 100 100010-5

10-4

10-3

10-2

10-1

100

-1.64(2)

S

P(S

)

510N


IV. Summary

cf

2.341.88(1)+exponenti

al

2.751.74(1)+exponenti

al

3.01 1.64(1)+(?)

4.30 1.65(1)

5.68 1.64(2)

(?)0cf

finitefc

SSP ~)(

cf

1.650(8)

SSP ~)(3/1cf

◆ Random Network

◆ Scale-free Network

◆ On SFN the system is not critical because all species are adapted.

3

◆ SFN is critical 3

◆ On SFN the system is critical and it’s behavior is similar to RN.

3

◆ is similar to Earthquake model on RN.

3, RNS.Lise and M.Paczuski PRL 88, 228301 (2002)

Documents

Punctuated Equilibrium and Criticality on Network Structures