Bak-Sneppen Evolution models on Random and Scale-free Networks
18
1 Satellite Meeting of STATPHYS 22(KIAS) Bak-Sneppen Evoluti on models on Random and Scale-free Netw orks I. Introduction II. Random Neighbor Model III. BS Evolution Model on Network Structures IV. Results V. Summary Sungmin Lee, Yup Kim Kyung Hee Univ.
description
Bak-Sneppen Evolution models on Random and Scale-free Networks. I. Introduction II. Random Neighbor Model III. BS Evolution Model on Network Structures IV. Results V. Summary. Sungmin Lee, Yup Kim Kyung Hee Univ. I. Introduction. The "punctuated equilibrium" theory. - PowerPoint PPT Presentation
Transcript of Bak-Sneppen Evolution models on Random and Scale-free Networks
1Satellite Meeting of STATPHYS 22(KIAS)
Bak-Sneppen Evolution models on Random and Scale
-free Networks
I. Introduction
II. Random Neighbor Model
III. BS Evolution Model
on Network Structures
IV. Results
V. Summary
Sungmin Lee, Yup KimKyung Hee Univ.
2Satellite Meeting of STATPHYS 22(KIAS)
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
3Satellite Meeting of STATPHYS 22(KIAS)
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)
4Satellite Meeting of STATPHYS 22(KIAS)
Gap and Critical fitness
)(min sf : The lowest fitness at step s
)](,),0(max[)( minmin sffsG
cfsG )(
5Satellite Meeting of STATPHYS 22(KIAS)
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)
6Satellite Meeting of STATPHYS 22(KIAS)
(1) Study for a characteristic of evolution when interacting structures of biospecies are Scale-free Networks or Random Networks
Motivation of this study
(2) Self-Organized Criticality
of Evolution and Punctuated
Equilibrium on Network
Structures (3) What is the best
structure for the adaptation of species-correlation? (Is there evolution-free
network?)
7Satellite Meeting of STATPHYS 22(KIAS)
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
8Satellite Meeting of STATPHYS 22(KIAS)
),(),()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
9Satellite Meeting of STATPHYS 22(KIAS)
- 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
10Satellite Meeting of STATPHYS 22(KIAS)
dkkPNk
max
)(1
dkkNk
max
1
11
~max Nk
kkP ~)(Scale-free network
dkkPkkk
max
0
22 )(
13
~~ 3max
2
Nkk
)(kP: degree distribution
- We predict the critical behavior of random network is similar to random neighbor model.
Random network
Scale-free network
4 kMean degree :
63 10~10N
05.0
c
c
f
ff
Num. Nodes :
- Condition
11Satellite Meeting of STATPHYS 22(KIAS)
IV. ResultsRandom Network
10-7 10-6 10-5 10-4 10-3
0.20
0.22
0.24
0.26
N fc(N)
--------------------------1000 0.257610000 0.22098100000 0.211041000000 0.20748
f c(N
)
1/N
5/1cf
Random Neighbormodel
51
11 Kcf
610N
100 101 102 103
10-6
10-5
10-4
10-3
10-2
10-1
100
f(x)=A*x-exp(-x/xc)
A=0.8 =1.5 x
c=738
P(S
)
S
Consistent with PRL 81,2380 (1998)
5.1
12Satellite Meeting of STATPHYS 22(KIAS)
10-6 10-5 10-4 10-3
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
1/N
f c(N
)
=3.50 =4.30 =5.70
3
finitefc
Scale-free Network
0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.150.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
1000 10000 100000 1000000
0.1
f c(N
)
1/ln(N)
N
fc
fc(N)
~ 1/<K2>
Is consistent withEurophys. Lett., 57, 765 (2002)
3
3
13211 ~~
Nkcf
13Satellite Meeting of STATPHYS 22(KIAS)
3
0cf
10-6 10-5 10-4 10-3
0.00
0.05
0.10
0.15
0.20
=2.75 =2.40 =2.15
f c(N
)
1/N
10-6 10-5 10-4 10-3
0.01
0.1
0.37(2)0.42(3)
0.27(2)
=2.75 =2.40 =2.15
f c(N
)
1/N
13211 ~~
Nkcf
)15.2(~ 739.01
N
)40.2(~ 428.01
N
)75.2(~ 142.01
N
14Satellite Meeting of STATPHYS 22(KIAS)
100 101 102 103
10-6
10-5
10-4
10-3
10-2
10-1
100
f(x)=A*x-exp(-x/xc)
A=0.8 =1.65 x
c=1000
S
P(S
)
610N30.4
70.5
100 101 102 103
10-6
10-5
10-4
10-3
10-2
10-1
100
f(x)=A*x-exp(-x/xc)
A=0.7 =1.5 x
c=1050
S
P(S
)610N
3
15Satellite Meeting of STATPHYS 22(KIAS)
50.3
100 101 102 103
10-6
10-5
10-4
10-3
10-2
10-1
100
A=0.85 B=1.65
f(x)=A*x-B
S
P(S
)
610N100 101 102 103
10-6
10-5
10-4
10-3
10-2
10-1
100
f(x)=0.63*x-2.09
g(x)=0.15*x-1.59
S
P(S
)
Europhys. Lett.,57, 765 (2002)
3
610N
16Satellite Meeting of STATPHYS 22(KIAS)
100 101 102 103
10-5
10-4
10-3
10-2
10-1
100
g(x)=0.04*x-1.2
f(x)=0.32*x-2.3
S
P(S
)15.2
75.2 610N
100 101 102 103
10-6
10-5
10-4
10-3
10-2
10-1
100
g(x)=0.1*x-1.47
f(x)=0.6*x-2.22
S
P(S
)610N
3
17Satellite Meeting of STATPHYS 22(KIAS)
100 101 102 103
10-6
10-5
10-4
10-3
10-2
10-1
100
P(S
)
S
100 101 102 103 104 105 106
10-5
10-4
10-3
10-2
10-1
100
P(S
)
S
** Star networks
510totalN
20starN
05.0
c
c
f
ff
18Satellite Meeting of STATPHYS 22(KIAS)
IV. Summary
cf
2.15
(two power-law regimes)
(star network)
2.3+1.2
2.40
2.27+1.32
2.75
2.22+1.47
3.00
logarithmic
2.06+1.59
3.50
1.65
4.30
1.65
5.70
1.5
0cf
finitefc
cf
1.5)/exp(~)( cSSSSP 5/1cf
◆ Random Network
◆ Scale-free Network
SSP ~)(
)/exp( cSSS
)(SP
~)(SP
)(SP
SSP ~)(
◆ We would like to remark that two power-law regimes are shown in BS model on small world (cond-mat/9905066)
)(~ 21
k
