Network growth under the constraint of synchronization stability Xingang Wang Department of Physics,...
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Transcript of Network growth under the constraint of synchronization stability Xingang Wang Department of Physics,...
Network growth under the constraint of synchronization stability
Xingang WangDepartment of Physics, ZJU
Oct. 17, 2010, Suzhou
Network is growing
External demand: techonological and information networks, etc.
Driving forces
Function requirement: social, economy and biology networks, etc.
1. Growth
2. Preferential attachement j jii kk /
1t
2t
1 nt Smooth growth ?
BA growth model (SFN)[1]
[1] R. Albert and A.-L. Barabasi, Rev. Mod. Phys. 74, 47 (2002).
Network growth in real world
9n 8n 4n
Intermitent growth
Uneven growth 1n
1tEcological networks [3]
0t 1t 2t 3t
Other examples: WWW, Internet, authorship, etc.
[2] P. Holme and B.J. Kim, Phys. Rev. E 65, 066109 (2002).[3] J.I. Perotti, et.al., Phys. Rev. Lett. 103. 108701 (2009).
Technological networks (power-grid)[2]
Dynamic growth !
),,(~/)( nFdttdn
ityFunctional System ;Links
FunctionNonlinear F
Outline:
1. Phenomenon
2. Properties
3. Mechanisms
4. Consequences
The model
1. Growth (BA)2. Preferential attachement (BA)
3. Synchronization stability (functionality)
Synchronizable
Non-synchronizable
n
jijjiii XHXHcXFX
1
)]()([)(
matrixadjacency }{,/ jiijiji akac
[4] A.E. Motter, et.al., EPL 69, 334 (2005).
The viewpoint from evolutionary network
Node dynamics
Growth dynamics
Structure
Time-scale separation
gt time unit for node addition
T charactering time for system dynamics (synchronization)
:Ttg No contraint, BA SFN
:Ttg Adiabatic growth, constraint activated
:Ttg Entangled dynamics
Master stability function (MSF)[5]
[5] M. Barahona and L.M. Pecora, PRL 89, 054101 (2002).
synchronizability 2/ nR
:...0 21 n Eigen-spectrum of C
Necessary condition: 12 / cRR0.0 0.5 1.0 1.5 2.0
-7
-6
-5
-4
-3
-2
-1
0
1
1.50.5
MSF of logistic map a=4
21
A schematic plot on network growth
n
2
~
n~
)~
(2 RRcc
)~
( RRccn
ccc
n R2/
cn
[6] A. Arenas, et.al., Phys. Rep. 469, 93 (2008).
kk
21
~,
21
~22
cnn ?
Questions:
1. Accepting probability
2. Where the new node is connected to
3. The properties of the generated network
)( 1 nn ttt
The boundary eigenvalues
Parameters: 4,8,100 cRkm
0 500 1000 1500 2000
1.56
1.62
1.68
n
n
BA SFN
R=4, Constrained
0 500 1000 1500 2000
0.4
0.6
0.8
2
n
BA SFN
R=4, Constrained
140cn
140cn
(a) (b)
Accepting probability (missing)
M the number of trying additions
0 500 1000 1500 2000
0
5
10
15
20
25
M
n]25,1[,2000 Mn
Intermittent, non-smooth growth
10 100
100
101
102
103
104
P(M
)
M
P(M
)4~)( MMP
Where missing occurs ?
10 1001E-6
1E-5
1E-4
1E-3
0.01
0.1
1
10
P_m
issi
ng
k
R=4 R=3.8
Emergence of super-node
0 500 1000 1500 20000
100
200
300
400
K_m
ax
R=4 R=3.8 BA SFN ()
n
Consequence of dynamic growth
10 100 1000
10-5
10-4
10-3
10-2
10-1
P(k
)
k
BA (SFN) R=4 R=3.8
BA
R=4
R=3.8
Super-node SFN (SN-SFN)
SN-SFN in practice
100 101 102 103
100
101
102
103
P(k
)
k
Internet
Internet at AS level[7] Stock market of New York[8]
[7] M.E.J. Newman, SIAM Rev. 45, 167 (2002).[8] G. Bonanno, et.al., Phys. Rev. E 68, 046130 (2003).
Super-node
Topological properties of SN-SFN
Network average diameter
100 1000
3
<d>
n
BA SFN R=4 R=3.8
cn100 1000
0.1
<c_
n>
n
BA SFN R=4 R=3.8
Averaged clustering coefficient
cn
Another question to BA SFN:
Is preferential attachement anecessary condition ?
Network growth with random attachement
1 10 100
1
10
100
1000
P'(M
)
M10 100
10-2
100
102
104
P'(K
)
K
Missing distribution Missing location
Dynamic growth still
Network growth with random attachement
10 100
10-5
10-4
10-3
10-2
10-1
P(k
)
k
8.2
Dynamics stability Preferential attachement?
SFN with random attachement
n
2
~
n~
)~
(2 RRcc
)~
( RRccn
cn
1)1/(,2 nnn
Star-network
hub
newhub k
1~1
1/1 ~ newii
Syn. Stability iik ~
0.0 0.3 0.6 0.9 1.2 1.5 1.80
200
400
600
N()
n=500n=1500n=2000
Variation of eigen-spectrum
01 Fast increase
11 Slow increaseSN-SFN
Direct simulations
Local dynamics )1(41 lll xxx
,500 thresholdationsynchronizTransient
1 10 100
100
101
102
103
104
P'(M
)
M
Missing distribution
0 100 200 300 400 500 600
30
60
90
120
150
k_m
ax
n
=10-3
=10-5
Super-node
Remarks & discussions
1. The use of synchronization constraint
2. The value of R
3. New viewpoint for network evolution
4. Specific form of growth dynamics
5. Long-time evolution
6. Dynamical basis for PA
Summary
1. Network growth
2. Preferential attachement
Dynamic
Growth dynamics
Thank you