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