Dynamics on Networks.
3. Synchronization
Ernesto Estrada
Department of Mathematics & Statistics,
University of Strathclyde
Glasgow G1 1XH, UK
www.estradalab.org
Σύγ χρόνος – sharing a common property in time
BIOLOGICAL
SOCIAL
• Ensemble of doves (wings in synchrony)
• Collective firing of neurons
• Fireflies, crickets, frogs
• Communities of consciousness and
crises
• Football (mexican wave: la ola, ...)
• Rhythmic applause
Video Sync
n identical dynamical units
each unit has an internal degree of freedom x i
 
x i  f xi   c Lij H x j ,
n
i  1,2,, n
j 1
xi
internal degree of freedom
c0
H
 L H x    H x   H x 
n
j 1
ij
i
jV i
L  Δ  D  A
i
j
 
L  Lij   nn
1
2
3
4
6
5
 1 1 0 0 0 0 
  1 3  1 0 0  1
 0 1 3 1 1 0 
L
0 0 1 2 1 0 
 0 0  1  1 3  1
 0 1 0 0 1 2 


x1 t   x2 t     xn t   st , as t  
•equilibrium point
•periodic orbit
•chaotic orbit
i
 i  s
xi  s   i
f xi   f s    i f s 
H xi   H s    i H s 


i  f s  i  c Lij H s  j
j
Let
1  2    n1  n  0.
It is known that the system of eqs. can
be decoupled by using the set of eigenvectors  i , which are an appropriate set
of linear combinations of the perturbations:
i   f s   ci H s  i
At short times, one can assumes that s almost does not vary and
i t    i0 exp  f s   ci H s t
   max  f s   H s 
s

0
1
2
1  2
Q :

 n1  1

Stanley Milgram
1933-1984
1,305 mi.
10
No.of Completed Chains
12
10
8
6
4
2
0
0
2
4
6
8
10
12
No. of Intermediadiaries needed to reach the
Target Person
dij  5
2  Number of transitive relations
Ci 
Number of possible transitive relations
n  1 n  k  1

d
2kn


C
4  k  1
3 k 2
ln n  
1
d

ln pn 2
 
Cp
15
d
1847
3.65
C
0.74
0.79
2.99
0.00027
16
d 10.5
2.65
2.25
C 0.69
0.28
0.05
17
d 926
18.7
12.4
C 0.3
0.08
0.0005
18
D. J. Watts
S. H.
Strogatz
19
1 ~ 2r  11  2 / 3 
n1 ~ 2 r r  12r  1 / 3n
2
n
nodes
k  2r
r  1 r  n
1
n2
Q :
~
 n1 r r  1
2

Q
np  2 p1  p n log n
np  2 p1  p n log n
Q
  2m / nn  1
regular
SW
random
pu 
kv
k
w
w
4/9
1/ 6
1/ 9
1/ 9
1/ 9
1 / 18
4/9
u
BAn, d 
pk  
2d d  1
k k  1k  2
~ k 3
pk   ak 
ln pk    ln k  ln a
pck   cak 
ln pck    ln k  ln ca
pck   pk 
dxi

 F xi   
dt
ki
 L H x 
ij

i
j

CK L




det K L  I  det K
 / 2
LK
 / 2
 I

dxi

 F xi   
dt
ki


ki H xi    H x j 
j ~i


 F xi   ki1  H i  H xi 
H i   H x j / ki
j ~i
If the network is sufficiently random, and if the system
is close to the synchronised state s, then
H i  H s 
and
dxi
 F xi   ki1  H s   H xi 
dt
Condition of synchronizability:
1 
i
1  k
  2 , i
as soon as one node has degree different from the others
the network is more difficult to synchronise.
 k
 max
 k min

Q

 k min
 k max
1 



if   1
1 



if   1
The minimum value of the eigenratio is then obtained
for   1.
Q
 3
 5
 7
We consider n planar rotors with angular phase xi and natural
frequency i coupled with strength K and evolving according
to:
dxi
 i  K  sin xi  x j 
dt
j ~i
The level of synchronization is quantified by the order
parameter:
n
1
ix j t 
i t 
r t e

e

n
j 1
n
1
ix j t 
i t 
r t e
 e
n j 1
j
xj
r 
The radius measures the coherence and  t  is the average
phase of the rotors.
Multiply both sides of
by
e
 ixl
. Thus,
n
1
ix j t 
i t 
r t e
 e
n j 1
1 n
r sin   xl    sin x j  xl 
n j 1
So that
dxl
0
 l  K r sin xl  
dt
K 0  Kn
r
1
0
r 0
 K  Kc

Kc
K  Kc
0

K0

K 0  Kc
8
6
synchronized
Kc
4
2
0
0
0.2

0.4
p
0.6
r n, K   n   / F K  Kc n1/
0.8
1

F is a scaling function and describes the divergence of the
typical correlation size K  Kc  .
order
parameter
Kc  c
k
k2
where c depends on the distribution
of the individual frequencies.
rlink
1
1

lim

2m j l ~ j t  t
rlink
K
  t
 e


i x j t  xl t 
1
Dij  Aij lim
t  t
K
  t
 e


i x j t  xl t 
K