Synchronization in large networks of coupled phase oscillators: The effect of network topology

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Synchronization in large networks of coupled
phase oscillators The effect of network topology
Edward Ott
University of Maryland
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Examples of Synchronized Oscillators

• Cellular clocks in the brain.
• Pacemaker cells in the heart.
• Pedestrians on a bridge.
• Electric circuits.
• Laser arrays.
• Oscillating chemical reactions.
• Bubbly fluids.
• Neutrino oscillations.
• Parkinsons disease.

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Male fireflies flashing in unison
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Circadian Rhythm
Incoherent
Coherent
Yamaguchi et al., Science 302 p.1408 (2003).
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Synchrony in the Brain
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Coupled Phase Oscillators
Change of variables
Limit cycle in phase space
Many such phase oscillators
i1,2,,N 1
Couple them
Assumption Attraction to limit cycle attractor
is strong.
Global coupling
Kuramoto 1975 (H.Daido PRL 1994)
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Framework

• N oscillators described only by their phase q. N
  is very large.

?n

• The oscillator frequencies are randomly chosen
  from a distribution g(?) with a single local
  maximum.

qn
g(?)
(We assume the mean frequency is zero)
?
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Kuramoto Model All-to-All Coupling
n 1, 2, ., N k (coupling constant)

• Assumes sinusoidal all-to-all coupling.
• Macroscopic coherence of the system is
  characterized by

order parameter
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Order Parameter Measures Coherence
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Results for the Kuramoto model
There is a transition to synchrony at a critical
value of the coupling constant.
r
Synchronization
Incoherence
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Example Order parameter in the incoherent case
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Example Order parameter in the coherent case
k
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The order parameter
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N?8
Introduce the distribution function F(q,w,t)
the fraction of oscillators with phases in the
range (q,qdq ) and freqs. in the range (w,wdw )

Conservation of number of oscillators
0
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(uniform distribution in angle)
0
0
Q. Is it stable?
A. Yes, for kltkc. No, for k gt kc.
perturbation
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Look for a solution of the form
Get
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g(w)
D
w
3 poles
Close integral in upper half plane
r
kc
k
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Models of coupled heterogeneous oscillators
All-to-all Network.
Sine coupled phase oscillators
Kuramoto model
Sine coupled phase oscillators
All-to-all Network.
More general network.
More general dynamics.
Ichinomiya, Phys. Rev. E 04 05 Restrepo et
al., Phys. Rev. E 05 Chaos 06
Ott et al 02 Pikovsky et al 96 Baek et al 04
Topaj et al 02
More general Network.
More general dynamics.
Restrepo et al., PRL06 Physica D06
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Why networks?
In recent years it has been realized that many
processes in nature can be described in terms of
interaction of elements in networks.
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d2 4
Nodes
n 1,2,,N
4
1
Links
2
dn Degree of a node number of links
6
5
d6 2
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Real world networks are often complex
Map of the Internet
http//www.caida.org/Papers/Nae/
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In order to study the effect of a network on the
emergence of synchronization, we will maintain
the phase dynamics, but will introduce a network
in the problem.
Sine coupled phase oscillators
All-to-all Network.
More general Network.
Sine coupled phase oscillators
What follows is based on Juan G. Restrepo, Ed
Ott, Brian R. Hunt, Physical Review E, 71
036151(2005). Juan G. Restrepo, Ed Ott, Brian R.
Hunt, Chaos (2006).
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A visual example
color phase
lt
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Kuramoto Model on a Network
The network is introduced by means of a matrix A
m is not connected to n Anm 0. PDF of
frequencies symmetric about 0.
The nonzero elements of A can have any positive
or negative value and correspond to the
interaction strength at each link.
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Order Parameter Description
Local order parameter for node n where
time average.
Global order parameter where is the node
degree PROBLEM Find r vrs. k
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Order Parameter Form of the Dynamics
and
yield
where
.
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Neglect of hn
Assumption The degree is large.
There are many terms in the determining
.
is noise-like.
Neglect hn compared to O(dn)
The effect of finite hn is treated in Restrepo,
Ott and Hunt Phys. Rev. E (2005), Sec 6.
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Time Averaged Approximation

Locked nodes
Putting this in the equation for
, using and
, gives the time
averaged approximation.
(Restrepo, Ott, Hunt, PRE 2005)
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Frequency Distribution Approximation
Assuming rn is independent of ?n we can average
over the frequencies to get
Knowledge of each individual frequency is not
needed.
The onset of synchronization corresponds to rn ?
0. We obtain an eigenvalue equation!
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Mean Field Approximation
Assuming rn r dn , we get the mean field
approximation ( Ichinomiya, 2004 )
Near the transition, it has the extra assumption
that the eigenvector u of A with the largest
eigenvalue satisfies un ? dn.
Only knowledge of the degree and frequency
distributions is required.
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Results

• For a large class of networks, there is still a
  transition to synchrony at a critical coupling
  strength kC.

• We have developed several approximations to the
  order parameter past the transition.

• The critical coupling strength is given by

Network
All-to-all Kuramoto model
is the largest eigenvalue of the matrix A
defining the network.
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Generation of Networks to Test Our Theory
We consider a network with a degree dist., p(d).

1. Specify the degree dn of each node n.
2. Imagine dn spokes stick out of node n.
3. Randomly connect pairs of spokes, avoiding self
   and double connections.

4
2
3
1
3
3
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Example Scale-Free Networks
We prescribe a degree distribution of the form
p(d) ? d -? , d ? 50, and N 2000.
r2
Simulation
Time averaged theory
Frequency distribution approximation
Mean field theory
1
2
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Example Scale-Free Networks
r2
Simulation
Time averaged theory
Frequency distribution approximation
Mean field theory
1
2
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Some Implications
Heterogeneity in the degree distribution
Red network
Green network
l lt l
For a given number of connections, a
heterogeneous network tends to synchronize easier
(smaller kC).
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Some Implications
Randomness in the interaction strengths
Red network
Green network
1
3
4
3
3
3
3
3
2
5
3
3
3
3
3
3
l lt l
For a given average interaction strength, a
network with higher randomness tends to
synchronize more easily.
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Some Implications
Degree-degree correlations
Disperse hubs
Highly connected core
l lt l
For the same degree distribution, a highly
connected core tends to synchronize more easily.
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Further Work
All-to-all Network.
Coupled phase oscillators (simple dynamics).
Kuramoto model (Kuramoto, 1984)
All-to-all Network.
More general network.
More general dynamics.
Coupled phase oscillators.
Ichinomiya, Phys. Rev. E 04 Restrepo et al.,
Phys. Rev E 05 Chaos06
Ott et al.,02 Pikovsky et al.96 Baek et al.,04
Topaj et al.01
More general Network.
More general dynamics.
Restrepo et al. Physica D 06
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Networks with General Node Dynamics
Uncoupled node dynamics
Could be periodic or chaotic. Kuramoto is a
special case
Main result
Q depends on the collection of node dynamical
behaviors (not on network topology). l Max.
eigenvalue of A depends on network topology (not
on node dynamics).
Restrepo, Hunt, Ott, PRL 06 Physica D 06
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(No Transcript)
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Approximating the largest eigenvalue of network
adjacency matrices

• Reference Paper with the above title Restrepo,
  Ott, Hunt, Phys. Rev. E 76, 056119 (2007).
• Markovian theoryP(din,dout din,dout )
  prob. a random out-link from a node with
  (din,dout ) connects to a node with (din, dout
  ).
• ? assortivity coefficient

high in-degree node
high out-degree node
low in-degree node
low out-degree node
? gt 1 Assortitive
? lt 1 Disassortitive
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Test of Markovian theory
Randomly generated As with N25,000, ltdgt 20.
P(d) ? d-2.5 Dashed line expansion about
?1. Solid black line actual eigenvalue. Squares
Markovian approximation.
P(din,dout ) ? (dindout )-2.5
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Summary

• For a large class of networks, there is a
  transition to synchrony at a critical coupling
  constant determined by the maximum eigenvalue of
  the adjacency matrix.
• A larger maximum eigenvalue of the adjacency
  matrix favors a lower threshold for
  synchronization.
• Heterogeneity in the degree distribution,
  randomness in the couplings, and positive degree
  correlations favors synchronization.
• Our papers can be obtained from
• http//www.chaos.umd.edu/umdsyncnets.html

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What is the effect of complex interaction
structure on dynamical processes taking place in
networks?
We will focus on the synchronization of coupled
heterogeneous oscillators.
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Effect of the nodes with small degree
So far we have been using the average value of
rn. However,
Finite degree dn
Fluctuations
What is the effect of the fluctuations?
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Perturbations from the incoherent state
The incoherent state is given by
We introduce perturbations
and assume .
Onset of synchronization
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Time fluctuations as noise
The original equations are
If the degree dn is large but finite and the
system is incoherent, the coupling term can be
approximated by a Gaussian random noise with
diffusion coefficient Dn.
Smaller degree dn
Larger diffusion coefficient Dn
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Theoretical results
We obtain the eigenvalue equation
If Dn 0, letting we get

Positive Dn corresponds to a smaller growth rate
and thus to a larger critical coupling strength.

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Comparison
We plot the order parameter r2 found numerically
and the growth rate s found from our theory for
three networks in which all nodes have degree
100, 50 and 20. (N 500)
r2
d 20
d 50
d 20
d 50
d 100
d 100
s
k/kc
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AN EXAMPLE
5000 Lorenz systems
Network N5000
References Restrepo, Ott, Hunt, Physica D
(2006) Ott, So, Barreto, Antonsen,
Physica D(2002)