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Complex Networks overview Proximity oscillator networks Results Bibliography
Numerical analysis of proximity oscillator networks
Giovanni Pugliese Carratelli - M58/30
DIETI - Univesity Federico II of Naples
26 September 2014
To follow knowledge like a sinking star,
Beyond the utmost bound of human thought.
–Ulysses, Lord Tennyson
Supervisor Prof. Franco Garofalo Co-Supervisors Dr. Piero De LellisEng. Francesco Lo Iudice
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
Complex Networksoverview
Proximity oscillatornetworks
Results
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
Complex Networksoverview
Proximity oscillatornetworks
Results
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
Introduction
Network: ensemble of interacting dynamical entities over a web of interconnectionsComplex: behaviour that cannot be explained in terms of the behaviour of each agent
Complex Networks model
xi (t) = fi (xi ) + gN∑i=1
aij (h(xi )− h(xj )), ∀i = 1, . . . ,N (1)
where
fi (xi ) is the independent dynamics of the i-th node
g∑N
i=1 aij (h(xi )− h(xj )) is the interaction term between nodes
g is the coupling gain
aij are the terms of the adjacency matrix A: defines the network topology
h is the output function
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
Topology
The structure of the network influences the dynamics!
Figure: All-to-all topology
Figure: Star topology
Figure: Ring topology
(c)
(a) (b)
Figure: In Fig.(a) an Erdos Renyi network, in Fig.(b) a Small worldnetwork, in Fig.(c) a Scale Free network
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
State dependent topologies
We aim to study the dynamics of a
Proximity Networks
Nodes are connected if their distance lies below a given threshold
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
State dependent topologies
We aim to study the dynamics of a
Proximity Networks
Nodes are connected if their distance lies below a given threshold
State dependent topologies
xi (t) = fi (xi ) + gN∑i=1
aij (x(t))(h(xi )− h(xj )), ∀i = 1, . . . ,N (2)
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
Complex Networksoverview
Proximity oscillatornetworks
Results
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
The Kuramoto model
The Kuramoto model
θi = ωi + gN∑j=1
sin(θi − θj ), ∀i = 1, . . . ,N (3)
Synchronization
A Kuramoto network is completely synchronized [1]⇐⇒
Frequency synchronization,limt→∞ θi (t) = ω, ∀i = 1, . . . ,N
Phase-locking,limt→∞ ||θi (t)− θj (t)|| = 0, ∀i , j = 1, . . . ,N
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
The restricted visibility Kuramoto model
We consider N heterogeneous Kuramoto oscillators moving on a circular route
The proposed model
θi = ωi + gN∑j=1
aij (t)sin(θij ), ∀i = 1, . . . ,N (4)
where, θij = θi − θj is the relative angular position
aij (t) =
{1, if min
{mod (θij ), mod (−θij )
}≤ θvis ≤ π
2
0, otherwise(5)
θvis
Figure: In red an oscillator for which aij = 0, in green an oscillator for which aij = 1
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
Equilibria topologies
Results in [2] hold, i.e. entrainment frequency
ω =
∑Ni=1 ωi
N(6)
is reached, phase-locking is achieved and thus the topology is steady with respect totime.Hence by imposing θi = ω, ∀i = 1, . . . ,N, in Eq. (4) we obtain
ω = ωi + gN∑j=1
aij sin (θji ), ∀i = 1, . . . ,N (7)
that can be recast to
ω − ωi
g=
N∑j=1
aij sin (θji ), ∀i = 1, . . . ,N (8)
a necessary condition for the exitance of a solution is that
g > gmin =maxi |ω − ωi |
N − 1(9)
Topology bifurcation
Multiple equilibria may exist, depending on g and the initial conditions of network (4)
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
Experimental plan
We are interested to
Verify that also for network (4) entrainment frequency is reached for sufficientvalues of g
Evaluate as many possible equilibria topologies for values by varying g and initialconditions and thus build a topology bifurcation diagram of a N = 5 node network
Show that for high values of g the reached steady-state topology is the all-to-alltopology
with these aims we
build a grid for g ranging from g = 0.1 to g = 12, with a pace ∆g = 0.1
use Montecarlo techniques to generate 20 initial conditions for each topology
Note that with N = 5, 2N2−N
2 = 210 permutations could be possible; although somematrixes are not valid topologies for the system (4).Thus we account for 20 conditions for 687 topologies leading to 12740 simulations tobe performed for all the values of the gain grid g , which leads to 1528800 simulations!
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
Complex Networksoverview
Proximity oscillatornetworks
Results
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
A simple example: the chain topology
Figure: Qualitative diagram of a chain topology for a 5 node network
0 5 10 15 20 25 30 35 40 45 50
1
2
3
Time[s]
ωi[
rad s
]
Figure: Synchronisation is not achieved for g = 1.0, N = 5, θvis = π10 ,∇1,i [ωi ] = 0.1 · 2π
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
A simple example: the chain topology
Figure: Qualitative diagram of a chain topology for a 5 node network
0
1
2
3
4
ωi[
rad s
]
π10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
π2
π
32π
2π
Time[s]
θ 1−θ i
[rad
]
Figure: All-to-all topology is achieved for g = 3.0, N = 5, θvis = π10 ,∇1,i [ωi ] = 0.1 · 2π
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
A simple example: the chain topology
Figure: Qualitative diagram of a chain topology for a 5 node network
1.5
2
2.5
ωi[
rad s
]
π10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
Time[s]
θ 1−θ i
[rad
]
Figure: Not All-to-all topology is achieved for g = 6.6, N = 5, θvis = π10 ,∇1,i [ωi ] = 0.1 · 2π
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
A simple example: the chain topology
Figure: Qualitative diagram of a chain topology for a 5 node network
0
2
4
6
8
ωi[
rad s
]
π10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
Time[s]
θ 1−θ i
[rad
]
Figure: All-to-all topology is achieved for g = 11.0, N = 5, θvis = π10 ,∇1,i [ωi ] = 0.1 · 2π
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
A simple example: the chain topology
Figure: Qualitative diagram of a chain topology for a 5 node network
0 2 4 6 8 10 12 14 16
0
5
10
15
20
g [ 1t
]
Narcs
Figure: Narcs diagram, with respect to g , for a fixed initial condition: the chainGiovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
Topology bifurcation diagram
0 2 4 6 8 10 12 14
0
5
10
15
20
g [ 1t
]
Narcs
Figure: N = 5 network topology bifurcation diagram
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
Topology bifurcation diagram
0 2 4 6 8 10 12 14
0
5
10
15
20
g [ 1t
]
Narcs
Figure: N = 5 network topology bifurcation diagram. The red area denotes values of g notsufficient for synchronization
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
Topology bifurcation diagram
0 2 4 6 8 10 12 14
0
5
10
15
20
g [ 1t
]
Narcs
Figure: N = 5 network topology bifurcation diagram. The red area denotes values of g notsufficient for synchronization. The blue area denotes values of g that no matter the initialcondition lead to the all-to-all topology.
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
Topology bifurcation diagram
0 2 4 6 8 10 12 14
0
5
10
15
20
g [ 1t
]
Narcs
Figure: N = 5 network topology bifurcation diagram. The red area denotes values of g notsufficient for synchronization. The blue area denotes values of g that no matter the initialcondition lead to the all-to-all topology. The area in green denotes values of g for which by varyingthe initial conditions more and more equilibria topologies appear.
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
Topology bifurcation diagram
0 2 4 6 8 10 12 14
0
5
10
15
20
g [ 1t
]
Narcs
Figure: N = 5 network topology bifurcation diagram. The red area denotes values of g notsufficient for synchronization. The blue area denotes values of g that independently for initialcondition lead to the all-to-all topology. The area in green denotes values of g for which by varyingthe initial conditions more and more equilibria topologies appear. The red area on the rightdenotes values of g that independently from the initial condition lead to the all-to-all topology.
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
Conclusions and future work
We have introduced a proximity rule in a network of heterogenous Kuramotooscillators, developing a restricted visibility model
We have illustrated how multiple equilibria topologies may exist
We have numerically shown the emergence of an interesting phenomenon that wecalled topological bifurcation topology bifurcation
The number of times the not all-to-all equilibrium topologies are seen isapproximately 5% of the total number of simulations
Statistical analysis have shown a dependance between the equilibria topologiesand the initial conditions
Future works will be devoted to investigate the possible emergence of thesephenomenon for different individual dynamics and coupling rules
We envision that control strategies may be developed to control both theindividual dynamics and the emerging topologies
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
Complex Networks overview Proximity oscillator networks Results Bibliography
A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou,“Synchronization in complex networks,” pp. 1–80, May 2008. [Online]. Available:http://arxiv.org/abs/0805.2976
F. Radicchi and H. Meyer-Ortmanns, “Reentrant synchronization and patternformation in pacemaker-entrained Kuramoto oscillators,” Physical Review E,vol. 74, no. 2, p. 026203, Aug. 2006. [Online]. Available:http://link.aps.org/doi/10.1103/PhysRevE.74.026203
Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
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