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 Lellis Eng. Francesco Lo Iudice Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks
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Numerical analysis of proximity oscillator networks · Numerical analysis of proximity oscillator networks Giovanni Pugliese Carratelli - M58/30 DIETI - Univesity Federico II of Naples
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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
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
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
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
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
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
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
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