Hub synchronization in scale-free networks

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 82, Issue 3, 2010, Pages

  Pereira, Tiago ^a  

^a UNIVERSIDADE FEDERAL DO ABC   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

COMPLEX NETWORKS; COUPLED OSCILLATORS; DEGREE DISTRIBUTIONS; GLOBAL SYNCHRONIZATION; SCALE FREE NETWORKS;

OSCILLATORS (ELECTRONIC);

SYNCHRONIZATION;

EID: 77957205798     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.82.036201     Document Type: Article

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20. The choice of normalized coupling is immaterial. The choice does not play any role in the analysis. Note that we use the same normalization for all nodes, so we could have written σ = α / k n, as is usually done in the literature.
21. In our context these are natural assumptions. Lyapunov regularity basically assures that the Lyapunov exponents exist. The integral separation is a generic property in the space of continuous bounded matrix valued functions. See for a detailed discussion.
22. Collected Lectures on the Preservation of Stability under Discretization, edited by, D. J. Estep, and, S. Tavener, (SIAM, Philadelphia, PA, 2002).
23. Without attempt at rigor, the local mean field argument is the following. First remember that ηn-1 = kn -1 Σj ( A (n-1 ) j - Anj ) E ( ξj ), since the oscillators are chaotic and unsynchronized (at least for small values of α) once can think of ξj as identically distributed random numbers. For kn 1, by the center limit theorem ηi =O ( kn -1/2 ).
24. L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations (Springer-Verlag, Berlin; Heidelberg, 2008).
25. The stability of global synchronization is formulated in terms of the spectrum of Laplacian matrix. Let L be the Laplacian matrix of the graph. The spectrum of L is real and can be ordered as 0= γ1 ≤ γ2 ≤≤ γn. Global synchronization is possible if γn / γ2 < α2 / α1, see for details. For this network we have, γn / γ2 ≈180, while α2 / α1 ≈35.
26. The unique solution of the homogeneous part of Eq. can be written in terms of the principal matrix ζ (t) =T (t,s ) ζ (s). Moreover, under our hypotheses the operator T (t,s ) admits a dichotomy being exponentially stable on the subspace S. Since ∥B∥ <δ, it follows that the Lyapunov exponents of the perturbed equation remain negative.
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