Synchronization of networks with prescribed degree distributions

Details Synchronization of networks with prescribed degree distributions Synchronization of networks with prescribed degree distributions

2004-07-09

arxiv.org/abs/nlin/0407024v2

IEEE Transactions on Circuits and Systems-I, Vol. 53 (1): 92-98, 2006

Nonlinear Sciences

Adaptation and Self-Organizing Systems

v2: A new theorem and a numerical example added. To appear in IEEE Trans. Circuits and Systems I: Fundamental Theory and Appli

Scientific paper

10.1109/TCSI.2005.854604

We show that the degree distributions of graphs do not suffice to characterize the synchronization of systems evolving on them. We prove that, for any given degree sequence satisfying certain conditions, there exists a connected graph having that degree sequence for which the first nontrivial eigenvalue of the graph Laplacian is arbitrarily close to zero. Consequently, complex dynamical systems defined on such graphs have poor synchronization properties. The result holds under quite mild assumptions, and shows that there exists classes of random, scale-free, regular, small-world, and other common network architectures which impede synchronization. The proof is based on a construction that also serves as an algorithm for building non-synchronizing networks having a prescribed degree distribution.

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