Synchronization of networks with prescribed degree distributions
Fatihcan M. Atay, Türker Biyikoglu, and Jürgen Jost
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Submission date: 17. Jun. 2004 (revised version: May 2005)
published in: IEEE transactions on circuits and systems / 1, 53 (2006) 1, p. 92-98
DOI number (of the published article): 10.1109/TCSI.2005.854604
with the following different title: On the synchronization of networks with prescribed degree distributions
Keywords and phrases: synchronization, networks, graph theory, laplacian
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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, a connected graph having that degree sequence exists for which the first nontrivial eigenvalue of the graph Laplacian is arbitrarily close to zero. Consequently, 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.