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Emergence of loop structure in scale-free networks and dynamical consequences

Xiaojuan Ma^1,2, Liang Huang¹, Ying-Cheng Lai^1,3, and Zhigang Zheng²

¹Department of Electrical Engineering, Arizona State University, Tempe, Arizona 85287, USA
²Department of Physics, Beijing Normal University, Beijing 100875, China
³Department of Physics, Arizona State University, Tempe, Arizona 85287, USA

Phys. Rev. E 79, 056106 – Published 14 May, 2009

DOI: https://doi.org/10.1103/PhysRevE.79.056106

Abstract

Previous work has revealed that the synchronizability of a scale-free network tends to be suppressed when its clustering coefficient is increased. We present a theory to explain this phenomenon. Our proposition is that, as the network becomes more strongly clustered, topological loop structure can emerge, generating a set of eigenvalues that are close to zero. As a result, the dynamics of synchronization tends to be dominated by the loop structure. As the clustering coefficient is increased, the size of the dominant loop increases, leading to continuous degradation of the network synchronizability. We provide analysis and numerical evidence to support the proposition and we speculate that the loop structure can provide a platform for controlling dynamical processes on scale-free networks with high clustering coefficients.

Article Text

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