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A step forward to pinning control of complex networks: Finding an optimal vertex to control

Synchronization in complex networks has been widely investigated recently. However, if a network cannot reach synchrony by itself, pinning control needs to be applied on a small fraction of vertices in the network. In the last decade, many pinning schemes were designed for choosing appropriate pinning vertices. However, the challenging problem as which vertices should be controlled to achieve the best performance for synchronization still has no solution. This paper focuses on the first step toward this challenging open problem. By using spectral analysis and algebraic graph theory tools to separate the network, it is found that an optimal vertex for pinning a network to achieve the best performance in synchronization must be in the matched pinning component or the matched cut vertex set.