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To understand how certain dynamical behaviors can or cannot persist as theunderlying network grows is a problem of increasing importance in complexdynamical systems as well as sustainability science and engineering. We addressthe question of whether a complex network of nonlinear oscillators can maintainits synchronization stability as it expands or grows. A network in the realworld can never be completely synchronized due to noise and/or externaldisturbances. This is especially the case when, mathematically, the transientsynchronous state during the growth process becomes marginally stable, as alocal perturbation can trigger a rapid deviation of the system from thevicinity of the synchronous state. In terms of the nodal dynamics, a largescale avalanche over the entire network can be triggered in the sense that theindividual nodal dynamics diverge from the synchronous state in a cascadingmanner within a short time period. Because of the high dimensionality of thenetworked system, the transient process for the system to recover to thesynchronous state can be extremely long. Introducing a tolerance threshold toidentify the desynchronized nodes, we find that, after an initial stage oflinear growth, the network typically evolves into a critical state where theaddition of a single new node can cause a group of nodes to losesynchronization, leading to synchronization collapse for the entire network. Astatistical analysis indicates that, the distribution of the size of thecollapse is approximately algebraic (power law), regardless of the fluctuationsin the system parameters. This is indication of the emergence of self-organizedcriticality. We demonstrate the generality of the phenomenon of synchronizationcollapse using a variety of complex network models, and uncover the underlyingdynamical mechanism through an eigenvector analysis.

Growth, collapse, and self-organized criticality in complex networks