  To understand how certain dynamical behaviors can or cannot persist as the
underlying network grows is a problem of increasing importance in complex
dynamical systems as well as sustainability science and engineering. We address
the question of whether a complex network of nonlinear oscillators can maintain
its synchronization stability as it expands or grows. A network in the real
world can never be completely synchronized due to noise and/or external
disturbances. This is especially the case when, mathematically, the transient
synchronous state during the growth process becomes marginally stable, as a
local perturbation can trigger a rapid deviation of the system from the
vicinity of the synchronous state. In terms of the nodal dynamics, a large
scale avalanche over the entire network can be triggered in the sense that the
individual nodal dynamics diverge from the synchronous state in a cascading
manner within a short time period. Because of the high dimensionality of the
networked system, the transient process for the system to recover to the
synchronous state can be extremely long. Introducing a tolerance threshold to
identify the desynchronized nodes, we find that, after an initial stage of
linear growth, the network typically evolves into a critical state where the
addition of a single new node can cause a group of nodes to lose
synchronization, leading to synchronization collapse for the entire network. A
statistical analysis indicates that, the distribution of the size of the
collapse is approximately algebraic (power law), regardless of the fluctuations
in the system parameters. This is indication of the emergence of self-organized
criticality. We demonstrate the generality of the phenomenon of synchronization
collapse using a variety of complex network models, and uncover the underlying
dynamical mechanism through an eigenvector analysis.
