  In this paper, the transition of synchronizing path of delay-coupled chaotic
oscillators in a scale-free network is highlighted. Mainly, through the
critical transmission delay makes chaotic oscillators be coupled on the edge of
stability, we find that the transition of synchronizing path is
\emph{abnormal}, which is characterized by the following evidences: (a)
synchronization process starts with low-degree rather than high-degree ones;
(b) the high-degree nodes don't undertake the role of hub; (c) the synchronized
subnetworks show a poor small-world property as a result of hubs absence; (d)
the clustering synchronization behavior emerges even community structure is
absent in the scale-free network. This abnormal synchronizing path suggests
that the diverse synchronization behaviors occur in the same topology, which
implies that the relationship between dynamics and structure of network is much
more complicated than the common sense that the structure is the foundation of
dynamics. Moreover, it also reveals the potential connection from the
transition of synchronization behavior to disorder in real complex networks,
e.g. Alzheimer disease.
