Abnormal synchronizing path of delay-coupled chaotic oscillators on the edge of stability

• We investigated the influence of transmission delay on the synchronizing path.
• Synchronization on the edge of stability starts with small-degree nodes.
• Synchronous behavior on the edge of stability shows poor small-world property.
• Cluster synchronization emerges on the edge of stability when no community exists in network.

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 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 do not 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.