Emergence of chaotic cluster synchronization in heterogeneous networks.

Many real-world complex systems rely on cluster synchronization to function properly. A cluster of nodes exhibits synchronous behavior, while others behave erratically. Predicting the emergence of these clusters and understanding the mechanism behind their structure and variation in response to a parameter change is a daunting task in networks that lack symmetry. We unravel the mechanism for the emergence of cluster synchronization in heterogeneous random networks. We develop heterogeneous mean-field approximation together with a self-consistent theory to determine the onset and stability of the cluster. Our analysis shows that cluster synchronization occurs in a wide variety of heterogeneous networks, node dynamics, and coupling functions. The results could lead to a new understanding of the dynamical behavior of networks ranging from neural to social.