Local Assortativity Affects the Synchronizability of Scale-Free Network

Synchronization is critical for system-level behavior in physical, chemical, biological, and social systems. Empirical evidence has shown that the network topology strongly impacts the synchronizability of the system, and the analysis of their relationship remains an open challenge. We know that the eigenvalue distribution determines a network's synchronizability, but analytical expressions that connect network topology and all relevant eigenvalues (e.g., the extreme values) remain elusive. Here, we accurately determine its synchronizability by proposing an analytical method to estimate the extreme eigenvalues using perturbation theory. Our analytical method exposes the role that global and local topology combine to influence synchronizability. We show that the smallest nonzero eigenvalue lambda((2)), which determines synchronizability, is estimated by the smallest degree augmented by the inverse degree difference in the least connected nodes. From this, we can conclude that there exists a clear negative relationship between lambda((2)) and the local assortativity of nodes with the smallest degree value. We validate the accuracy of our framework within the setting of a scale-free network and can be driven by commonly used ordinary differential equations (e.g., 3-D Rosler dynamics or Hindmarsh-Rose neuronal circuit). From the results, we demonstrate that the synchronizability of the network can be tuned by rewiring the connections of these particular nodes while maintaining the general degree profile of the network.

Automated summary

Synchronization is critical for system-level behavior in various systems, and the relationship between network topology and synchronizability remains an open challenge. We propose an analytical method using perturbation theory to accurately determine synchronizability by estimating extreme eigenvalues. Our method reveals the combined influence of global and local topology on synchronizability and shows a negative relationship between the smallest nonzero eigenvalue lambda((2)) and the local assortativity of nodes with the smallest degree value. We validate our framework in a scale-free network driven by ordinary differential equations.